diff b34550f..b1f9bc4 zero_distillable_entanglement_v2.txt
diff --git a/zero_distillable_entanglement_v2.txt b/zero_distillable_entanglement_v2.txt
index 8e97222..9d8cce0 100644
--- a/zero_distillable_entanglement_v2.txt
+++ b/zero_distillable_entanglement_v2.txt
@@ -11,7 +11,7 @@
-\hypersetup{pdftitle={Entanglement with zero distillable secret key in local dimension ten},pdfauthor={OpenAI}}
+hypersetup{pdftitle={Entanglement with zero distillable secret key in local dimension ten},pdfauthor={OpenAI}}
@@ -19,7 +19,7 @@
-\newcommand{\Mat}[1]{M_{#1}(\mathbb C)}
+newcommand{Mat}[1]{M_{#1}(mathbb C)}bbC)
@@ -31,149 +31,29 @@
-# Entanglement with zero distillable secret key\\ in local dimension ten
+# Entanglement with zero distillable secret key in local dimension ten
-\maketitle
+maketitle
Abstract:
-We construct an entangled density operator \(\rho\) on \(\C^{10} ⊗ \C^{10}\)
+We construct an entangled density operator (rho) on (C^{10} ⊗ C^{10})
whose range contains no nonzero product vector and whose distillable
-secret key is zero. More precisely: for every integer \(n\ge1\) and every
-admitted protocol on \(\rho^{ ⊗ n}\), the unhalved trace-norm distance
+secret key is zero. More precisely: for every integer (nge1) and every
+admitted protocol on (rho^{ ⊗ n}), the unhalved trace-norm distance
from any completed bit output to an ideal secret bit (with arbitrary
-eavesdropper state) is at least \(1/5\). The construction disproves the
+eavesdropper state) is at least (1/5). The construction disproves the
unrestricted two-map PPT-composition conjecture of Christandl,
-M\"uller-Hermes, and Wolf. A separate trace-preserving PPT channel on
-\(\Mat{21}\) has a square that is not entanglement breaking.
-
-
-# Preface for younger readers
-
-This paper proves a surprising theorem and then does some frighteningly
-careful arithmetic. The theorem says: there is a specific entangled
-quantum state from which no protocol can squeeze out even one secret
-bit. The arithmetic makes the state explicit, so no one has to take the
-existence on faith.
-
-The following paragraphs are an on-ramp. Every formal object below has a
-plain-language counterpart in this preface. If the notation later starts
-to groan, come back here.
-
-Quantum states.: A quantum system of dimension \(d\) is modeled
-by the unit sphere in \(\C^d\) up to global phase. A qubit is \(d=2\).
-If \(\{e_0,e_1\}\) is the standard basis of \(\C^2\), the vector
-\((e_0+e_1)/\sqrt2\) is a superposition: measuring in that basis gives
-\(0\) or \(1\) with equal probability.
-
-When the state is not known---because someone flipped a coin and
-prepared \(|e_0>\) or \(|e_1>\) so that we only know a
-probability---the right object is a density operator. This is a
-positive semidefinite matrix \(\rho\) with \(\tr\rho=1\). If the state
-is pure, prepared from a known unit vector \(|\psi>\), then
-\(\rho=|\psi><\psi|\). Measuring an observable \(A\) yields
-expectation \(\tr(A\rho)\). Density operators are the universal currency
-of quantum statistics: two preparations lead to the same state precisely
-when they assign the same matrix \(\rho\).
-
-Bipartite states.: Suppose two distant laboratories Alice and
-Bob share a quantum system that lives on a tensor product
-\(\C^a ⊗ \C^b\). The joint density operator \(\rho\) records the
-statistics of all joint measurements one could perform there.
-
-A state is separable if it can be written as a finite sum
-\[
-\rho=\sum_{r} p_r\; |\psi_r><\psi_r|_A
- ⊗ |\phi_r><\phi_r|_B,
-\]
-with \(p_r\ge0\) and \(\sum_r p_r=1\). In English: some classical random
-variable chooses a label \(r\), and each laboratory prepares a state
-depending only on \(r\). Separable states therefore carry only classical
-correlations.
-
-A state is entangled if it is not separable. Entanglement is the extra
-correlation that lets Bell-inequality experiments beat any classical
-model of the same statistics. It is not the same as classical randomness
-shared through a hidden variable.
-
-Secret-key distillation.: Alice and Bob share a state \(\rho\).
-They can perform any sequence of local quantum operations (measurements,
-unitaries, ancillas) and exchange classical messages over an
-authenticated public channel. Question: can they produce identical secret
-random bits?
-
-The adversary is an eavesdropper Eve. The most pessimistic and most
-natural model is: Eve started with a purification of \(\rho\), a pure
-state \(|\Psi><\Psi|\) on \(\C^a ⊗ \C^b ⊗ \C^E\)
-whose restriction to \(\C^a ⊗ \C^b\) is \(\rho\). She also sees the
-full public transcript. An ideal secret bit is one for which the joint
-state of Alice, Bob, and Eve has the form
-\[
-\gamma = \frac12\sum_{z\in\{0,1\}} |z,z>< z,z| ⊗ \sigma,
-\]
-for some eavesdropper state \(\sigma\). That is: half the time Alice and
-Bob hold \(0 ⊗ 0\) and Eve holds \(\sigma\); half the time
-\(1 ⊗ 1\) and Eve holds \(\sigma\). Eve sees the bit but, because
-the state is identical in both branches, learns nothing new from Alice
-and Bob's announcement.
-
-The output of a real protocol \(\tau\) is compared to \(\gamma\) by the
-unnormalized trace norm \(\norm{\tau-\gamma}_1\). This is twice the
-traditional "trace distance" and has the advantage that
-\(0\le\norm{\tau-\gamma}_1\le2\). If it is close to zero, Eve learns
-essentially nothing.
-
-The distillable secret key \(K_D(\rho)\) is the supremum over secure
-protocols of the asymptotic rate of bits produced per copy of \(\rho\),
-in the limit of many copies. Ordinary entropic protocols (privacy
-amplification) show that some states have \(K_D>0\).
-
-The question.: Every separable \(\rho\) has \(K_D=0\):
-Eve's purification is a summation over product states, so she already
-knows everything about each branch and no announcement can surprise her.
-
-The converse is a priori not obvious. If \(\rho\) is entangled, does it
-have \(K_D>0\)? By the mid-2000s this was a standard open problem. It
-appears as Problem 24 in Kr\"uger and Werner's 2005 collection, in the
-form: "Secret key from all entangled states?", attributed to
-P.~Horodecki. The intuition was: entanglement is a resource; surely
-some useful amount of it must survive public discussion.
-
-What this paper proves.: It does not. There is a specific
-entangled \(\rho\) on \(\C^{10} ⊗ \C^{10}\) with \(K_D(\rho)=0\). In
-fact we prove the sharper statement: any completed protocol on any number
-of copies that outputs one bit must be at unhalved trace-norm distance at
-least \(1/5\) from every ideal bit. The conventional trace distance
-(the one with a factor \(1/2\)) is therefore at least \(1/10\), uniformly
-in the number of copies, the number of rounds of public discussion, and
-the eavesdropper's initial purification.
-
-The construction is not magical. It comes from a completely positive map
-whose Choi matrix---an ordinary matrix built from the map---has a very
-special feature: all its "product" directions, which separability would
-force into its range, are visibly absent from that range. If a protocol
-could extract key, a finite combinatorial estimate of how much coherence
-can coexist with that pattern would have to break. It does not. The rest
-of the paper makes the state explicit: a specific \(6\times4\) integer
-matrix pencil, twenty specific singular directions, an exact modular
-certificate at primes \(41\), \(131\), \(139\), an irreducible
-degree-twenty polynomial, and Galois group \(S_{20}\).
-
-The paper has one technical core (the operational obstruction, via a
-four-effect coherence inequality) and two independent realizations
-(the twenty-direction integer pencil of Section~sec:certificate,
-and the projection--combinatorial construction of
-Sections~sec:projection--sec:combinatorial). Both provide
-independently certified entangled members of the operational class. The
-second is enormous in dimension but conceptually simple; the first is
-small but certified by exact finite arithmetic.
+M"uller-Hermes, and Wolf. A separate trace-preserving PPT channel on
+(Mat{21}) has a square that is not entanglement breaking.
+
# What we prove, and why it is surprising
-Fix two finite-dimensional complex Hilbert spaces \(\C^a\) and \(\C^b\).
-A density operator \(\rho\) on \(\C^a ⊗ \C^b\) is a positive
+Fix two finite-dimensional complex Hilbert spaces (C^a) and (C^b).
+A density operator (rho) on (C^a ⊗ C^b) is a positive
semidefinite matrix of trace one. It is separable if it is a
finite convex combination of positive product matrices, and
entangled otherwise.
@@ -183,7 +63,7 @@ instruments, private ancillas, private randomness, and authenticated
two-way classical communication. The input is the only shared private
resource. An eavesdropper Eve starts with a purification of the input and
sees the complete public transcript. Distillable secret key
-\(K_D(\rho)\) is defined as the supremum of bits-per-copy rates that
+(K_D(rho)) is defined as the supremum of bits-per-copy rates that
these protocols can extract, with zero asymptotically vanishing error.
Section~sec:transcript gives the precise operational model.
@@ -198,33 +78,33 @@ The paper produces an explicit counterexample.
[Entangled state with zero secret key]
-There is an explicitly specified entangled density operator \(\rho\) on
-\(\C^{10} ⊗ \C^{10}\) whose range contains no nonzero product vector
-and for which \(K_D(\rho)=0\). More precisely: for every integer \(n\ge1\)
-and every admitted protocol on \(\rho^{ ⊗ n}\) that produces a
-completed bit output \(\tau_{K_AK_BE'}\) with \(E'\) retaining an initial
+There is an explicitly specified entangled density operator (rho) on
+(C^{10} ⊗ C^{10}) whose range contains no nonzero product vector
+and for which (K_D(rho)=0). More precisely: for every integer (nge1)
+and every admitted protocol on (rho^{ ⊗ n}) that produces a
+completed bit output (tau_{K_AK_BE'}) with (E') retaining an initial
purification and the full public transcript, we have, for every
-eavesdropper state \(\sigma_{E'}\),
+eavesdropper state (sigma_{E'}),
-\norm{
-\tau_{K_AK_BE'}-\frac12\sum_{i=0}^1 |i,i>< i,i| ⊗ \sigma_{E'}
-}_1\ge\frac15.
+norm{
+tau_{K_AK_BE'}-frac12sum_{i=0}^1 |i,i>< i,i| ⊗ sigma_{E'}
+}_1gefrac15.
-Thus the traditional trace distance to an ideal bit is at least \(1/10\).
+Thus the traditional trace distance to an ideal bit is at least (1/10).
-The bound \(1/5\) is uniform in \(n\), in the number of rounds, in the
+The bound (1/5) is uniform in (n), in the number of rounds, in the
protocol itself, and in the eavesdropper's purification. Proposition~%
prop:operational-extensions presents three natural extensions:
conditioning on a positive-probability transcript event, vanishing abort
probabilities, and trace-norm limits of completed actual outputs whose
Eve registers retain the full information.
-The state is derived from maps. A linear map \(F:\Mat a\to\Mat b\) is
-completely positive (CP) if \(id_k ⊗ F\) is
-positive for every \(k\ge1\). It is PPT if both \(F\) and
-\(T_b\circ F\) are CP, where \(T_d(X)=X^{\T}\) is the (complex-linear)
-transpose on \(\Mat d\). A CP map \(F\) is entanglement breaking
+The state is derived from maps. A linear map (F:Mat atoMat b) is
+completely positive (CP) if (id_k ⊗ F) is
+positive for every (kge1). It is PPT if both (F) and
+(T_bcirc F) are CP, where (T_d(X)=X^{T}) is the (complex-linear)
+transpose on (Mat d). A CP map (F) is entanglement breaking
(EB) if applying it to one subsystem of any positive bipartite input
always outputs a separable state. Equivalently, its
Choi matrix is separable~[cite]. Our second main result is a
@@ -233,153 +113,126 @@ counterexample to composition.
[Two PPT maps whose composition is not EB]
There are explicitly specified PPT maps
-\(\Phi_1,\Phi_2:\Mat{10}\to\Mat{10}\) such that \(Z=J(\Phi_2\circ\Phi_1)\)
+(Phi_1,Phi_2:Mat{10}toMat{10}) such that (Z=J(Phi_2circPhi_1))
is nonzero and its range contains no nonzero product vector. Hence
-\(\Phi_2\circ\Phi_1\) is not entanglement breaking.
+(Phi_2circPhi_1) is not entanglement breaking.
-The maps \(\Phi_1,\Phi_2\) come from a \(6\times4\) real integer matrix
-pencil \(M(x)=x_0M_0+x_1M_1+x_2M_2+x_3M_3\). Section~%
+The maps (Phi_1,Phi_2) come from a (6times4) real integer matrix
+pencil (M(x)=x_0M_0+x_1M_1+x_2M_2+x_3M_3). Section~%
sec:certificate constructs them; Section~sec:geometry
proves the tensor-criterion that turns them into Theorem~%
thm:main-pair. Theorem~thm:main-state is then the
corresponding statement for the density operator
-\(\rho=Z/\tr Z\).
+(rho=Z/tr Z).
A trace-preserving channel version of the same phenomenon is stated next.
[Trace-preserving PPT channel whose square is not EB]
There is an explicitly specified trace-preserving PPT channel
-\(Θ:\Mat{21}\to\Mat{21}\) whose square \(Θ\circΘ\) is not
+(Θ:Mat{21}toMat{21}) whose square (ΘcircΘ) is not
entanglement breaking.
-The dimension \(21\) is \(2\cdot10+1\): the pair from Theorem~%
+The dimension (21) is (2cdot10+1): the pair from Theorem~%
thm:main-pair is embedded, rescaled, and completed by an absorbing
-flag, following Christandl--M\"uller-Hermes--Wolf~[cite] and
+flag, following Christandl--M"uller-Hermes--Wolf~[cite] and
Filippov~[cite]. Section~sec:applications presents the
details.
-Why this is surprising.: For two decades, evidence pointed the
-other way. Horodecki, Horodecki, Horodecki, and Oppenheim~[cite] proved that some PPT entangled (bound)
-states have positive secret-key cost, via the private-state paradigm of
-Refs.~[cite]. Devetak and Winter~%
-[cite] gave one-way positive-key criteria. On the collapse
-side, An and Lee~[cite] proved that a qutrit CP map with an
-undistillable Choi matrix composes with a low-Schmidt Choi matrix into an
-EB map in both orders; Park~[cite] recently proved finite EB
-index for every PPT channel. A failure at the second iterate is
-consistent with eventual EB. Meanwhile the CMHW paper~[cite]
-proved \(PPT^2\) composition in several important cases (all maps on
-\(\Mat 3\); Gaussian channels; diagonal unitary covariant and conjugate
-diagonal unitary covariant PPT maps). Our example violates the general
-statement but not any of those specific affirmative classes.
-
-Organisation.: Section~sec:foundations establishes map
-conventions, the represented class \(\mathscr C\), and the Choi
-transfer identity. Sections~sec:coherence--sec:secret-key
-prove the operational theorem: the coherence inequality, the transcript
-lemma, and the \(1/5\) gap. Section~sec:geometry constructs the
-dimension-ten pair from a tensor criterion; Section~sec:certificate
-supplies the exact integer certificate; Section~sec:applications
-derives the state and channel statements.
-Sections~sec:projection--sec:combinatorial give an
-independent construction from projections and the Mermin--Peres square.
-The appendices collect exact arithmetic and supplementary checks.
-
# Foundations: maps, PPT, and the represented class
-All matrix algebras are finite-dimensional over \(\C\), with fixed
-orthonormal bases and product bases. We write \(X^*\) for conjugate
-transpose, \(\overline X\) for entrywise conjugation,
-\(T_d(X)=X^{\T}\) for the complex-linear transpose on \(\Mat d\). For a
-linear map \(F\), its Hilbert--Schmidt adjoint \(F^\dagger\) satisfies
-\(\tr(Y^*F(X))=\tr((F^\dagger(Y))^*X)\). If \(V\) is a matrix, we write
-\(\Ad_V(X)=VXV^*\).
+All matrix algebras are finite-dimensional over (C), with fixed
+orthonormal bases and product bases. We write (X^*) for conjugate
+transpose, (overlineoverline XX) for entrywise conjugation,
+(T_d(X)=X^{T}) for the complex-linear transpose on (Mat d). For a
+linear map (F), its Hilbert--Schmidt adjoint (F^dagger) satisfies
+(tr(Y^*F(X))=tr((F^dagger(Y))^*X)). If (V) is a matrix, we write
+(Ad_V(X)=VXV^*).
Our Choi matrix convention is
-J(F)=\sum_{i,j=1}^a E_{ij} ⊗ F(E_{ij})
+J(F)=sum_{i,j=1}^a E_{ij} ⊗ F(E_{ij})
=(id_a ⊗ F)(Ω_aΩ_a^*),
- \qquad Ω_a=\sum_{i=1}^a e_i ⊗ e_i.
+ qquad Ω_a=sum_{i=1}^a e_i ⊗ e_i.
-So \(\norm{Ω_a}^2=a\), and the first tensor factor is the input
+So (norm{Ω_a}^2=a), and the first tensor factor is the input
reference. This fixes every dimensional factor in the transfer formula
and in every channel corollary. It is the map--operator correspondence
-of Jamio\l{}kowski and Choi~[cite].
+of Jamiol{}kowski and Choi~[cite].
[PPT stability]
-Let \(F:\Mat a\to\Mat b\) be CP. Then \(F^\dagger\) and
-\(T_b\circ F\circ T_a\) are CP. Moreover
-\[
-F\circ T_a\ is CP\iff T_b\circ F\ is CP.
-\]
-If \(F\) is PPT, so are \(T_b\circ F\), \(F\circ T_a\), its adjoint,
-\(T_b\circ F\circ T_a\), and \(F^\sharp:=T_a\circ F^\dagger\circ T_b\).
-Tensor products of PPT maps are PPT. If \(P:\Mat{a'}\to\Mat a\) and
-\(Q:\Mat b\to\Mat{b'}\) are CP, then \(Q\circ F\circ P\) is PPT whenever
-\(F\) is PPT.
+Let (F:Mat atoMat b) be CP. Then (F^dagger) and
+(T_bcirc Fcirc T_a) are CP. Moreover
+[
+Fcirc T_a is CPiff T_bcirc F is CP.
+]
+If (F) is PPT, so are (T_bcirc F), (Fcirc T_a), its adjoint,
+(T_bcirc Fcirc T_a), and (F^sharp:=T_acirc F^daggercirc T_b).
+Tensor products of PPT maps are PPT. If (P:Mat{a'}toMat a) and
+(Q:Mat btoMat{b'}) are CP, then (Qcirc Fcirc P) is PPT whenever
+(F) is PPT.
Proof.
-Write \(F(X)=\sum_r A_r X A_r^*\) with Kraus matrices
-\(A_r\in\C^{b\times a}\). Then
-\(F^\dagger(Y)=\sum_r A_r^*YA_r\) and
-\((T_b\circ F\circ T_a)(X)=\sum_r A_rbarXA_rbar^{\,*}\),
+Write (F(X)=sum_r A_r X A_r^*) with Kraus matrices
+(A_rinC^{btimes a}). Then
+(F^dagger(Y)=sum_r A_r^*YA_r) and
+((T_bcirc Fcirc T_a)(X)=sum_r A_rbarXA_rbar^{,*}),
which proves both CP assertions. Applying the simultaneous-transpose
-statement to \(F\circ T_a\) or \(T_b\circ F\) gives the equivalence.
+statement to (Fcirc T_a) or (T_bcirc F) gives the equivalence.
These arguments all work with rectangular Kraus matrices.
-Transpose is Hilbert--Schmidt self-adjoint, so when \(F\) is PPT the
-identities \(T_a\circ F^\dagger=(F\circ T_a)^\dagger\) and
-\(F^\dagger\circ T_b=(T_b\circ F)^\dagger\) are CP. That proves PPT for
+Transpose is Hilbert--Schmidt self-adjoint, so when (F) is PPT the
+identities (T_acirc F^dagger=(Fcirc T_a)^dagger) and
+(F^daggercirc T_b=(T_bcirc F)^dagger) are CP. That proves PPT for
the adjoint; simultaneous transposition gives the simultaneous
transpose and the transpose-adjoint. The output transpose of
-\(T_b\circ F\) is \(F\), and of \(F\circ T_a\) is
-\(T_b\circ F\circ T_a\): both CP by hypothesis. Finally
-\(T_{b'}\circ Q\circ F\circ P
-=(T_{b'}\circ Q\circ T_b)\circ(T_b\circ F)\circ P\) is a composition of
-CP maps; the untransposed composite \(Q\circ F\circ P\) is CP too.
-Products follow from \(T_{b_1b_2}=T_{b_1} ⊗ T_{b_2}\).
+(T_bcirc F) is (F), and of (Fcirc T_a) is
+(T_bcirc Fcirc T_a): both CP by hypothesis. Finally
+(T_{b'}circ Qcirc Fcirc P
+=(T_{b'}circ Qcirc T_b)circ(T_bcirc F)circ P) is a composition of
+CP maps; the untransposed composite (Qcirc Fcirc P) is CP too.
+Products follow from (T_{b_1b_2}=T_{b_1} ⊗ T_{b_2}).
[Choi characterization of CP and of EB]
-A linear map \(F:\Mat a\to\Mat b\) is CP iff \(J(F)\succeq0\). For a CP
+A linear map (F:Mat atoMat b) is CP iff (J(F)succeq0). For a CP
map, the following are equivalent:
-\item \(J(F)\) is separable;
-\item for every \(k\ge1\) and \(W\succeq0\) in \(\Mat k ⊗ \Mat a\),
- \((id_k ⊗ F)(W)\) is separable.
+item (J(F)) is separable;
+item for every (kge1) and (Wsucceq0) in (Mat k ⊗ Mat a),
+ ((id_k ⊗ F)(W)) is separable.
-Also \(J(T_b\circ F)=(id_a ⊗ T_b)J(F)\) and
-\(J(F\circ T_a)=(T_a ⊗ id_b)J(F)\).
+Also (J(T_bcirc F)=(id_a ⊗ T_b)J(F)) and
+(J(Fcirc T_a)=(T_a ⊗ id_b)J(F)).
Proof.
The forward CP implication is immediate from (eq:choi-definition).
-Conversely, write \(J(F)=\sum_r v_rv_r^*\); each \(v_r\) has a unique
-\(A_r\in\C^{b\times a}\) with \(v_r=(I_a ⊗ A_r)Ω_a\). Comparing
-the \((i,j)\) block gives \(F(E_{ij})=\sum_r A_rE_{ij}A_r^*\), hence the
+Conversely, write (J(F)=sum_r v_rv_r^*); each (v_r) has a unique
+(A_rinC^{btimes a}) with (v_r=(I_a ⊗ A_r)Ω_a). Comparing
+the ((i,j)) block gives (F(E_{ij})=sum_r A_rE_{ij}A_r^*), hence the
Kraus representation. This is Choi's criterion~[cite].
-If \(J(F)\) is separable, spectral decompositions of its positive
-factors express it as \(\sum_r a_ra_r^* ⊗ b_rb_r^*\), and blockwise
+If (J(F)) is separable, spectral decompositions of its positive
+factors express it as (sum_r a_ra_r^* ⊗ b_rb_r^*), and blockwise
comparison gives the measure-and-prepare formula
-\(F(X)=\sum_r\tr[(a_ra_r^*)^{\T}X]\,b_rb_r^*\). With
-\(E_r=(a_ra_r^*)^{\T}\succeq0\), for \(W\succeq0\)
-\[
-\tr_a[(I_k ⊗ E_r)W]
-=\tr_a[(I_k ⊗ √(E_r))W(I_k ⊗ √(E_r))]\succeq0.
-\]
-Therefore \((id_k ⊗ F)(W)\) is a sum of positive products
-with second factor \(b_rb_r^*\). The converse follows by testing
-\(W=Ω_aΩ_a^*\). The two transpose identities are immediate
+(F(X)=sum_rtr[(a_ra_r^*)^{T}X],b_rb_r^*). With
+(E_r=(a_ra_r^*)^{T}succeq0), for (Wsucceq0)
+[
+tr_a[(I_k ⊗ E_r)W]
+=tr_a[(I_k ⊗ √(E_r))W(I_k ⊗ √(E_r))]succeq0.
+]
+Therefore ((id_k ⊗ F)(W)) is a sum of positive products
+with second factor (b_rb_r^*). The converse follows by testing
+(W=Ω_aΩ_a^*). The two transpose identities are immediate
from reindexing matrix units. This is entanglement breaking in the cone
-sense~\cite[Theorem 4]{HSR2003}.
+sense~cite[Theorem 4]{HSR2003}.
[Nonseparability from range]
@@ -391,43 +244,43 @@ no nonzero product vector is nonseparable.
Proof.
Refine a separable decomposition by spectral decompositions of its
-positive factors. It becomes \(Z=\sum_r v_rv_r^*\) with each \(v_r\)
-a product vector \(a_r ⊗ b_r\). Since
-\(\ker Z=∩_r v_r^\perp\), every \(v_r\) lies in
-\((\ker Z)^\perp=\ran Z\). If \(Z\ne0\), at least one \(v_r\) is
+positive factors. It becomes (Z=sum_r v_rv_r^*) with each (v_r)
+a product vector (a_r ⊗ b_r). Since
+(ker Z=∩_r v_r^perp), every (v_r) lies in
+((ker Z)^perp=ran Z). If (Zne0), at least one (v_r) is
nonzero. This is the range criterion of Horodecki~[cite].
-[The represented class \(\mathscr C\)]
+[The represented class (mathscr CscrC)]
-A density operator \(\omega\) on \(\C^a ⊗ \C^b\) belongs to
-\(\mathscr C\) if there exist positive integers \(r,s\), a vector
-\(\chi\in\C^r ⊗ \C^s\), and linear maps
-\(\mathcal L:\Mat r\to\Mat a\), \(\mathcal M:\Mat s\to\Mat b\) such
+A density operator (omega) on (C^a ⊗ C^b) belongs to
+(mathscr CscrC) if there exist positive integers (r,s), a vector
+(chiinC^r ⊗ C^s), and linear maps
+(mathcal L:calL:Mat rtoMat a), (mathcal M:calM:Mat stoMat b) such
that
-\omega=(\mathcal L ⊗ \mathcal M)(\chi\chi^*),
-\qquad
-\mathcal L,\ \mathcal L\circ T_r,\ \mathcal M,\ \mathcal M\circ T_s
+omega=(mathcal LcalL ⊗ mathcal M)(calM)(chichi^*),
+qquad
+mathcal L,calL, mathcal LcalLcirc T_r, mathcal M,calM, mathcal McalMcirc T_s
are CP.
-Only \(\omega\) is normalized. The vector \(\chi\) may be unnormalized
+Only (omega) is normalized. The vector (chi) may be unnormalized
and the maps may not preserve trace.
This is a representation property, not a physical claim. Lemma~%
-lem:ppt-stability shows that \(\mathcal L,\mathcal M\) are also
-PPT in the output-transpose convention. In particular \(\omega\) itself
+lem:ppt-stability shows that (mathcal L,calL,mathcal McalM) are also
+PPT in the output-transpose convention. In particular (omega) itself
has positive partial transpose, since
-\((T_a ⊗ id_b)(\omega)
-=((T_a\circ\mathcal L) ⊗ \mathcal M)(\chi\chi^*)\succeq0\).
+((T_a ⊗ id_b)(omega)
+=((T_acircmathcal L)calL) ⊗ mathcal M)(calM)(chichi^*)succeq0).
[Closure under tensor products]
-\(\mathscr C\) is closed under finite tensor products (after grouping
+(mathscr CscrC) is closed under finite tensor products (after grouping
all first-party systems and all second-party systems). In particular
-\(\omega^{ ⊗ n}\in\mathscr C\) for every \(\omega\in\mathscr C\)
-and \(n\ge1\).
+(omega^{ ⊗ n}inmathscr CscrC) for every (omegainmathscr CscrC)
+and (nge1).
Proof.
@@ -435,44 +288,44 @@ Tensor the representing vectors and local maps separately. Tensor
products of CP maps are CP and input transpose in the grouped product
basis is the tensor product of the individual input transposes. The
required four CP conditions are preserved. The output is
-\(\omega^{ ⊗ n}\) (trace one). Local permutation unitaries and
+(omega^{ ⊗ n}) (trace one). Local permutation unitaries and
their conjugates are CP pre- and post-compositions and preserve
everything.
[Choi transfer]
-Let \(F:\Mat a\to\Mat b\) and \(G:\Mat b\to\Mat c\) be CP maps and set
-\(F^\sharp=T_a\circ F^\dagger\circ T_b\). Then
+Let (F:Mat atoMat b) and (G:Mat btoMat c) be CP maps and set
+(F^sharp=T_acirc F^daggercirc T_b). Then
-J(G\circ F)=(F^\sharp ⊗ G)(Ω_bΩ_b^*).
+J(Gcirc F)=(F^sharp ⊗ G)(Ω_bΩ_b^*).
-If \(F,G\) are PPT and \(Z:=J(G\circ F)\ne0\), then \(\lambda:=\tr Z>0\)
+If (F,G) are PPT and (Z:=J(Gcirc F)ne0), then (lambda:=tr Z>0)
and
-(Z)/(\lambda)=(F^\sharp ⊗ G)
-((Ω_bΩ_b^*)/(\lambda))\in\mathscr C
+(Z)/(lambda)=(F^sharp ⊗ G)
+((Ω_bΩ_b^*)/(lambda))inmathscr CscrC
-on \(\C^a ⊗ \C^c\).
+on (C^a ⊗ C^c).
Proof.
-If \(A_r\) and \(B_s\) are Kraus matrices for \(F\) and \(G\), then
-\(F^\sharp(Y)=\sum_r A_r^{\T}YA_rbar=\sum_r A_r^{\T}Y(A_r^{\T})^*\),
-so the Kraus matrices of \(F^\sharp\) are the transposes \(A_r^{\T}\).
+If (A_r) and (B_s) are Kraus matrices for (F) and (G), then
+(F^sharp(Y)=sum_r A_r^{T}YA_rbar=sum_r A_r^{T}Y(A_r^{T})^*),
+so the Kraus matrices of (F^sharp) are the transposes (A_r^{T}).
For each pair, direct expansion gives
-\((A_r^{\T} ⊗ B_s)Ω_b=(I_a ⊗ B_sA_r)Ω_a\).
+((A_r^{T} ⊗ B_s)Ω_b=(I_a ⊗ B_sA_r)Ω_a).
Summing the corresponding rank-one outer products gives
(eq:choi-transfer).
-Now suppose \(F,G\) are PPT. Besides \(F^\sharp\) and \(G\) CP, Lemma~%
-lem:ppt-stability gives \(F^\sharp\circ T_b=T_a\circ F^\dagger
-=(F\circ T_a)^\dagger\) CP and \(G\circ T_b\) CP, which are exactly
+Now suppose (F,G) are PPT. Besides (F^sharp) and (G) CP, Lemma~%
+lem:ppt-stability gives (F^sharpcirc T_b=T_acirc F^dagger
+=(Fcirc T_a)^dagger) CP and (Gcirc T_b) CP, which are exactly
the input-transpose conditions in (eq:represented-class). Finally
-\(Z=\sum_{r,s}|(I_a ⊗ B_sA_r)Ω_a><(I_a ⊗ B_sA_r)Ω_a|\)
-and \(\lambda=\sum_{r,s}\norm{B_sA_r}_F^2\). Since \(Z\ne0\),
-\(\lambda>0\), and \(\chi=Ω_b/\sqrt\lambda\) proves
-(eq:choi-transfer-normalized). Its squared norm is \(b/\lambda\),
+(Z=sum_{r,s}|(I_a ⊗ B_sA_r)Ω_a><(I_a ⊗ B_sA_r)Ω_a|)
+and (lambda=sum_{r,s}norm{B_sA_r}_F^2). Since (Zne0),
+(lambda>0), and (chi=Ω_b/sqrtlambda) proves
+(eq:choi-transfer-normalized). Its squared norm is (b/lambda),
which is allowed.
@@ -480,7 +333,7 @@ which is allowed.
The first step of the operational obstruction is a purely algebraic
-inequality about any state in \(\mathscr C\) and four arbitrary positive
+inequality about any state in (mathscr CscrC) and four arbitrary positive
local matrices. It compares the "coherence" between two equal-bit
branches with the weights of branches in which the bits disagree. No
pointwise normalization is assumed: later the matrices will be densities
@@ -489,40 +342,40 @@ required.
[Positive block estimate]
-Let \(H\) and \(J\) be square matrices of possibly different sizes and
-\(Z\) have corresponding rectangular size. If
-\(H&Z\\ Z^*&J\succeq0\), then
-\(\norm{Z}_1\le√(\tr H\,\tr J)\). This holds also for singular
-\(H,J\).
+Let (H) and (J) be square matrices of possibly different sizes and
+(Z) have corresponding rectangular size. If
+(H&Z Z^*&Jsucceq0), then
+(norm{Z}_1le√(tr H,tr J)). This holds also for singular
+(H,J).
Proof.
Spectral decomposition of the positive block matrix writes it as a finite
-sum of outer products of block columns \(\binom{a_k}{b_k}\). So
-\(H=\sum_k a_ka_k^*\), \(J=\sum_k b_kb_k^*\), \(Z=\sum_k a_kb_k^*\).
-Since a rank-one \(ab^*\) has trace norm \(\norm{a}\norm{b}\), triangle
+sum of outer products of block columns (binom{a_k}{b_k}). So
+(H=sum_k a_ka_k^*), (J=sum_k b_kb_k^*), (Z=sum_k a_kb_k^*).
+Since a rank-one (ab^*) has trace norm (norm{a}norm{b}), triangle
and C-S give
-\(\norm{Z}_1\le\sum_k\norm{a_k}\norm{b_k}
-\le(\sum_k\norm{a_k}^2)^{1/2}(\sum_k\norm{b_k}^2)^{1/2}
-=√(\tr H\,\tr J)\).
+(norm{Z}_1lesum_knorm{a_k}norm{b_k}
+le(sum_knorm{a_k}^2)^{1/2}(sum_knorm{b_k}^2)^{1/2}
+=√(tr H,tr J)).
[Four-effect inequality]
-Let \(\omega\in\mathscr C\) act on \(\C^p ⊗ \C^q\). For arbitrary
-\(X_0,X_1\succeq0\) in \(\Mat p\) and \(Y_0,Y_1\succeq0\) in \(\Mat q\),
-put \(D_{ij}=X_i ⊗ Y_j\) and \(p_{ij}=\tr(D_{ij}\omega)\), with
-\(i,j\in\{0,1\}\). Then
+Let (omegainmathscr CscrC) act on (C^p ⊗ C^q). For arbitrary
+(X_0,X_1succeq0) in (Mat p) and (Y_0,Y_1succeq0) in (Mat q),
+put (D_{ij}=X_i ⊗ Y_j) and (p_{ij}=tr(D_{ij}omega)), with
+(i,jin{0,1}). Then
-\norm{\sqrt{D_{00}}\,\omega\sqrt{D_{11}}}_1
-\le\sqrt{p_{00}p_{01}}+\sqrt{p_{10}p_{11}}+2\sqrt{p_{01}p_{10}}.
+norm{sqrt{D_{00}},omegasqrt{D_{11}}}_1
+lesqrt{p_{00}p_{01}}+sqrt{p_{10}p_{11}}+2sqrt{p_{01}p_{10}}.
No normalization is assumed.
Proof. [Strategy]
-Represent \(\omega=(\mathcal L ⊗ \mathcal M)(\chi\chi^*)\) with
-\(\mathcal L,\mathcal M\) input-transpose CP. Pull the \(X_i,Y_j\) back
+Represent (omega=(mathcal LcalL ⊗ mathcal M)(calM)(chichi^*)) with
+(mathcal L,calL,mathcal McalM) input-transpose CP. Pull the (X_i,Y_j) back
by adjoints to positive matrices on the input spaces. A congruence
regularizes the two pullback pairs so they commute with sum equal to a
projection. Then a direct block bound and a reverse block bound (via
@@ -534,86 +387,86 @@ terms on the RHS.
Proof.
Fix a representation
-\(\omega=(\mathcal L ⊗ \mathcal M)(\chi\chi^*)\) with
-\(\mathcal L\circ T_r\), \(\mathcal M\circ T_s\) CP. Pull back:
-\(A_i=\mathcal L^\dagger(X_i)\in\Mat r\),
-\(B_j=\mathcal M^\dagger(Y_j)\in\Mat s\). These are positive.
+(omega=(mathcal LcalL ⊗ mathcal M)(calM)(chichi^*)) with
+(mathcal LcalLcirc T_r), (mathcal McalMcirc T_s) CP. Pull back:
+(A_i=mathcal L^calL^dagger(X_i)inMat r),
+(B_j=mathcal M^calM^dagger(Y_j)inMat s). These are positive.
Normalization on the support.:
-Let \(H=A_0+A_1\) and \(\Pi\) be the orthogonal projection onto
-\(\ran H\). If \(a\in\ker H\), then \(a^*A_i a=\norm{√(A_i)a}^2\)
-sums to zero over \(i\), so \(A_i\) kills \(\ker H\) and
-\(A_i=\Pi A_i\Pi\). Let \(S\) be \(H^{-1/2}\) on \(\ran H\) and the
-identity on \(\ker H\). Then the \(A_i'=S^*A_iS\) are positive, supported
-on \(\Pi\), and satisfy \(A_0'+A_1'=\Pi\), so they commute.
+Let (H=A_0+A_1) and (Pi) be the orthogonal projection onto
+(ran H). If (ainker H), then (a^*A_i a=norm{√(A_i)a}^2)
+sums to zero over (i), so (A_i) kills (ker H) and
+(A_i=Pi A_iPi). Let (S) be (H^{-1/2}) on (ran H) and the
+identity on (ker H). Then the (A_i'=S^*A_iS) are positive, supported
+on (Pi), and satisfy (A_0'+A_1'=Pi), so they commute.
This congruence is achievable inside the representation without changing
-\(\omega\): with \(\Ad_S(U)=SUS^*\), replace
-\(\mathcal L\) by \(\mathcal L\circ\Ad_S\) and \(\chi\) by
-\((S^{-1} ⊗ I_s)\chi\). The output \(\omega\) is unchanged, the
-pullbacks become \(A_i'\), the map is still CP, and
-\(\mathcal L'\circ T_r=(\mathcal L\circ T_r)\circ\Ad_{\overline S}\) is
+(omega): with (Ad_S(U)=SUS^*), replace
+(mathcal LcalL) by (mathcal LcalLcircAd_S) and (chi) by
+((S^{-1} ⊗ I_s)chi). The output (omega) is unchanged, the
+pullbacks become (A_i'), the map is still CP, and
+(mathcal L'calL'circ T_r=(mathcal LcalLcirc T_r)circAd_{overlineoverline S}S) is
CP. Suppress the primes.
Two block estimates.:
-Let \(C=\mathcal L ⊗ \mathcal M\). For vectors
-\(u,z\in\C^r ⊗ \C^s\) put
-\(p_{ij}(u)=u^*(A_i ⊗ B_j)u\) and
-\(Z(u,z)=\sqrt{D_{00}}\,C(uz^*)\sqrt{D_{11}}\). CP of \(C\) applied to
-the rank-one \(\binom{u}{z}\binom{u}{z}^*\) gives a positive block.
-Conjugating by \(diag(\sqrt{D_{00}},\sqrt{D_{11}})\) and
+Let (C=mathcal LcalL ⊗ mathcal McalM). For vectors
+(u,zinC^r ⊗ C^s) put
+(p_{ij}(u)=u^*(A_i ⊗ B_j)u) and
+(Z(u,z)=sqrt{D_{00}},C(uz^*)sqrt{D_{11}}). CP of (C) applied to
+the rank-one (binom{u}{z}binom{u}{z}^*) gives a positive block.
+Conjugating by (diag(sqrt{D_{00}},sqrt{D_{11}})) and
applying Lemma~lem:positive-block yields
-\norm{Z(u,z)}_1\le\sqrt{p_{00}(u)p_{11}(z)}.
+norm{Z(u,z)}_1lesqrt{p_{00}(u)p_{11}(z)}.
For the reverse estimate, the global input transpose
-\(T_{rs}=T_r ⊗ T_s\) makes \(C\circ T_{rs}\) CP. Apply the same
-block argument to this map on the ordered pair \((\overline z,\overline u)\)
+(T_{rs}=T_r ⊗ T_s) makes (Ccirc T_{rs}) CP. Apply the same
+block argument to this map on the ordered pair ((overlineoverline z,z,overlineoverline u)u))
and the same output filters. Since
-\((\overline z\,\overline u^*)^\T=uz^*\),
-\((\overline z\,\overline z^*)^\T=zz^*\), and
-\((\overline u\,\overline u^*)^\T=uu^*\), the off-diagonal output is
+((overlineoverline zz,overlineoverline u^*)^u^*)^T=uz^*),
+((overlineoverline zz,overlineoverline z^*)^z^*)^T=zz^*), and
+((overlineoverline uu,overlineoverline u^*)^u^*)^T=uu^*), the off-diagonal output is
unchanged but the two diagonal weights swap:
-\norm{Z(u,z)}_1\le\sqrt{p_{00}(z)p_{11}(u)}.
+norm{Z(u,z)}_1lesqrt{p_{00}(z)p_{11}(u)}.
It is the transpose on both input factors that supplies this reverse
estimate.
Splitting the input vector.:
-Choose a common eigenbasis of the commuting \(A_0,A_1\). Let \(P\) project
-onto the eigenvectors where the \(A_0\)-eigenvalue is at least the
-\(A_1\)-eigenvalue, and set \(v=(P ⊗ I_s)\chi\),
-\(w=((I_r-P) ⊗ I_s)\chi\). Since \(PA_i(I_r-P)=0\), cross terms in
-the \(p_{ij}\) vanish:
+Choose a common eigenbasis of the commuting (A_0,A_1). Let (P) project
+onto the eigenvectors where the (A_0)-eigenvalue is at least the
+(A_1)-eigenvalue, and set (v=(P ⊗ I_s)chi),
+(w=((I_r-P) ⊗ I_s)chi). Since (PA_i(I_r-P)=0), cross terms in
+the (p_{ij}) vanish:
-p_{ij}=p_{ij}(v)+p_{ij}(w),\quad
-p_{11}(v)\le p_{01}(v),\quad p_{00}(w)\le p_{10}(w).
+p_{ij}=p_{ij}(v)+p_{ij}(w),quad
+p_{11}(v)le p_{01}(v),quad p_{00}(w)le p_{10}(w).
-The first comparison follows because \(A_0-A_1\succeq0\) on \(\ran P\),
-so its tensor with \(B_1\) has nonnegative quadratic form on \(v\); the
-second follows similarly on \(\ker P\) with \(B_0\).
+The first comparison follows because (A_0-A_1succeq0) on (ran P),
+so its tensor with (B_1) has nonnegative quadratic form on (v); the
+second follows similarly on (ker P) with (B_0).
Assembling four terms.:
Diagonal terms: (eq:direct-block-bound) with (eq:spectral-weights)
gives
-\[
-\norm{Z(v,v)}_1\le\sqrt{p_{00}(v)p_{11}(v)}\le\sqrt{p_{00}p_{01}},
-\qquad
-\norm{Z(w,w)}_1\le\sqrt{p_{00}(w)p_{11}(w)}\le\sqrt{p_{10}p_{11}}.
-\]
-Off-diagonal: \(Z(v,w)\) is bounded by (eq:reversed-block-bound) and
-\(Z(w,v)\) by (eq:direct-block-bound), and both right sides become
-\(\sqrt{p_{10}p_{01}}\). Finally
-\(\sqrt{D_{00}}\omega\sqrt{D_{11}}
-=Z(v,v)+Z(v,w)+Z(w,v)+Z(w,w)\), so the triangle inequality gives
+[
+norm{Z(v,v)}_1lesqrt{p_{00}(v)p_{11}(v)}lesqrt{p_{00}p_{01}},
+qquad
+norm{Z(w,w)}_1lesqrt{p_{00}(w)p_{11}(w)}lesqrt{p_{10}p_{11}}.
+]
+Off-diagonal: (Z(v,w)) is bounded by (eq:reversed-block-bound) and
+(Z(w,v)) by (eq:direct-block-bound), and both right sides become
+(sqrt{p_{10}p_{01}}). Finally
+(sqrt{D_{00}}omegasqrt{D_{11}}
+=Z(v,v)+Z(v,w)+Z(w,v)+Z(w,w)), so the triangle inequality gives
(eq:coherence).
By Lemma~lem:tensor-closure this applies to arbitrary joint
positive matrices on all of Alice's and all of Bob's copies of a state in
-\(\mathscr C\). Its lack of normalization is exactly what lets us apply it
+(mathscr CscrC). Its lack of normalization is exactly what lets us apply it
at each complete public transcript.
# Complete public transcripts
@@ -630,34 +483,34 @@ product structure.
## Finite-record case
-Let \(R\) be the input state on \(\C^a ⊗ \C^b\).
+Let (R) be the input state on (C^a ⊗ C^b).
In fixed product bases, Eve's canonical purification is
-|\Psi_R>=\sum_h \sqrt R\,|h>_{AB}|h>_E,
-\qquad E\cong A ⊗ B.
+|Psi_R>=sum_h sqrt R,|h>_{AB}|h>_E,
+qquad Econg A ⊗ B.
Suppose the protocol is finite (only finitely many steps, all classical
-records finite) and produces bits \(i,j\in\{0,1\}\) almost surely on
-\(R\). At a fixed complete transcript \(t\), Alice's instrument branches
-consistent with \(t\) and final bit \(i\) compose into a CP map
-\(\mathcal A_i^t\) on her input; Bob's into \(\mathcal B_j^t\) on his.
+records finite) and produces bits (i,jin{0,1}) almost surely on
+(R). At a fixed complete transcript (t), Alice's instrument branches
+consistent with (t) and final bit (i) compose into a CP map
+(mathcal A_i^tcalA_i^t) on her input; Bob's into (mathcal B_j^tcalB_j^t) on his.
Because the compatible private histories sum independently inside each
laboratory (the other party's unannounced outcomes cannot affect what a
laboratory does), the joint branch is
-\(\mathcal A_i^t ⊗ \mathcal B_j^t\). Public-source weights are
-scalars fixed by \(t\); absorb the product into either local map.
+(mathcal A_i^tcalA_i^t ⊗ mathcal B_j^tcalB_j^t). Public-source weights are
+scalars fixed by (t); absorb the product into either local map.
-Set \(X_i(t)=(\mathcal A_i^t)^\dagger(I)\) and
-\(Y_j(t)=(\mathcal B_j^t)^\dagger(I)\). These are positive matrices, and
-the effect at branch \((i,j)\) is \(D_{ij}(t)=X_i(t) ⊗ Y_j(t)\).
+Set (X_i(t)=(mathcal A_i^t)^calA_i^t)^dagger(I)) and
+(Y_j(t)=(mathcal B_j^t)^calB_j^t)^dagger(I)). These are positive matrices, and
+the effect at branch ((i,j)) is (D_{ij}(t)=X_i(t) ⊗ Y_j(t)).
Tracing out the private memories leaves Eve with the density
-\[
-\tau_{ij}(t)=(\sqrt R\,D_{ij}(t)\sqrt R)^\T,
-\]
-where the transpose is in the product basis identifying \(E\) with
-\(A ⊗ B\). The same \(X_0,X_1\) and \(Y_0,Y_1\) work for all four
+[
+tau_{ij}(t)=(sqrt R,D_{ij}(t)sqrt R)^T,
+]
+where the transpose is in the product basis identifying (E) with
+(A ⊗ B). The same (X_0,X_1) and (Y_0,Y_1) work for all four
branches. Private instrument labels are summed, not transmitted to Eve.
-No pointwise normalization is required at a fixed \(t\).
+No pointwise normalization is required at a fixed (t).
## Admitted protocol model
@@ -670,69 +523,69 @@ transfers to the actual input via a filter on the references.
[Admitted local protocols]
-\(A,B\) are finite-dimensional complex input spaces. An admitted protocol
+(A,B) are finite-dimensional complex input spaces. An admitted protocol
consists of local quantum instruments, local memories and randomness, and
authenticated two-way public classical communication. Data and joint
laws satisfy:
-\item
+item
Initial resources. A recorded standard Borel public variable
-\(C_0\), possibly constant, with a law fixed once the protocol is chosen.
+(C_0), possibly constant, with a law fixed once the protocol is chosen.
It is independent of the input plus any purifying or untouched reference.
-Conditional on \(C_0\), the joint state of input-reference and the two
+Conditional on (C_0), the joint state of input-reference and the two
initial private auxiliaries factors position by position: input-reference
state times one local resource per laboratory. Shared initial randomness
-is included in \(C_0\). No preshared secret key.
-\item
+is included in (C_0). No preshared secret key.
+item
Local instruments. Each instrument assigns CP operations to
measurable outcome sets, countably additively on states, with measurable
outcome laws and measurable conditional local updates, and total
operation trace-preserving. Outcome space is standard Borel. Instruments
may depend measurably on local private history and preceding public
record.
-\item
+item
Local aggregation and chronological history. A local subroutine
completed before a later public step may be aggregated into one
instrument that retains its continuing memory. Public communication stays
in the public schedule. Genuine classical history, including private
outcomes and public records, is a finite sequence or an
-\(\mathbb N\)-indexed sequence in standard Borel coordinate spaces. Local
+(mathbb NbbN)-indexed sequence in standard Borel coordinate spaces. Local
choices use only that laboratory's preceding private history and the
preceding public record; anything affecting the other party is
communicated through a public step.
-\item
+item
Public steps and the complete record. Public discussion is
-correspondingly finite or \(\mathbb N\)-indexed. Each message is
+correspondingly finite or (mathbb NbbN)-indexed. Each message is
produced by its sender's local process from that sender's private
history and the preceding public record. A later public-source draw
-uses a measurable probability kernel \(\kappa_m(t,dc)\) depending only on
+uses a measurable probability kernel (kappa_m(t,dc)) depending only on
the preceding public record and identical for every input; the draw
multiplies by the scalar kernel and acts as the identity on quantum
registers. Public choices of speaker and stopping decisions use these
-same two rules. \(T\) records \(C_0\), all public messages, draws, speaker
+same two rules. (T) records (C_0), all public messages, draws, speaker
choices, and stopping decisions. Private outcomes and quantum memories
stay home.
-\item
+item
Reference-compatible prefix laws. The specified instruments act
on arbitrary positive trace-class inputs. Unnormalized branch laws give
countably additive positive matrix-valued measures on any untouched
finite-dimensional reference for finite prefixes. Integrating the next
record recovers the preceding reference measure after tracing out the
current laboratory memories.
-\item
+item
Terminal outputs. A terminal classical output may be a measurable
-function of that laboratory's instrument history and received \(T\). A
+function of that laboratory's instrument history and received (T). A
terminal quantum readout is supplied with its incoming memory (via one
fixed measurable local CP process on those histories) and a selected
local instrument on that memory, selected using only that laboratory's
-private history and \(T\). Together with the schedule, these give one
+private history and (T). Together with the schedule, these give one
countably additive unnormalized instrument law on the complete history
and terminal outcomes for all inputs and finite untouched references.
Each terminal readout includes a tag for missing or non-invoked. Summing
either laboratory's terminal outcomes removes that local refinement;
summing both and tracing memory recovers the reference measure on the
complete history.
-\item
+item
Completion on a specified input. A completed bit-output protocol
on a specified input has a bit in each laboratory almost surely on that
input, with normalized joint output. More generally a completed
@@ -752,22 +605,22 @@ classical bit can depend on an entire infinite record.
[Common local factors at a transcript]
-Let \(R\) be a density operator on \(A ⊗ B\), and let an admitted
-protocol produce bits \(i,j\in\{0,1\}\) almost surely on \(R\). Give Eve
+Let (R) be a density operator on (A ⊗ B), and let an admitted
+protocol produce bits (i,jin{0,1}) almost surely on (R). Give Eve
the canonical purification~(eq:canonical-purification) and let her
-retain \(E\) and the complete transcript \(T\) with values in a standard
-Borel space. Then there exist a finite positive measure \(\mu\) on the
-transcript space and measurable positive matrices \(X_0(t),X_1(t)\in\Mat a\)
-encoding Alice's side and \(Y_0(t),Y_1(t)\in\Mat b\) encoding Bob's side,
-such that the positive matrix-valued measure of Eve in branch \((i,j)\)
+retain (E) and the complete transcript (T) with values in a standard
+Borel space. Then there exist a finite positive measure (mu) on the
+transcript space and measurable positive matrices (X_0(t),X_1(t)inMat a)
+encoding Alice's side and (Y_0(t),Y_1(t)inMat b) encoding Bob's side,
+such that the positive matrix-valued measure of Eve in branch ((i,j))
has density
-\tau_{ij}(t)=(\sqrt R\,[X_i(t) ⊗ Y_j(t)]\sqrt R)^\T
-\quadwith respect to \mu,
+tau_{ij}(t)=(sqrt R,[X_i(t) ⊗ Y_j(t)]sqrt R)^T
+quadwith respect to mu,
-the same \(X_i,Y_j\) work for all four branches outside one
-\(\mu\)-null set, and
-\(\sum_{i,j}\int\tr\tau_{ij}(t)\,d\mu(t)=1\).
+the same (X_i,Y_j) work for all four branches outside one
+(mu)-null set, and
+(sum_{i,j}inttrtau_{ij}(t),dmu(t)=1).
Proof. [Strategy]
@@ -781,14 +634,14 @@ through to the actual input.
Proof.
-Set \(a=\dim A\), \(b=\dim B\), and attach untouched references
-\(A_0\cong A\) and \(B_0\cong B\). Run the protocol on the product of the
+Set (a=dim A), (b=dim B), and attach untouched references
+(A_0cong A) and (B_0cong B). Run the protocol on the product of the
normalized local probes
-|\Phi_a>_{AA_0} ⊗ |\Phi_b>_{BB_0},
-\quad
-|\Phi_a>=a^{-1/2}\sum_{r=1}^a |r,r>,\quad
-|\Phi_b>=b^{-1/2}\sum_{s=1}^b |s,s>.
+|Phi_a>_{AA_0} ⊗ |Phi_b>_{BB_0},
+quad
+|Phi_a>=a^{-1/2}sum_{r=1}^a |r,r>,quad
+|Phi_b>=b^{-1/2}sum_{s=1}^b |s,s>.
Assign bit zero in a laboratory if its prescribed bit is never produced,
retaining bits already produced. This is a measurable local default rule;
@@ -796,37 +649,37 @@ we remove its contribution after recovering the actual input.
Conditional independence of local tapes.:
Take a finite informationally complete POVM on each reference (basis
-projections, and \((|r>+|s>)/\sqrt2\) and
-\((|r>+i|s>)/\sqrt2\) rank-one projections, normalized into
+projections, and ((|r>+|s>)/sqrt2) and
+((|r>+i|s>)/sqrt2) rank-one projections, normalized into
a POVM). These tests commute with all laboratory operations because the
references are untouched.
-In this tested product-probe experiment, sample \(C_0\) with its fixed
-law, then conditional on \(C_0\) simulate the whole run using independent
-local random tapes \(U\) (Alice) and \(V\) (Bob), plus fresh public
+In this tested product-probe experiment, sample (C_0) with its fixed
+law, then conditional on (C_0) simulate the whole run using independent
+local random tapes (U) (Alice) and (V) (Bob), plus fresh public
uniform random variables for public-source draws.
At each local instrument, sample the conditional outcome law from a fresh
uniform variable on the appropriate tape, and apply the stipulated
conditional update. Only the recorded public-source draw enters the
transcript. Induction over finite prefixes gives the genuine joint
finite-prefix laws. Standard-Borel kernel sampling is standard
-\cite[App.~A1, Thm.~A1.2, Lemma 3.22]{Kallenberg2002}.
+cite[App.~A1, Thm.~A1.2, Lemma 3.22]{Kallenberg2002}.
Each defaulted bit and each reference POVM outcome is a measurable
-function of one local tape and \(T\). Let \(\mathcal F_m=\sigma(T_m)\).
+function of one local tape and (T). Let (mathcal F_m=calF_m=sigma(T_m)).
A step sent by Alice makes the new record measurable with respect to
-\(\sigma(U)\vee\mathcal F_m\); Bob's symmetric. A public-source step
+(sigma(U)veemathcal F_mcalF_m); Bob's symmetric. A public-source step
has the sampling kernel as its regular conditional law. Induction shows
-\(U\perp V\mid\mathcal F_m\) for every \(m\), and passing to
-\(\mathcal F_∞=\sigma(T)\) in \(L^1\) gives
-\(\mathbb E[u(U)v(V)\mid T]=\mathbb E[u(U)\mid T]\mathbb E[v(V)\mid T]\).
+(Uperp Vmidmathcal F_mcalF_m) for every (m), and passing to
+(mathcal F_∞=calF_∞=sigma(T)) in (L^1) gives
+(mathbb E[u(U)v(V)bbE[u(U)v(V)mid T]=mathbb E[u(U)bbE[u(U)mid T]mathbb E[v(V)bbE[v(V)mid T]).
Using regular conditional probability kernels on the standard Borel
-tape and transcript spaces~\cite[Thm.~6.3--6.4]{Kallenberg2002} and
+tape and transcript spaces~cite[Thm.~6.3--6.4]{Kallenberg2002} and
testing against countable generating algebras, we get
-\(\mathsf P_{U,V\mid T=t}=\mathsf P_{U\mid T=t} ⊗ \mathsf P_{V\mid T=t}\)
-on a single full-measure set of \(t\). Alice's defaulted bit + reference
-POVM outcome is conditionally independent of Bob's, given \(t\).
+(mathsf P_sfP_{U,Vmid T=t}=mathsf P_sfP_{Umid T=t} ⊗ mathsf P_sfP_{Vmid T=t})
+on a single full-measure set of (t). Alice's defaulted bit + reference
+POVM outcome is conditionally independent of Bob's, given (t).
Reference densities and product structure.:
Return to the run without performing the reference POVM on the probes.
@@ -834,97 +687,97 @@ The fixed initial law and specified finite-prefix instruments give a
positive matrix-valued reference measure at every prefix; summing the
next local outcome recovers it, as does integrating a public-source
outcome. Countable extension in finite dimension
-\cite[Thm.~6.14]{Kallenberg2002} constructs the complete-history positive
-matrix-valued reference measure. Let \(W\) denote the resulting reference
+cite[Thm.~6.14]{Kallenberg2002} constructs the complete-history positive
+matrix-valued reference measure. Let (W) denote the resulting reference
measure on the history space; when a terminal quantum readout is used,
-its instrument refines this to \(\widehat W\), agreeing on the preterminal
+its instrument refines this to (widehatwidehat WW), agreeing on the preterminal
history.
-Push \(\widehat W\) forward to the defaulted-bit pair plus transcript.
-Call the transport measures \(W_{ij}\). The probe transcript law
-\(\mu(S)=\sum_{i,j}\tr W_{ij}(S)\) is a probability measure. Squeeze and
-Radon--Nikodym give matrix-entry densities \(W_{ij}(t)\) with
-\(\sum_{i,j}\tr W_{ij}(t)=1\) for \(\mu\)-a.e. \(t\), positivity
+Push (widehatwidehat WW) forward to the defaulted-bit pair plus transcript.
+Call the transport measures (W_{ij}). The probe transcript law
+(mu(S)=sum_{i,j}tr W_{ij}(S)) is a probability measure. Squeeze and
+Radon--Nikodym give matrix-entry densities (W_{ij}(t)) with
+(sum_{i,j}tr W_{ij}(t)=1) for (mu)-a.e. (t), positivity
a.e., and
-\[
-U_i(t)=\sum_j\tr_{B_0}W_{ij}(t),\quad
-V_j(t)=\sum_i\tr_{A_0}W_{ij}(t),\quad
-\sum_i\tr U_i(t)=\sum_j\tr V_j(t)=1.
-\]
+[
+U_i(t)=sum_jtr_{B_0}W_{ij}(t),quad
+V_j(t)=sum_itr_{A_0}W_{ij}(t),quad
+sum_itr U_i(t)=sum_jtr V_j(t)=1.
+]
The conditional POVM statistics and their factorized form give
-\[
-\tr[(E_r ⊗ G_s)W_{ij}(t)]=\tr[E_rU_i(t)]\tr[G_sV_j(t)]
-\]
+[
+tr[(E_r ⊗ G_s)W_{ij}(t)]=tr[E_rU_i(t)]tr[G_sV_j(t)]
+]
almost everywhere.
-Since \(E_r ⊗ G_s\) span the Hermitian matrices on
-\(A_0 ⊗ B_0\), we conclude
+Since (E_r ⊗ G_s) span the Hermitian matrices on
+(A_0 ⊗ B_0), we conclude
-W_{ij}(t)=U_i(t) ⊗ V_j(t)\quadfor all (i,j)
- and \mu-a.e. t.
+W_{ij}(t)=U_i(t) ⊗ V_j(t)quadfor all (i,j)
+ and mu-a.e. t.
Recovering the actual purification.:
-Identify \(A_0 ⊗ B_0\) with \(E\) in the product basis and set
-\(K=√(ab)\,(\sqrt R)^\T\). The elementary identity
-\((I ⊗ S^\T)\sum_h|h,h>=(S ⊗ I)\sum_h|h,h>\)
-shows that \(I_{AB} ⊗ K\) maps the product probes
-(eq:local-probes) to \(|\Psi_R>\) exactly. The same initial
+Identify (A_0 ⊗ B_0) with (E) in the product basis and set
+(K=√(ab),(sqrt R)^T). The elementary identity
+((I ⊗ S^T)sum_h|h,h>=(S ⊗ I)sum_h|h,h>)
+shows that (I_{AB} ⊗ K) maps the product probes
+(eq:local-probes) to (|Psi_R>) exactly. The same initial
law and local auxiliaries are used on probes and actual input. On each
finite cylinder of histories the reference matrix for the filtered input
-is \(K W(\,\cdot\,)K^*\) because the filter on the untouched references
+is (K W(,cdot,)K^*) because the filter on the untouched references
commutes with every local instrument. Every public-source update also
commutes. Uniqueness of finite measures on cylinders extends this to the
full history sigma field; the same commutation holds for any appended
terminal instrument by clause~protocol:terminal.
-Let \(N\) be the event that some prescribed bit is missing before the
-default rule. On the actual input \(R\), clause~protocol:completion
-gives \(\tr[K\widehat W(N)K^*]=0\), so \(K\widehat W(N)K^*=0\), so
+Let (N) be the event that some prescribed bit is missing before the
+default rule. On the actual input (R), clause~protocol:completion
+gives (tr[Kwidehatwidehat W(N)K^*]=0W(N)K^*]=0), so (Kwidehatwidehat W(N)K^*=0W(N)K^*=0), so
default bit tags correspond to zero filtered measure on the actual
input. Therefore
-\[
-\tau_{ij}(t)=K(U_i(t) ⊗ V_j(t))K^*\quadat \mu-a.e. t.
-\]
+[
+tau_{ij}(t)=K(U_i(t) ⊗ V_j(t))K^*quadat mu-a.e. t.
+]
Expanding,
-\(K(U_i ⊗ V_j)K^*=ab(\sqrt R)^\T(U_i ⊗ V_j)(\sqrt R)^\T
-=(\sqrt R(aU_i^\T ⊗ bV_j^\T)\sqrt R)^\T\).
-Take \(X_i(t)=aU_i(t)^\T\) and \(Y_j(t)=bV_j(t)^\T\), adjusting on a
+(K(U_i ⊗ V_j)K^*=ab(sqrt R)^T(U_i ⊗ V_j)(sqrt R)^T
+=(sqrt R(aU_i^T ⊗ bV_j^T)sqrt R)^T).
+Take (X_i(t)=aU_i(t)^T) and (Y_j(t)=bV_j(t)^T), adjusting on a
null set if necessary. Transposition preserves positivity, giving
-(eq:eve-blocks). No inverse of \(R\) appears.
+(eq:eve-blocks). No inverse of (R) appears.
[Recovery from another purifying space]
-Let \(R\) and \(|\Psi_R>\) be as in Lemma~lem:transcript, and
-let \(|\psi>\in A ⊗ B ⊗ \widetilde E\) be any
+Let (R) and (|Psi_R>) be as in Lemma~lem:transcript, and
+let (|psi>in A ⊗ B ⊗ widetilde E) be any
purification. There is a channel
-\(\mathcal R:\mathcal T(\widetilde E)\to\mathcal T(E)\) mapping
-\(|\psi>\) to \(|\Psi_R>\). If Alice and Bob run the same
+(mathcal R:calR:mathcal T(calT(widetilde E)tomathcal T(E)calT(E)) mapping
+(|psi>) to (|Psi_R>). If Alice and Bob run the same
admitted protocol and Eve retains her purification and full transcript,
-\(\mathcal R ⊗ id_T\) recovers the canonical output of
-Lemma~lem:transcript. Moreover \(\mathcal R\) maps every ideal
-secret-bit law with Eve in \(\widetilde ET\) to one with Eve in \(ET\).
+(mathcal RcalR ⊗ id_T) recovers the canonical output of
+Lemma~lem:transcript. Moreover (mathcal RcalR) maps every ideal
+secret-bit law with Eve in (widetilde ET) to one with Eve in (ET).
Proof.
-Let \(\mathcal S=\supp(R^\T)⊆ E\). Schmidt decomposition gives
-an isometry \(V:\mathcal S\to\widetilde E\) with
-\(|\psi>=(I_{AB} ⊗ V)|\Psi_R>\).
-Choose any density \(\zeta\) on \(E\) and set
-\[
-\mathcal R(\xi)=V^*\xi V+\tr[(I_{\widetilde E}-VV^*)\xi]\,\zeta.
-\]
+Let (mathcal S=calS=supp(R^T)⊆ E). Schmidt decomposition gives
+an isometry (V:mathcal ScalStowidetilde E) with
+(|psi>=(I_{AB} ⊗ V)|Psi_R>).
+Choose any density (zeta) on (E) and set
+[
+mathcal R(calR(xi)=V^*xi V+tr[(I_{widetilde E}-VV^*)xi],zeta.
+]
Compression is CP; the second term is a positive trace functional times a
-fixed state. Trace is preserved, so \(\mathcal R\) is a channel. Its
-first term inverts \(V\) on the actual purifying support; the second
+fixed state. Trace is preserved, so (mathcal RcalR) is a channel. Its
+first term inverts (V) on the actual purifying support; the second
vanishes there. Local operations keep each conditional purifying state
-on \(\ran V\); public-source steps and identity on register \(T\) do not
+on (ran V); public-source steps and identity on register (T) do not
disturb this. Recovery is exact in every transcript and bit branch, and
-the formula extends to trace-class inputs on larger \(\widetilde E\).
+the formula extends to trace-class inputs on larger (widetilde E).
An ideal bit has form
-\(\frac12\sum_{i}|i,i>< i,i| ⊗ \sigma\); conjugating by
-\(\mathcal R ⊗ id_T\) replaces \(\sigma\) by a normalized
+(frac12sum_{i}|i,i>< i,i| ⊗ sigma); conjugating by
+(mathcal RcalR ⊗ id_T) replaces (sigma) by a normalized
state and keeps the form. Trace-norm contraction transfers any lower
bound from the canonical to the general case.
@@ -933,220 +786,220 @@ bound from the canonical to the general case.
The coherence inequality applies at each transcript. Take
-\(\mathcal S\) = the trace-norm distance from the actual transcript joint
+(mathcal ScalS) = the trace-norm distance from the actual transcript joint
state to the ideal secret-bit transcript joint state. Two forces compete:
correctness forces little mass in unequal-bit branches, and secrecy
forces the equal-bit branches to have matching Eve densities of at least
half mass each. A Cauchy--Schwarz calculation turns these into incompatible
-requirements. Below \(\eta < 1/5\) they cannot both hold.
+requirements. Below (eta < 1/5) they cannot both hold.
Throughout, the trace norm is unhalved:
-\(\norm{H}_1=\tr√(H^*H)\). A classical-quantum state is a positive
+(norm{H}_1=tr√(H^*H)). A classical-quantum state is a positive
matrix-valued measure of total trace one. If a Hermitian difference of
-such measures has density \(H(t)\) with respect to a common finite
-dominating measure \(\lambda\), then
-\(\norm{H}_1=\int\norm{H(t)}_1\,d\lambda(t)\), independent of the
+such measures has density (H(t)) with respect to a common finite
+dominating measure (lambda), then
+(norm{H}_1=intnorm{H(t)}_1,dlambda(t)), independent of the
dominating measure by homogeneity.
[Distillable secret key]
-Let \(\omega\) be a bipartite state, with Eve initially holding a
+Let (omega) be a bipartite state, with Eve initially holding a
purification. A key-distillation sequence uses an admitted protocol of
-Definition~def:protocol on each \(\omega^{ ⊗ n}\), allowing
-each party to process all its \(n\) copies jointly. The normalized
-classical output at each party is a string of fixed length \(\ell_n\).
-Eve's register \(E'\) retains the initial purification and the complete
+Definition~def:protocol on each (omega^{ ⊗ n}), allowing
+each party to process all its (n) copies jointly. The normalized
+classical output at each party is a string of fixed length (ell_n).
+Eve's register (E') retains the initial purification and the complete
public transcript. For any normalized state or classical-quantum law
-\(\sigma_{E'}\), the ideal length-\(\ell\) key is
-\[
-\gamma^{(\ell)}_\sigma=2^{-\ell}\sum_{z\in\{0,1\}^\ell}
-|z,z>< z,z| ⊗ \sigma_{E'}.
-\]
+(sigma_{E'}), the ideal length-(ell) key is
+[
+gamma^{(ell)}_sigma=2^{-ell}sum_{zin{0,1}^ell}
+|z,z>< z,z| ⊗ sigma_{E'}.
+]
The sequence is secure if there are normalized comparison laws
-\(\sigma_{E'}^{(n)}\) with
-\(\norm{\tau^{(n)}_{K_AK_BE'}-\gamma^{(\ell_n)}_{\sigma^{(n)}}}_1\to0\).
-Its asymptotic lower rate is \(liminf_n \ell_n/n\). The distillable
-secret key \(K_D(\omega)\) is the supremum of these rates over secure
+(sigma_{E'}^{(n)}) with
+(norm{tau^{(n)}_{K_AK_BE'}-gamma^{(ell_n)}_{sigma^{(n)}}}_1to0).
+Its asymptotic lower rate is (liminf_n ell_n/n). The distillable
+secret key (K_D(omega)) is the supremum of these rates over secure
sequences (empty key achieves rate zero).
[Root fidelity and trace norm]
-For square complex matrices \(A,B\) of the same size,
+For square complex matrices (A,B) of the same size,
-f(A^*A,B^*B):=\norm{√(A^*A)√(B^*B)}_1=\norm{AB^*}_1.
+f(A^*A,B^*B):=norm{√(A^*A)√(B^*B)}_1=norm{AB^*}_1.
-For positive matrices \(H,J\) of the same size,
+For positive matrices (H,J) of the same size,
-f(H,J)\ge\frac{\tr H+\tr J-\norm{H-J}_1}{2}.
+f(H,J)gefrac{tr H+tr J-norm{H-J}_1}{2}.
-Also \(f(H^\T,J^\T)=f(H,J)\).
+Also (f(H^T,J^T)=f(H,J)).
Proof.
-Extend the partial isometries in the polar decompositions \(A=U√(A^*A)\)
-and \(B=V√(B^*B)\) to unitaries. Then
-\(AB^*=U√(A^*A)√(B^*B)V^*\); unitary invariance of the trace
-norm gives (eq:polar-fidelity) (valid also for singular \(A,B\)).
-
-For (eq:fidelity-lower), put \(Z=\sqrt H-\sqrt J\). Then
-\(H-J=Z\sqrt H+\sqrt J Z\). The spectral sign
-\(sgn(Z)\) is a Hermitian contraction, so trace-norm
+Extend the partial isometries in the polar decompositions (A=U√(A^*A))
+and (B=V√(B^*B)) to unitaries. Then
+(AB^*=U√(A^*A)√(B^*B)V^*); unitary invariance of the trace
+norm gives (eq:polar-fidelity) (valid also for singular (A,B)).
+
+For (eq:fidelity-lower), put (Z=sqrt H-sqrt J). Then
+(H-J=Zsqrt H+sqrt J Z). The spectral sign
+(sgn(Z)) is a Hermitian contraction, so trace-norm
duality and cyclicity give
-\(\norm{H-J}_1\ge\tr[sgn(Z)(H-J)]=\tr[|Z|(\sqrt H+\sqrt J)]\ge \tr Z^2\),
+(norm{H-J}_1getr[sgn(Z)(H-J)]=tr[|Z|(sqrt H+sqrt J)]ge tr Z^2),
where the last inequality uses
-\(<\sqrt H+\sqrt J>\ge|<\sqrt H-\sqrt J>|
-=|z|\) in an eigenvector of \(Z\). Finally
-\(\tr Z^2=\tr H+\tr J-2\tr(\sqrt H\sqrt J)\) and
-\(0\le\tr(\sqrt H\sqrt J)\le\norm{\sqrt H\sqrt J}_1=f(H,J)\).
+(<sqrt H+sqrt J>ge|<sqrt H-sqrt J>|
+=|z|) in an eigenvector of (Z). Finally
+(tr Z^2=tr H+tr J-2tr(sqrt Hsqrt J)) and
+(0letr(sqrt Hsqrt J)lenorm{sqrt Hsqrt J}_1=f(H,J)).
The transpose identity follows from functional calculus.
[Uniform obstruction to a secret bit]
-Let \(\omega\in\mathscr C\). For every integer \(n\ge1\), let an
-admitted completed protocol on \(\omega^{ ⊗ n}\) produce normalized
-bit registers \(K_A,K_B\), and let its output \(\tau_{K_AK_BE'}\)
-retain in \(E'\) an initial purification held by Eve and the complete
-public transcript. For every normalized \(\sigma_{E'}\),
+Let (omegainmathscr CscrC). For every integer (nge1), let an
+admitted completed protocol on (omega^{ ⊗ n}) produce normalized
+bit registers (K_A,K_B), and let its output (tau_{K_AK_BE'})
+retain in (E') an initial purification held by Eve and the complete
+public transcript. For every normalized (sigma_{E'}),
-\norm{\tau_{K_AK_BE'}-\tfrac12\sum_{i=0}^1
-|i,i>< i,i| ⊗ \sigma_{E'}}_1\ge\frac15.
+norm{tau_{K_AK_BE'}-tfrac12sum_{i=0}^1
+|i,i>< i,i| ⊗ sigma_{E'}}_1gefrac15.
-Thus the traditional trace distance is at least \(1/10\), and
-\(K_D(\omega)=0\) in Definition~def:distillable-key.
+Thus the traditional trace distance is at least (1/10), and
+(K_D(omega)=0) in Definition~def:distillable-key.
Proof.
-Group Alice's and Bob's factors, and set \(R=\omega^{ ⊗ n}\). Lemma~%
-lem:tensor-closure gives \(R\in\mathscr C\). Apply Eve's classical
+Group Alice's and Bob's factors, and set (R=omega^{ ⊗ n}). Lemma~%
+lem:tensor-closure gives (Rinmathscr CscrC). Apply Eve's classical
readout of the transcript and Lemma~lem:purification-recovery;
trace-norm contraction reduces the claim to the canonical purification and
-the full transcript \(T\).
+the full transcript (T).
-Apply Lemma~lem:transcript: there exist \(\mu\), positive
-\(X_i(t),Y_j(t)\), and Eve densities \(\tau_{ij}(t)\) of
+Apply Lemma~lem:transcript: there exist (mu), positive
+(X_i(t),Y_j(t)), and Eve densities (tau_{ij}(t)) of
(eq:eve-blocks). Fix any normalized positive matrix-valued
-comparison law \(\sigma_{ET}\). Let \(\nu(S)=\tr\sigma_{ET}(S)\), use
-\(\lambda=\mu+\nu\), and replace each actual density and both \(X_i(t)\)
-by their product with \(h=d\mu/d\lambda\), leaving \(Y_j(t)\)
+comparison law (sigma_{ET}). Let (nu(S)=trsigma_{ET}(S)), use
+(lambda=mu+nu), and replace each actual density and both (X_i(t))
+by their product with (h=dmu/dlambda), leaving (Y_j(t))
unchanged. Equation~(eq:eve-blocks) and all probabilities are
-preserved. The ideal law has positive density \(\sigma(t)\) with
-\(\int\tr\sigma(t)\,d\lambda(t)=1\). Now rename \(\lambda\) as \(\mu\).
+preserved. The ideal law has positive density (sigma(t)) with
+(inttrsigma(t),dlambda(t)=1). Now rename (lambda) as (mu).
This common measure keeps singular parts of either law; no support
assumption.
For these common densities set
-\(D_{ij}(t)=X_i(t) ⊗ Y_j(t)\),
-\(p_{ij}(t)=\tr\tau_{ij}(t)=\tr[D_{ij}(t)R]\),
-\(P_{ij}=\int p_{ij}(t)\,d\mu(t)\). All \(P_{ij}\ge0\) and
-\(\sum_{i,j}P_{ij}=1\). With \(A=\sqrt{D_{00}}\sqrt R\) and
-\(B=\sqrt{D_{11}}\sqrt R\), transpose invariance and
+(D_{ij}(t)=X_i(t) ⊗ Y_j(t)),
+(p_{ij}(t)=trtau_{ij}(t)=tr[D_{ij}(t)R]),
+(P_{ij}=int p_{ij}(t),dmu(t)). All (P_{ij}ge0) and
+(sum_{i,j}P_{ij}=1). With (A=sqrt{D_{00}}sqrt R) and
+(B=sqrt{D_{11}}sqrt R), transpose invariance and
(eq:polar-fidelity) give
-f(\tau_{00}(t),\tau_{11}(t))
-=\norm{\sqrt{D_{00}(t)}R\sqrt{D_{11}(t)}}_1.
+f(tau_{00}(t),tau_{11}(t))
+=norm{sqrt{D_{00}(t)}Rsqrt{D_{11}(t)}}_1.
Apply Lemma~lem:coherence a.e. and scalar Cauchy--Schwarz after
integration:
-\mathfrak f:=\int f(\tau_{00}(t),\tau_{11}(t))\,d\mu(t)
-\le\sqrt{P_{00}P_{01}}+\sqrt{P_{10}P_{11}}+2\sqrt{P_{01}P_{10}}.
+mathfrak f:=frakf:=int f(tau_{00}(t),tau_{11}(t)),dmu(t)
+lesqrt{P_{00}P_{01}}+sqrt{P_{10}P_{11}}+2sqrt{P_{01}P_{10}}.
No pointwise POVM normalization was used.
-Let \(\eta\) be the unhalved trace-norm distance to the ideal bit after
+Let (eta) be the unhalved trace-norm distance to the ideal bit after
the common transcript readout, and define
-\(e=P_{01}+P_{10}\),
-\(q=\int\norm{\tau_{00}-\tau_{11}}_1d\mu\),
-\(a_i=\int\norm{\tau_{ii}-\tfrac12\sigma(t)}_1d\mu\).
+(e=P_{01}+P_{10}),
+(q=intnorm{tau_{00}-tau_{11}}_1dmu),
+(a_i=intnorm{tau_{ii}-tfrac12sigma(t)}_1dmu).
The unequal-bit blocks of the ideal state are zero. Additivity of trace
norm across bit blocks, then triangle within equal-bit blocks, gives
-\eta=e+a_0+a_1,\qquad q\le a_0+a_1=\eta-e.
+eta=e+a_0+a_1,qquad qle a_0+a_1=eta-e.
Trace conservation gives:
-since actual equal-bit branches have total trace \(1-e\) while ideal
+since actual equal-bit branches have total trace (1-e) while ideal
have total trace one,
-\(a_0+a_1\ge|e|\), hence by (eq:linked-errors),
+(a_0+a_1ge|e|), hence by (eq:linked-errors),
-e\le\eta/2.
+eleeta/2.
Integration of (eq:fidelity-lower) gives
-\(\mathfrak f\ge(P_{00}+P_{11}-q)/2=(1-e-q)/2\ge(1-\eta)/2\). On the
+(mathfrak ffrakfge(P_{00}+P_{11}-q)/2=(1-e-q)/2ge(1-eta)/2). On the
other side, Cauchy--Schwarz on the first two terms and AM-GM on the
last give
-\(\sqrt{P_{00}P_{01}}+\sqrt{P_{10}P_{11}}\le√(e(1-e))\) and
-\(2\sqrt{P_{01}P_{10}}\le e\), so
+(sqrt{P_{00}P_{01}}+sqrt{P_{10}P_{11}}le√(e(1-e))) and
+(2sqrt{P_{01}P_{10}}le e), so
-(1-\eta)/(2)\le\mathfrak f\le√(e(1-e))+e,\qquad e\le\frac\eta2.
+(1-eta)/(2)lemathfrak ffrakfle√(e(1-e))+e,qquad elefraceta2.
-If \(\eta<1/5\), then \(e<1/10\), and on \(0\le e\le1/10\) the function
-\(√(e(1-e))+e\) is increasing with value \(2/5\) at \(e=1/10\).
-The right side is strictly below \(2/5\), while the left is strictly
+If (eta<1/5), then (e<1/10), and on (0le ele1/10) the function
+(√(e(1-e))+e) is increasing with value (2/5) at (e=1/10).
+The right side is strictly below (2/5), while the left is strictly
above. This proves (eq:uniform-secret-bit-gap).
-Finally, if a secure sequence had \(\ell_n\ge1\) for infinitely many
-\(n\), the local classical channels retaining only the first bit would
-map each \(\gamma^{(\ell_n)}_{\sigma^{(n)}}\) to an ideal bit; trace-norm
+Finally, if a secure sequence had (ell_nge1) for infinitely many
+(n), the local classical channels retaining only the first bit would
+map each (gamma^{(ell_n)}_{sigma^{(n)}}) to an ideal bit; trace-norm
contraction would give completed bit outputs approaching an ideal bit,
-contradicting the gap. Thus every secure sequence has \(\ell_n=0\) for
-all sufficiently large \(n\), and \(K_D(\omega)=0\).
+contradicting the gap. Thus every secure sequence has (ell_n=0) for
+all sufficiently large (n), and (K_D(omega)=0).
[Conditioning, aborts, completed-output limits]
-Consider inputs \(\omega^{ ⊗ n}\) with \(\omega\in\mathscr C\) and
-\(n\ge1\), and admitted protocols whose normalized completed outputs
+Consider inputs (omega^{ ⊗ n}) with (omegainmathscr CscrC) and
+(nge1), and admitted protocols whose normalized completed outputs
retain an arbitrary initial purification and the full transcript.
-\item If \(A\) is a measurable transcript event with probability
- \(r>0\), the normalized bit output conditioned on \(A\) has
- distance at least \(1/5\) from every ideal bit law for the
- accepted output. The bound is uniform in \(r>0\).
-\item Suppose a sequence of completed protocols has a local abort symbol
+item If (A) is a measurable transcript event with probability
+ (r>0), the normalized bit output conditioned on (A) has
+ distance at least (1/5) from every ideal bit law for the
+ accepted output. The bound is uniform in (r>0).
+item Suppose a sequence of completed protocols has a local abort symbol
available. If the probability that at least one party aborts
tends to zero, and the normalized non-aborting outputs approach
ideal non-aborting bit laws, then replacing each abort by the
- fixed bit \(0\) gives completed bit outputs approaching ideal
- bits. Such a sequence is impossible for inputs in \(\mathscr C\).
-\item Let \(\tau_k\) be completed normalized bit outputs to which the
+ fixed bit (0) gives completed bit outputs approaching ideal
+ bits. Such a sequence is impossible for inputs in (mathscr CscrC).
+item Let (tau_k) be completed normalized bit outputs to which the
gap applies, possibly on different output spaces. Suppose their
bit registers are kept fixed and their Eve registers embedded in
- a common \(F\) with embeddings \(\mathcal R_k\). If
- \(\widehat\tau_k\to\tau\) in trace norm, then
- \(\norm{\tau-\tfrac12\sum_i|i,i>< i,i| ⊗ \sigma_F}_1
- \ge 1/5\) for every normalized \(\sigma_F\).
+ a common (F) with embeddings (mathcal R_kcalR_k). If
+ (widehattau_ktotau) in trace norm, then
+ (norm{tau-tfrac12sum_i|i,i>< i,i| ⊗ sigma_F}_1
+ ge 1/5) for every normalized (sigma_F).
Proof.
(1) Restrict the canonical factors of Lemma~lem:transcript to
-\(A\), and replace both \(X_i(t)\) by \(X_i(t)/r\), leaving \(Y_j(t)\)
-unchanged. All four blocks are divided by \(r\), and every probability
+(A), and replace both (X_i(t)) by (X_i(t)/r), leaving (Y_j(t))
+unchanged. All four blocks are divided by (r), and every probability
and root fidelity scale by the same factor. The proof of
(eq:fidelity-sandwich) applies without change. No lower bound on
-\(r\) is used.
-
-(2) Let \(\delta_k\) be the probability of at least one abort,
-\(\tau_k^{ok}\) the normalized non-aborting state (when
-\(\delta_k<1\)), \(\xi_k\) the normalized aborting part. Product local
-channels \(\mathcal Q\) sending each party's own abort symbol to \(0\)
-fix the ideal bit \(\gamma_k\) and give
-\(\norm{\mathcal Q(\tau_k)-\gamma_k}_1
-\le(1-\delta_k)\norm{\tau_k^{ok}-\gamma_k}_1+2\delta_k\to0\),
+(r) is used.
+
+(2) Let (delta_k) be the probability of at least one abort,
+(tau_k^{ok}) the normalized non-aborting state (when
+(delta_k<1)), (xi_k) the normalized aborting part. Product local
+channels (mathcal QcalQ) sending each party's own abort symbol to (0)
+fix the ideal bit (gamma_k) and give
+(norm{mathcal Q(calQ(tau_k)-gamma_k}_1
+le(1-delta_k)norm{tau_k^{ok}-gamma_k}_1+2delta_kto0),
contradicting Theorem~thm:zero-key applied to the completed protocol
with the local replacement appended.
-(3) Recovery channel \(\mathcal R_k\) transfers ideal bits:
-\(\mathcal R_k:\sigma\mapsto\sigma_{\mathcal R_k}\). Contraction and the
+(3) Recovery channel (mathcal R_kcalR_k) transfers ideal bits:
+(mathcal R_k:calR_k:sigmamapstosigma_{mathcal R_k}calR_k). Contraction and the
per-protocol gap give
-\(\norm{\widehat\tau_k-\gamma_\sigma}_1\ge\norm{\tau_k-\gamma_{\mathcal R_k(\sigma)}}_1\ge1/5\).
+(norm{widehattau_k-gamma_sigma}_1genorm{tau_k-gamma_{mathcal R_k(calR_k(sigma)}}_1ge1/5).
Then
-\(\norm{\tau-\gamma_\sigma}_1\ge 1/5-\norm{\widehat\tau_k-\tau}_1\to 1/5\).
-No convergence on auxiliary probes and no factorization of \(\tau\) is
+(norm{tau-gamma_sigma}_1ge 1/5-norm{widehattau_k-tau}_1to 1/5).
+No convergence on auxiliary probes and no factorization of (tau) is
used.
@@ -1154,260 +1007,260 @@ used.
We now construct the dimension-ten pair. The argument has one geometric
-step: a map \(L:\Mat 4\to\Mat 4\) with twenty specific projective
-directions at which \(L(xx^*)\) is singular. The symmetric square of
-\(\C^4\) records the quadratic monomials of \(x\); the exterior square
-\(∧^2\C^4\) converts rank loss of \(L(xx^*)\) into a product
-vector in the kernel of a \(6\times6\) intermediate Choi matrix.
+step: a map (L:Mat 4toMat 4) with twenty specific projective
+directions at which (L(xx^*)) is singular. The symmetric square of
+(C^4) records the quadratic monomials of (x); the exterior square
+(∧^2C^4) converts rank loss of (L(xx^*)) into a product
+vector in the kernel of a (6times6) intermediate Choi matrix.
Two products in the kernel is all we need: the composite Choi matrix
-\(Z\) is then positive with each \(\hat x ⊗ \hat x\) in its kernel.
+(Z) is then positive with each (hathat xx ⊗ hathat xx) in its kernel.
Any product vector would have to be inside the kernel; two nonzero
quadratics whose zero sets together cover all twenty directions cannot
-both vanish on nine of them (because \(9+9<20\)).
+both vanish on nine of them (because (9+9<20)).
-For a vector \(x\), write \(P_x=xx^*\), with no normalization. A
-projective direction \([x]\) is the complex line through \(x\ne0\).
+For a vector (x), write (P_x=xx^*), with no normalization. A
+projective direction ([x]) is the complex line through (xne0).
[Tensor criterion]
-Let \(L:\Mat 4\to\Mat 4\) be PPT. Suppose
-\(\mathcal X=\{x_1,...,x_{20}\}\subset\C^4∖\{0\}\) represents
+Let (L:Mat 4toMat 4) be PPT. Suppose
+(mathcal X=calX={x_1,...,x_{20}}subsetC^4∖{0}) represents
twenty distinct projective directions with:
-\item \(\rank L(P_x)<4\) for every \(x\in\mathcal X\), and every nonzero
- homogeneous quadratic in \(\C[x_0,x_1,x_2,x_3]\) vanishes on at
- most nine members of \(\mathcal X\);
-\item \(L(P_y)\) is positive definite for some \(y\in\C^4\).
+item (rank L(P_x)<4) for every (xinmathcal XcalX), and every nonzero
+ homogeneous quadratic in (C[x_0,x_1,x_2,x_3]) vanishes on at
+ most nine members of (mathcal XcalX);
+item (L(P_y)) is positive definite for some (yinC^4).
Then there are explicitly constructed PPT maps
-\(\Phi_1,\Phi_2:\Mat{10}\to\Mat{10}\) with
-\(Z=J(\Phi_2\circ\Phi_1)\ne0\) and
-\(\ran Z\cap\{u ⊗ v\}=\{0\}\). In particular \(\Phi_2\circ\Phi_1\)
+(Phi_1,Phi_2:Mat{10}toMat{10}) with
+(Z=J(Phi_2circPhi_1)ne0) and
+(ran Zcap{u ⊗ v}={0}). In particular (Phi_2circPhi_1)
is not entanglement breaking.
-Section~sec:certificate constructs the input \(L\). The proof below
+Section~sec:certificate constructs the input (L). The proof below
is the mechanism of the tensor criterion.
Proof.
Symmetric and exterior tensors.
-Index an orthonormal basis of \(\C^{10}\) by pairs \(i\le j<4\), and of
-\(\C^6\) by \(i<j<4\). Define
+Index an orthonormal basis of (C^{10}) by pairs (ile j<4), and of
+(C^6) by (i<j<4). Define
U e_{ii}&=e_i ⊗ e_i,&
-U e_{ij}&=(e_i ⊗ e_j+e_j ⊗ e_i)/(\sqrt2)\quad(i<j),
-\notag\\
-V e_{ij}&=(e_i ⊗ e_j-e_j ⊗ e_i)/(\sqrt2)\quad(i<j).
+U e_{ij}&=(e_i ⊗ e_j+e_j ⊗ e_i)/(sqrt2)quad(i<j),
+notag
+V e_{ij}&=(e_i ⊗ e_j-e_j ⊗ e_i)/(sqrt2)quad(i<j).
Their ranges are the symmetric and antisymmetric tensor subspaces. Put
-\hat x=U^*(x ⊗ x)
-=\sum_i x_i^2\;e_{ii}+\sqrt2\sum_{i<j}x_ix_j\;e_{ij}.
+hathat x=U^*(xx=U^*(x ⊗ x)
+=sum_i x_i^2;e_{ii}+sqrt2sum_{i<j}x_ix_j;e_{ij}.
-Then \(U\hat x=x ⊗ x\) and \(UP_{\hat x}U^*=P_x ⊗ P_x\).
-The coordinates of \(\hat x\) are the ten quadratic monomials of \(x\).
-A nonzero \(u\in\C^{10}\) with \(u^*\hat x=0\) at ten distinct points is
+Then (Uhathat x=xx=x ⊗ x) and (UP_{hathat x}xU^*=P_x ⊗ P_x).
+The coordinates of (hathat xx) are the ten quadratic monomials of (x).
+A nonzero (uinC^{10}) with (u^*hathat x=0x=0) at ten distinct points is
a nonzero homogeneous quadratic vanishing there, which the hypothesis
-forbids. So any ten \(\hat x\)'s are independent; equivalently the ten
-\(\hat x\)'s span \(\C^{10}\) for any ten of the \(x\in\mathcal X\).
-
-Identify \(\C^6=∧^2\C^4\) with \(e_{ij}=e_i\wedge e_j\) and this
-basis orthonormal. Then \(V(a\wedge b)=(a ⊗ b-b ⊗ a)/\sqrt2\).
-For \(i<j\) let \(k<l\) be complements and \(ε_{ijkl}\) be the
-sign of the ordering \((i,j,k,l)\) of \((0,1,2,3)\). Define the real
-signed permutation \(K\) of \(\C^6\) by \(K e_{ij}=ε_{ijkl}e_{kl}\);
+forbids. So any ten (hathat xx)'s are independent; equivalently the ten
+(hathat xx)'s span (C^{10}) for any ten of the (xinmathcal XcalX).
+
+Identify (C^6=∧^2C^4) with (e_{ij}=e_iwedge e_j) and this
+basis orthonormal. Then (V(awedge b)=(a ⊗ b-b ⊗ a)/sqrt2).
+For (i<j) let (k<l) be complements and (ε_{ijkl}) be the
+sign of the ordering ((i,j,k,l)) of ((0,1,2,3)). Define the real
+signed permutation (K) of (C^6) by (K e_{ij}=ε_{ijkl}e_{kl});
explicitly
-\[
+[
(Ke_{01},Ke_{02},Ke_{03},Ke_{12},Ke_{13},Ke_{23})
=(e_{23},-e_{13},e_{12},e_{03},-e_{02},e_{01}),
-\]
-so \(K=K^*=K^{-1}\). The conjugation \(w\mapsto K\overline w\) is the
+]
+so (K=K^*=K^{-1}). The conjugation (wmapsto Koverlineoverline ww) is the
complementary-pair exterior dual used by Schliemann, Loss, and
MacDonald~[cite].
-Define \(S,R:\Mat{10}\to\Mat 6\) by
+Define (S,R:Mat{10}toMat 6) by
-S(A)=V^*(L ⊗ L)(UAU^*)V,\qquad
-R(A)=K\,S(A)^\T\,K^*.
+S(A)=V^*(L ⊗ L)(UAU^*)V,qquad
+R(A)=K,S(A)^T,K^*.
-Embeddings and compressions are CP; \(L\) PPT gives \(L ⊗ L\) PPT;
-Lemma~lem:ppt-stability makes \(S\) PPT. Input-transpose CP of
-\(S\) plus simultaneous transpose gives \(T_6\circ S\) PPT, so \(R\)
+Embeddings and compressions are CP; (L) PPT gives (L ⊗ L) PPT;
+Lemma~lem:ppt-stability makes (S) PPT. Input-transpose CP of
+(S) plus simultaneous transpose gives (T_6circ S) PPT, so (R)
is PPT.
The composite Choi matrix.
The rectangular maps give an endomorphism
-\(R^\dagger\circ S\circ T_{10}\) of \(\Mat{10}\). To express it as a
-composition of two endomorphisms, let \(E:\C^6\to\C^{10}\) be the real
+(R^daggercirc Scirc T_{10}) of (Mat{10}). To express it as a
+composition of two endomorphisms, let (E:C^6toC^{10}) be the real
coordinate inclusion into the first six positions and define
-\Phi_1(A)=E\,S(A^\T)\,E^*,\qquad
-\Phi_2(B)=R^\dagger(E^* B E).
+Phi_1(A)=E,S(A^T),E^*,qquad
+Phi_2(B)=R^dagger(E^* B E).
-The stability lemma gives \(\Phi_1,\Phi_2\) PPT. Since \(E^*E=I_6\),
-\(\Phi_2\circ\Phi_1=R^\dagger\circ S\circ T_{10}\) with Choi matrix
-\(Z=J(\Phi_2\circ\Phi_1)\ge0\).
+The stability lemma gives (Phi_1,Phi_2) PPT. Since (E^*E=I_6),
+(Phi_2circPhi_1=R^daggercirc Scirc T_{10}) with Choi matrix
+(Z=J(Phi_2circPhi_1)ge0).
Singular outputs and kernel products.
-The Choi pairing gives, for arbitrary \(a,b\) of compatible sizes,
-\(\tr[(a ⊗ b)J(F)]=\tr[bF(a^\T)]\). For Hermitian
-\(a,b\in\Mat{10}\),
+The Choi pairing gives, for arbitrary (a,b) of compatible sizes,
+(tr[(a ⊗ b)J(F)]=tr[bF(a^T)]). For Hermitian
+(a,binMat{10}),
-\tr[(a ⊗ b)Z]=\tr[bR^\dagger(S(a))]=\tr[R(b)S(a)].
+tr[(a ⊗ b)Z]=tr[bR^dagger(S(a))]=tr[R(b)S(a)].
-For a Hermitian \(B\) on \(\C^4\),
-\(V^*(B ⊗ B)V=∧^2B\). If \(B_x=L(P_x)\succeq0\) and
-\(W_x=\ran B_x\), then
-\(S(P_{\hat x})=∧^2 B_x\) with range
-\(∧^2 W_x\). Transposition conjugates a range, so
-\(\ran R(P_{\hat x})=K∧^2 W_xbar\).
+For a Hermitian (B) on (C^4),
+(V^*(B ⊗ B)V=∧^2B). If (B_x=L(P_x)succeq0) and
+(W_x=ran B_x), then
+(S(P_{hathat x}x)=∧^2 B_x) with range
+(∧^2 W_x). Transposition conjugates a range, so
+(ran R(P_{hathat x}x)=K∧^2 W_xbar).
-For exterior two-vectors \(w,u\), with Hermitian inner product
-conjugate-linear in the first argument, the definition of \(K\) gives
+For exterior two-vectors (w,u), with Hermitian inner product
+conjugate-linear in the first argument, the definition of (K) gives
-< K\overline w,u>
-=[w\wedge u]_{0123},
+< Koverlineoverline w,u>w,u>
+=[wwedge u]_{0123},
-the coefficient of \(e_0\wedge e_1\wedge e_2\wedge e_3\). If
-\(w,u\in∧^2 W\) and \(\dim W<4\), then this wedge vanishes; so
-\(S(P_{\hat x})\) and \(R(P_{\hat x})\) have orthogonal ranges for every
-\(x\in\mathcal X\) (including \(\dim W_x<2\), when both are zero).
+the coefficient of (e_0wedge e_1wedge e_2wedge e_3). If
+(w,uin∧^2 W) and (dim W<4), then this wedge vanishes; so
+(S(P_{hathat x}x)) and (R(P_{hathat x}x)) have orthogonal ranges for every
+(xinmathcal XcalX) (including (dim W_x<2), when both are zero).
Hence
-\[
-(\hat x ⊗ \hat x)^*Z(\hat x ⊗ \hat x)
-=\tr[R(P_{\hat x})S(P_{\hat x})]=0
-\]
-for \(x\in\mathcal X\). Positive \(Z\) with \(v^*Zv=0\) implies
-\(Zv=0\), so
+[
+(hathat xx ⊗ hathat x)^*Z(x)^*Z(hathat xx ⊗ hathat x)x)
+=tr[R(P_{hathat x}x)S(P_{hathat x}x)]=0
+]
+for (xinmathcal XcalX). Positive (Z) with (v^*Zv=0) implies
+(Zv=0), so
-\hat x ⊗ \hat x\in\ker Z\qquad(x\in\mathcal X).
+hathat xx ⊗ hathat xxinker Zqquad(xinmathcal X).calX).
-For the second hypothesis' input \(y\), \(S(P_{\hat y})=∧^2L(P_y)\)
-is positive definite, as is \(R(P_{\hat y})=KS(P_{\hat y})^\T K^*\), so
-\(\tr[R(P_{\hat y})S(P_{\hat y})]>0\), giving \(Z\ne0\) by
+For the second hypothesis' input (y), (S(P_{hathat y}y)=∧^2L(P_y))
+is positive definite, as is (R(P_{hathat y}y)=KS(P_{hathat y}y)^T K^*), so
+(tr[R(P_{hathat y}y)S(P_{hathat y}y)]>0), giving (Zne0) by
(eq:choi-pairing).
Excluding product vectors from the range.
-Suppose \(u ⊗ v\in\ran Z\) is nonzero. Since \(Z\) is Hermitian,
-\(\ran Z\perp\ker Z\), so (eq:kernel-vectors) gives
-\(p_u(x)p_v(x)=0\) at every \(x\in\mathcal X\), where
-\(p_u(x)=u^*\hat x\), \(p_v(x)=v^*\hat x\). Each is a homogeneous
-quadratic in \(x\); nonzero \(u,v\) mean nonzero \(p_u,p_v\). Each
-vanishes on at most nine of \(\mathcal X\). Two zero sets cover at most
-\(9+9<20\) points, contradiction. So no nonzero product vector is in
-\(\ran Z\). By Lemma~lem:product-range \(Z\) is nonseparable;
-Lemma~lem:choi-separable gives \(Z\) not EB.
+Suppose (u ⊗ vinran Z) is nonzero. Since (Z) is Hermitian,
+(ran Zperpker Z), so (eq:kernel-vectors) gives
+(p_u(x)p_v(x)=0) at every (xinmathcal XcalX), where
+(p_u(x)=u^*hathat xx), (p_v(x)=v^*hathat xx). Each is a homogeneous
+quadratic in (x); nonzero (u,v) mean nonzero (p_u,p_v). Each
+vanishes on at most nine of (mathcal XcalX). Two zero sets cover at most
+(9+9<20) points, contradiction. So no nonzero product vector is in
+(ran Z). By Lemma~lem:product-range (Z) is nonseparable;
+Lemma~lem:choi-separable gives (Z) not EB.
# An integer pencil and its exact certificate
-We now supply \(L:\Mat 4\to\Mat 4\) satisfying Proposition~%
+We now supply (L:Mat 4toMat 4) satisfying Proposition~%
prop:tensor-criterion. Everything is explicit and integer.
-The twenty singular directions \([x]\) of a \(6\times 4\) integer pencil
-\(M(x)=x_0M_0+x_1M_1+x_2M_2+x_3M_3\) are certified by exact modular
+The twenty singular directions ([x]) of a (6times 4) integer pencil
+(M(x)=x_0M_0+x_1M_1+x_2M_2+x_3M_3) are certified by exact modular
arithmetic at three primes. The ranks and the quadratic-evaluation bound
are obtained by a single algebraic computation (the algebra generated by
the maximal minors of the pencil) whose quotient is proved to be a
-degree-twenty field with Galois group \(S_{20}\).
+degree-twenty field with Galois group (S_{20}).
-## The pencil and \(L\)
+## The pencil and (L)
-Use indices \(0,1,2,3\) and define real \(6\times 4\) matrices
+Use indices (0,1,2,3) and define real (6times 4) matrices
M_0&=
-6&0&0&0\\0&6&0&0\\0&0&6&0\\0&0&0&6\\0&0&0&0\\0&0&0&0
+6&0&0&00&6&0&00&0&6&00&0&0&60&0&0&00&0&0&0
,&
M_1&=
--12&0&0&0\\0&-6&0&0\\0&0&6&0\\0&0&0&12\\-6&0&6&6\\-6&-6&6&0
-,\\[2mm]
+-12&0&0&00&-6&0&00&0&6&00&0&0&12-6&0&6&6-6&-6&6&0
+,[2mm]
M_2&=
-0&6&-2&0\\6&6&0&2\\-2&0&10&0\\0&2&0&24\\0&0&0&6\\0&-6&6&-6
+0&6&-2&06&6&0&2-2&0&10&00&2&0&240&0&0&60&-6&6&-6
,&
M_3&=
-0&0&-4&-3\\0&-2&-3&-4\\-4&-3&11&6\\-3&-4&6&25\\6&6&0&0\\0&-6&6&6
+0&0&-4&-30&-2&-3&-4-4&-3&11&6-3&-4&6&256&6&0&00&-6&6&6
.
-For \(x\in\C^4\), \(M(x)=\sum_i x_iM_i\); then \(L\) is defined on
+For (xinC^4), (M(x)=sum_i x_iM_i); then (L) is defined on
matrix units by
-L(E_{ij})=M_i^{\T}M_j\qquad(0\le i,j<4).
+L(E_{ij})=M_i^{T}M_jqquad(0le i,j<4).
[The pencil is PPT]
-\(L\) is PPT. For every \(x\in\C^4\),
+(L) is PPT. For every (xinC^4),
-L(P_x)=M(x)^{\T}M(x)bar=M(x)^*M(x),\qquad
-\rank L(P_x)=\rank M(x).
+L(P_x)=M(x)^{T}M(x)bar=M(x)^*M(x),qquad
+rank L(P_x)=rank M(x).
-Moreover \(L(P_{e_0})=36I_4\).
+Moreover (L(P_{e_0})=36I_4).
Proof.
Direct integer multiplication gives
-\(M_i^{\T}M_j=M_j^{\T}M_i\) for all \(i,j\)
-(block symmetry). For each row \(r\), let \(Q_r\) be the \(4\times4\)
-matrix with \(i\)-th column equal to row \(r\) of \(M_i^{\T}\); then
-\(L(A)=\sum_{r=1}^6 Q_rAQ_r^*\), so \(L\) is CP. Block symmetry also
-makes every \(M_i^{\T}M_j\) symmetric, so \(T_4\circ L=L\), giving PPT.
+(M_i^{T}M_j=M_j^{T}M_i) for all (i,j)
+(block symmetry). For each row (r), let (Q_r) be the (4times4)
+matrix with (i)-th column equal to row (r) of (M_i^{T}); then
+(L(A)=sum_{r=1}^6 Q_rAQ_r^*), so (L) is CP. Block symmetry also
+makes every (M_i^{T}M_j) symmetric, so (T_4circ L=L), giving PPT.
-For complex \(x\):
-\(L(P_x)=\sum_{i,j}x_ix_jbarM_i^{\T}M_j
-=M(x)^{\T}M(x)bar\), and interchanging \(i,j\) with block
-symmetry gives \(M(x)^*M(x)\). Since
-\(v^*M(x)^*M(x)v=\norm{M(x)v}^2\), \(\ker L(P_x)=\ker M(x)\), giving
-the rank identity. Finally \(M_0^{\T}M_0=36I_4\).
+For complex (x):
+(L(P_x)=sum_{i,j}x_ix_jbarM_i^{T}M_j
+=M(x)^{T}M(x)bar), and interchanging (i,j) with block
+symmetry gives (M(x)^*M(x)). Since
+(v^*M(x)^*M(x)v=norm{M(x)v}^2), (ker L(P_x)=ker M(x)), giving
+the rank identity. Finally (M_0^{T}M_0=36I_4).
[Twenty directions]
-The projective directions \([x]\in\mathbb P^3(\C)\) with
-\(\rank M(x)<4\) consist of exactly twenty distinct points, each with
-unique representative \(x=(1,t_1,t_2,t_3)\). Every nonzero homogeneous
-quadratic in \(\C[x_0,x_1,x_2,x_3]\) vanishes on at most nine of them.
+The projective directions ([x]inmathbb P^3(bbP^3(C)) with
+(rank M(x)<4) consist of exactly twenty distinct points, each with
+unique representative (x=(1,t_1,t_2,t_3)). Every nonzero homogeneous
+quadratic in (C[x_0,x_1,x_2,x_3]) vanishes on at most nine of them.
-The equation \(M(x)y=0\) with \(x,y\ne0\) relates to bilinear kernel
-geometry of \(4\times4\) PPT states whose matrix and partial transpose
-both have rank six~\cite[Sect.~2.2,7]{HansenEtAl2012}. The twenty
-product rays \(x ⊗ y\) there are different from the twenty
-\(x ⊗ x\) here; the specific pencil matters.
+The equation (M(x)y=0) with (x,yne0) relates to bilinear kernel
+geometry of (4times4) PPT states whose matrix and partial transpose
+both have rank six~cite[Sect.~2.2,7]{HansenEtAl2012}. The twenty
+product rays (x ⊗ y) there are different from the twenty
+(x ⊗ x) here; the specific pencil matters.
Proof. [Proof of Proposition~prop:twenty-directions]
-The maximal \(4\times4\) minors of \(M(x)\) vanish exactly on
-\(\rank M(x)<4\). We chain three pieces:
+The maximal (4times4) minors of (M(x)) vanish exactly on
+(rank M(x)<4). We chain three pieces:
-\item[A] The maximal-minor ideal defines a nonzero affine quotient
-spanned by the twenty monomials of degree \(\le3\), so the direction set
-is contained in the affine chart \(x_0=1\). (Lemma~%
+item[A] The maximal-minor ideal defines a nonzero affine quotient
+spanned by the twenty monomials of degree (le3), so the direction set
+is contained in the affine chart (x_0=1). (Lemma~%
lem:six-bilinear supplies a nonzero simultaneous vanishing point of
-the six rows of \(M(x)y\) for some \(x\ne0,y\ne0\).)
-\item[B] A \(20\times20\) multiplication operator \(N\) on that span
-encodes the quotient. Its characteristic polynomial \(f(z)=\det(zI_{20}-N)\)
-is monic with coefficients in \(\Z_{(p)}\) for each of the primes
-\(p=41,131,139\). Exact modular computation gives: \(\det H\equiv40\) mod \(41\),
-\(23\) mod \(131\), \(9\) mod \(139\); \(f\) irreducible mod 41 (so
-over \(\Q\)); factor degrees \(1,2,17\) mod 131; \(1,19\) mod 139. The
+the six rows of (M(x)y) for some (xne0,yne0).)
+item[B] A (20times20) multiplication operator (N) on that span
+encodes the quotient. Its characteristic polynomial (f(z)=det(zI_{20}-N))
+is monic with coefficients in (Z_{(p)}) for each of the primes
+(p=41,131,139). Exact modular computation gives: (det Hequiv40) mod (41),
+(23) mod (131), (9) mod (139); (f) irreducible mod 41 (so
+over (Q)); factor degrees (1,2,17) mod 131; (1,19) mod 139. The
local Frobenius lemma then gives Galois elements of cycle types
-\((1,2,17)\) and \((1,19)\). Irreducibility makes the Galois group
-transitive; a transposition (seventeenth power of \((1,2,17)\)) plus
-transitivity on ordered pairs gives \(S_{20}\).
-\item[C] With \(S_{20}\) acting, the evaluation matrix
-\(\mathcal E_{j,\alpha}=(t^{(j)})^\alpha\) for \(|\alpha|\le2\) has
-rank ten over \(\C\). Any nonzero quadratic's dehomogenization
-\(q(1,t)\) is nonzero exactly when \(q\) is nonzero; if it vanished on
+((1,2,17)) and ((1,19)). Irreducibility makes the Galois group
+transitive; a transposition (seventeenth power of ((1,2,17))) plus
+transitivity on ordered pairs gives (S_{20}).
+item[C] With (S_{20}) acting, the evaluation matrix
+(mathcal E_calE_{j,alpha}=(t^{(j)})^alpha) for (|alpha|le2) has
+rank ten over (C). Any nonzero quadratic's dehomogenization
+(q(1,t)) is nonzero exactly when (q) is nonzero; if it vanished on
ten of the twenty triples, the rank-ten evaluation matrix would force
-\(q=0\). So no nonzero quadratic vanishes on ten of the twenty.
+(q=0). So no nonzero quadratic vanishes on ten of the twenty.
@@ -1417,184 +1270,184 @@ The three algebraic facts plugged into the proof above are stated here.
[Six bilinear equations]
-Any six complex bilinear forms on \(\C^4\times\C^4\) have a common zero
-\((x,y)\) with \(x\ne0\), \(y\ne0\).
+Any six complex bilinear forms on (C^4timesC^4) have a common zero
+((x,y)) with (xne0), (yne0).
Proof.
-Let \(u_{ij}\) denote \(X_iY_j\) in the polynomial ring
-\(\mathcal R=\C[u_{ij}]\cong\C[X_iY_j]\subset\C[X,Y]\) (a Noetherian
-domain). Its closed points are exactly rank \(\le1\) matrices with the
-relations \(u_{ij}u_{k\ell}=u_{i\ell}u_{kj}\).
-For \(P_{a,b}=(u_{ij}:i\ge a or j\ge b)\), \(P_{a,b}\) is prime
+Let (u_{ij}) denote (X_iY_j) in the polynomial ring
+(mathcal R=calR=C[u_{ij}]congC[X_iY_j]subsetC[X,Y]) (a Noetherian
+domain). Its closed points are exactly rank (le1) matrices with the
+relations (u_{ij}u_{kell}=u_{iell}u_{kj}).
+For (P_{a,b}=(u_{ij}:ige a or jge b)), (P_{a,b}) is prime
and the chain
-\(0=P_{4,4}\subset P_{3,4}\subset P_{2,4}\subset P_{1,4}
-\subset P_{1,3}\subset P_{1,2}\subset P_{1,1}\subset\mathfrak m\)
-is strict, so \(ht\mathfrak m\ge7\).
-Let \(\ell_1,...,\ell_6\) be the six forms and
-\(\mathfrak a=(\ell_1,...,\ell_6)\). If the only common zero on
+(0=P_{4,4}subset P_{3,4}subset P_{2,4}subset P_{1,4}
+subset P_{1,3}subset P_{1,2}subset P_{1,1}subsetmathfrak mfrakm)
+is strict, so (htmathfrak mfrakmge7).
+Let (ell_1,...,ell_6) be the six forms and
+(mathfrak a=(fraka=(ell_1,...,ell_6)). If the only common zero on
rank-one matrices were zero, the Nullstellensatz would give
-\(√(\mathfrak a)=\mathfrak m\), making \(\mathfrak m\) minimal over
+(√(mathfrak a)=fraka)=mathfrak mfrakm), making (mathfrak mfrakm) minimal over
a six-generator ideal. Krull's height theorem would give
-\(ht\mathfrak m\le6\), contradiction. So some nonzero
-rank-one \(xy^{\T}\) annihilates all six forms.
+(htmathfrak mfrakmle6), contradiction. So some nonzero
+rank-one (xy^{T}) annihilates all six forms.
[Monic local reduction]
-Let \(p\) be prime and \((L,R_p)\) be one of \((\Q,\Z_{(p)})\) or
-\((\Q_p,\Z_p)\). If a monic \(P\in R_p[z]\) factors as \(P=gh\) with
-monic \(g,h\in L[z]\), then \(g,h\in R_p[z]\), their reductions mod \(p\)
-retain degree. Consequently, if \(P\) has positive degree and irreducible
-reduction in \(\F_p[z]\), then \(P\) is irreducible over \(L\).
+Let (p) be prime and ((L,R_p)) be one of ((Q,Z_{(p)})) or
+((Q_p,Z_p)). If a monic (Pin R_p[z]) factors as (P=gh) with
+monic (g,hin L[z]), then (g,hin R_p[z]), their reductions mod (p)
+retain degree. Consequently, if (P) has positive degree and irreducible
+reduction in (F_p[z]), then (P) is irreducible over (L).
Proof.
-Normalize \(v_p(p)=1\). For \(a=\sum_i a_iz^i\ne0\), set
-\(\nu(a)=\min_{a_i\ne0}v_p(a_i)\). Scaling by \(p^{-\nu}\) makes
+Normalize (v_p(p)=1). For (a=sum_i a_iz^ine0), set
+(nu(a)=min_{a_ine0}v_p(a_i)). Scaling by (p^{-nu}) makes
coefficients integral with least valuation zero; reductions are nonzero
-and multiply in the domain \(\F_p[z]\), so
-\(\nu(ab)=\nu(a)+\nu(b)\). Monic \(g,h\) have \(\nu\le0\); monic
-\(P\in R_p[z]\) has \(\nu(P)=0\). Additivity forces
-\(\nu(g)=\nu(h)=0\), so both have integral coefficients. Their leading
+and multiply in the domain (F_p[z]), so
+(nu(ab)=nu(a)+nu(b)). Monic (g,h) have (nule0); monic
+(Pin R_p[z]) has (nu(P)=0). Additivity forces
+(nu(g)=nu(h)=0), so both have integral coefficients. Their leading
coefficients remain one, so degrees are preserved; a monic constant is
-\(1\) and has degree zero. A nontrivial factorization over \(L\) would
-factor \(P\) over \(\F_p[z]\) nontrivially, contradiction.
+(1) and has degree zero. A nontrivial factorization over (L) would
+factor (P) over (F_p[z]) nontrivially, contradiction.
[Locally integral Frobenius]
-Let \(F\in\Q[z]\) be monic, separable, of positive degree, with
-coefficients in \(\Z_{(p)}\). If its reduction modulo \(p\) is
-squarefree with irreducible factor degrees \(s_1,...,s_h\), then the
+Let (FinQ[z]) be monic, separable, of positive degree, with
+coefficients in (Z_{(p)}). If its reduction modulo (p) is
+squarefree with irreducible factor degrees (s_1,...,s_h), then the
Galois group of its rational splitting field contains an element of
-cycle type \((s_1,...,s_h)\).
+cycle type ((s_1,...,s_h)).
Proof.
-Embed coefficients in \(\Z_p\), put \(s=lcm(s_1,...,s_h)\)
-and choose a monic irreducible \(\overline g\in\F_p[X]\) of degree \(s\).
-Lift to monic \(g\in\Z_p[X]\); Lemma~lem:monic-local-reduction
-gives \(g\) irreducible over \(\Q_p\). Let
-\(\mathcal O=\Z_p[\alpha]\cong\Z_p[X]/(g)\) where \(\alpha\) is a root;
-\(\mathcal O\) is a free rank-\(s\) \(\Z_p\)-module with
-\(\mathcal O/p\mathcal O\simeq\F_{p^s}\), and every element with nonzero
+Embed coefficients in (Z_p), put (s=lcm(s_1,...,s_h))
+and choose a monic irreducible (overlineoverline gginF_p[X]) of degree (s).
+Lift to monic (ginZ_p[X]); Lemma~lem:monic-local-reduction
+gives (g) irreducible over (Q_p). Let
+(mathcal O=calO=Z_p[alpha]congZ_p[X]/(g)) where (alpha) is a root;
+(mathcal OcalO) is a free rank-(s) (Z_p)-module with
+(mathcal O/pcalO/pmathcal OcalOsimeqF_{p^s}), and every element with nonzero
residue is a unit.
-Hensel-style lifting: every simple residue root of a \(\Z_p[z]\)
-polynomial lifts uniquely to a root in \(\mathcal O\).
-Since \(\overline F\) is squarefree with roots in \(\F_{p^s}\), we get
-\(\deg F\) distinct roots in \(\mathcal O\). Finite fields are perfect,
-so \(\overline\alpha^p\) is a simple root of \(\overline g\) and lifts to
-some \(\beta\in\mathcal O\). Sending \(\alpha\mapsto\beta\) gives a
-\(\Q_p\)-embedding \(E\to E\) with \(E=\Q_p(\alpha)\); call it \(\sigma\).
-Finiteness makes it an automorphism, and \(\sigma(\mathcal O)=\mathcal O\)
-because the residue action is a permutation. If \(\rho_r\) is the lifted
-root of \(F\) with residue \(r\), uniqueness gives
-\(\sigma(\rho_r)=\rho_{r^p}\). So \(\sigma\)'s permutation of lifted
-roots has cycle lengths \(s_1,...,s_h\), exactly the Frobenius cycle
+Hensel-style lifting: every simple residue root of a (Z_p[z])
+polynomial lifts uniquely to a root in (mathcal OcalO).
+Since (overlineoverline FF) is squarefree with roots in (F_{p^s}), we get
+(deg F) distinct roots in (mathcal OcalO). Finite fields are perfect,
+so (overlinealpha^p) is a simple root of (overlineoverline gg) and lifts to
+some (betainmathcal OcalO). Sending (alphamapstobeta) gives a
+(Q_p)-embedding (Eto E) with (E=Q_p(alpha)); call it (sigma).
+Finiteness makes it an automorphism, and (sigma(mathcal O)=calO)=mathcal OcalO)
+because the residue action is a permutation. If (rho_r) is the lifted
+root of (F) with residue (r), uniqueness gives
+(sigma(rho_r)=rho_{r^p}). So (sigma)'s permutation of lifted
+roots has cycle lengths (s_1,...,s_h), exactly the Frobenius cycle
lengths on their residues. The rational splitting subfield is preserved,
giving the required rational Galois element.
-## The multiplication operator and \(f(z)\)
-
-Let \(\mathcal A=\Q[t_1,t_2,t_3]/I\), \(I\) generated by the fifteen
-maximal minors \(p_\rho(t)=\Delta_\rho(1,t)\). Monomial order:
-first by total degree, then lexicographically. Let \(v\) be the twenty
-monomials of degree \(\le3\), \(w\) the fifteen of degree four, with
-coefficients \(D\in\Z^{15\times20}\), \(H\in\Z^{15\times15}\):
-\((p_\rho(t))_\rho=Dv(t)+Hw(t)\).
-
-Exact coefficients yield \(\det H\equiv40\not\equiv0\pmod{41}\), so
-\(H\) is invertible. Define \(C=-H^{-1}D\); in \(\mathcal A\), \(w=Cv\)
-expresses degree-four monomials as combinations of degree \(\le3\)
-monomials. Multiplication by \(t_1\) is the formal \(20\times20\) matrix
-\(N\) on the spanned \(v\):
-\[
-Ne_\alpha=e_{\alpha+(1,0,0)}\ (for |\alpha|\le2),\quad
-Ne_\alpha=\textstyle\sum_{|\beta|\le3}C_{\alpha+(1,0,0),\beta}e_\beta
-\ (for |\alpha|=3).
-\]
-Put \(f(z)=\det(zI_{20}-N)\). Then \(f\) has coefficients in
-\(\Z_{(p)}\) for \(p=41,131,139\); Newton's identities compute the
+## The multiplication operator and (f(z))
+
+Let (mathcal A=calA=Q[t_1,t_2,t_3]/I), (I) generated by the fifteen
+maximal minors (p_rho(t)=Delta_rho(1,t)). Monomial order:
+first by total degree, then lexicographically. Let (v) be the twenty
+monomials of degree (le3), (w) the fifteen of degree four, with
+coefficients (DinZ^{15times20}), (HinZ^{15times15}):
+((p_rho(t))_rho=Dv(t)+Hw(t)).
+
+Exact coefficients yield (det Hequiv40notequiv0pmod{41}), so
+(H) is invertible. Define (C=-H^{-1}D); in (mathcal AcalA), (w=Cv)
+expresses degree-four monomials as combinations of degree (le3)
+monomials. Multiplication by (t_1) is the formal (20times20) matrix
+(N) on the spanned (v):
+[
+Ne_alpha=e_{alpha+(1,0,0)} (for |alpha|le2),quad
+Ne_alpha=textstylesum_{|beta|le3}C_{alpha+(1,0,0),beta}e_beta
+ (for |alpha|=3).
+]
+Put (f(z)=det(zI_{20}-N)). Then (f) has coefficients in
+(Z_{(p)}) for (p=41,131,139); Newton's identities compute the
twenty-one characteristic coefficients and the modular gcd degrees
-\(d_p(k)=\deg\gcd(\overline f,z^{p^k}-z)\) without leaving those rings.
+(d_p(k)=deggcd(overlineoverline f,z^f,z^{p^k}-z)) without leaving those rings.
Exact modular data (see Section~app:arithmetic for recorded values):
{ccll}
-\toprule
-\(p\) & \(\det H\bmod p\) & computed gcd degrees & deduced factor degrees\\
-\midrule
-41 & 40 & \(0\) for \(1\le k<20\), \(20\) at \(k=20\) & \(20\)\\
-131 & 23 & \(1,3,18\) at \(k=1,2,17\) & \(1,2,17\)\\
-139 & 9 & \(1,20\) at \(k=1,19\) & \(1,19\)\\
-\bottomrule
+toprule
+(p) & (det Hbmod p) & computed gcd degrees & deduced factor degrees
+midrule
+41 & 40 & (0) for (1le k<20), (20) at (k=20) & (20)
+131 & 23 & (1,3,18) at (k=1,2,17) & (1,2,17)
+139 & 9 & (1,20) at (k=1,19) & (1,19)
+bottomrule
-Since \(p>20\), divisions in Newton's identities are valid.
+Since (p>20), divisions in Newton's identities are valid.
Irreducibility mod 41 + Lemma~lem:monic-local-reduction gives
-\(f\) irreducible over \(\Q\). Applying Lemma~%
-lem:local-frobenius at \(p=131,139\) gives Galois elements of
-cycle types \((1,2,17)\) and \((1,19)\). Irreducibility gives
-transitivity; a transposition from the 17th power of \((1,2,17)\) plus
+(f) irreducible over (Q). Applying Lemma~%
+lem:local-frobenius at (p=131,139) gives Galois elements of
+cycle types ((1,2,17)) and ((1,19)). Irreducibility gives
+transitivity; a transposition from the 17th power of ((1,2,17)) plus
2-transitivity (from the 19-cycle with stabilizer transitive on the other
-19) gives every transposition, hence \(S_{20}\).
+19) gives every transposition, hence (S_{20}).
## The affine quotient is a field
-The \(w=Cv\) relations force \(\mathcal A\) to be spanned by the twenty
-low monomials. Independence follows from \(f\) irreducible: if
-\(\pi:\Q^{20}\to\mathcal A\) with \(e_\alpha\mapsto t^\alpha\), then
-\(\pi N=\mu_1\pi\) where \(\mu_1\) is multiplication by \(t_1\).
-Irreducibility makes \(N\) have no proper invariant subspace, so
-\(\ker\pi=0\). The twenty low monomials are a \(\Q\)-basis of
-\(\mathcal A\), and \(N\) is multiplication by \(t_1\). Since the minimal
-polynomial of \(N\) equals \(f\), the subalgebra generated by \(t_1\) is
-already all of \(\mathcal A\):
+The (w=Cv) relations force (mathcal AcalA) to be spanned by the twenty
+low monomials. Independence follows from (f) irreducible: if
+(pi:Q^{20}tomathcal AcalA) with (e_alphamapsto t^alpha), then
+(pi N=mu_1pi) where (mu_1) is multiplication by (t_1).
+Irreducibility makes (N) have no proper invariant subspace, so
+(kerpi=0). The twenty low monomials are a (Q)-basis of
+(mathcal AcalA), and (N) is multiplication by (t_1). Since the minimal
+polynomial of (N) equals (f), the subalgebra generated by (t_1) is
+already all of (mathcal AcalA):
-\mathcal A\simeq\Q[z]/(f),\qquad z\mapsto t_1.
+mathcal AcalAsimeqQ[z]/(f),qquad zmapsto t_1.
-In particular \(\mathcal A\) is a field, and there are
-\(h_2,h_3\in\Q[z]\) of degree below twenty with \(t_2=h_2(t_1)\),
-\(t_3=h_3(t_1)\).
+In particular (mathcal AcalA) is a field, and there are
+(h_2,h_3inQ[z]) of degree below twenty with (t_2=h_2(t_1)),
+(t_3=h_3(t_1)).
Separability (char 0) gives twenty distinct complex roots
-\(\lambda_1,...,\lambda_{20}\). Extending scalars:
+(lambda_1,...,lambda_{20}). Extending scalars:
-\C[t_1,t_2,t_3]/I\C[t_1,t_2,t_3]\simeq\C[z]/(f)
-\simeq\prod_{j=1}^{20}\C,
+C[t_1,t_2,t_3]/IC[t_1,t_2,t_3]simeqC[z]/(f)
+simeqprod_{j=1}^{20}C,
by CRT. The common zeros are exactly
-\(t^{(j)}=(\lambda_j,h_2(\lambda_j),h_3(\lambda_j))\), all distinct and
+(t^{(j)}=(lambda_j,h_2(lambda_j),h_3(lambda_j))), all distinct and
reduced.
## Quadratic evaluations via Galois
-Let \(\mathbb K\subset\C\) be a splitting field of \(f\),
-\(G=Gal(\mathbb K/\Q)\). The evaluation matrix
-\(\mathcal E_{j,\alpha}=(t^{(j)})^\alpha\) for
-\(1\le j\le20\), \(|\alpha|\le2\) has column rank ten over
-\(\mathbb K\). Under the \(S_{20}\) action, \(\sigma\in G\) permutes
-triples and preserves all \(h_2,h_3\) rational structure; so
-\(\sigma\mathcal E_{j,\alpha}=\mathcal E_{\sigma(j),\alpha}\).
-\(S_{20}\) transitive on ten-element subsets transports the nonzero
-determinant of one \(10\times10\) submatrix to every \(10\times10\)
-submatrix. Hence \(\mathcal E\) has column rank ten over \(\mathbb K\)
-and over \(\C\).
-
-A nonzero homogeneous quadratic \(q\) dehomogenizes to a nonzero
-\(q(1,t)\); if it vanished at ten triples, the invertible evaluation
-matrix would force \(q=0\). So \(q\) vanishes on at most nine.
+Let (mathbb KbbKsubsetC) be a splitting field of (f),
+(G=Gal(mathbb K/bbK/Q)). The evaluation matrix
+(mathcal E_calE_{j,alpha}=(t^{(j)})^alpha) for
+(1le jle20), (|alpha|le2) has column rank ten over
+(mathbb KbbK). Under the (S_{20}) action, (sigmain G) permutes
+triples and preserves all (h_2,h_3) rational structure; so
+(sigmamathcal E_calE_{j,alpha}=mathcal E_calE_{sigma(j),alpha}).
+(S_{20}) transitive on ten-element subsets transports the nonzero
+determinant of one (10times10) submatrix to every (10times10)
+submatrix. Hence (mathcal EcalE) has column rank ten over (mathbb KbbK)
+and over (C).
+
+A nonzero homogeneous quadratic (q) dehomogenizes to a nonzero
+(q(1,t)); if it vanished at ten triples, the invertible evaluation
+matrix would force (q=0). So (q) vanishes on at most nine.
## Putting it together
Combine Lemmas~lem:pencil-ppt and~%
prop:twenty-directions: Proposition~prop:tensor-criterion
-applies with \(m=10\), giving the \(20\)-direction pair
-\(\Phi_1,\Phi_2\). The composed Choi matrix
-\(Z=J(\Phi_2\circ\Phi_1)\ne0\) has no nonzero product in its range, so
-\(\Phi_2\circ\Phi_1\) is not EB. The state \(\rho=Z/\tr Z\) is
+applies with (m=10), giving the (20)-direction pair
+(Phi_1,Phi_2). The composed Choi matrix
+(Z=J(Phi_2circPhi_1)ne0) has no nonzero product in its range, so
+(Phi_2circPhi_1) is not EB. The state (rho=Z/tr Z) is
entangled. This proves Theorems~thm:main-pair
and~thm:main-state. The exact-arithmetic realisation is in
Section~app:arithmetic.
@@ -1603,128 +1456,128 @@ Section~app:arithmetic.
We now connect the dimension-ten pair to the operational theorem, and
-produce the trace-preserving channel on \(\Mat{21}\).
+produce the trace-preserving channel on (Mat{21}).
Proof. [Proof of Theorem~thm:main-state]
-Let \(\Phi_1,\Phi_2\) be from (eq:phi-pair),
-\(Z=J(\Phi_2\circ\Phi_1)\). Theorem~thm:main-pair gives \(Z\ne0\),
-\(\ran Z\) product-free, \(Z\ge0\). So \(\lambda=\tr Z>0\) and
-\(\rho=Z/\lambda\) has the same range, hence is entangled
-(Lemma~lem:product-range). Both \(\Phi_1,\Phi_2\) are PPT.
-Lemma~lem:choi-transfer with \(a=b=c=10\) gives
-\[
-\rho=(\Phi_1^\sharp ⊗ \Phi_2)
-(\frac{Ω_{10}Ω_{10}^*}{\lambda})
-\in\mathscr C.
-\]
+Let (Phi_1,Phi_2) be from (eq:phi-pair),
+(Z=J(Phi_2circPhi_1)). Theorem~thm:main-pair gives (Zne0),
+(ran Z) product-free, (Zge0). So (lambda=tr Z>0) and
+(rho=Z/lambda) has the same range, hence is entangled
+(Lemma~lem:product-range). Both (Phi_1,Phi_2) are PPT.
+Lemma~lem:choi-transfer with (a=b=c=10) gives
+[
+rho=(Phi_1^sharp ⊗ Phi_2)
+(frac{Ω_{10}Ω_{10}^*}{lambda})
+inmathscr C.scrC.
+]
Theorem~thm:zero-key and Proposition~%
prop:operational-extensions give (eq:main-gap) and
-\(K_D(\rho)=0\). All matrices defining \(\rho\) are explicit by
+(K_D(rho)=0). All matrices defining (rho) are explicit by
(eq:explicit-matrices) and (eq:phi-pair).
-\subsection{Trace-preserving PPT channel on \(\Mat{21}\)}
+subsection{Trace-preserving PPT channel on (Mat{21})}
-We want \(Θ:\Mat{21}\to\Mat{21}\) a trace-preserving PPT channel
-whose square is not EB. The pair \((\Phi_1,\Phi_2)\) need not be trace
+We want (Θ:Mat{21}toMat{21}) a trace-preserving PPT channel
+whose square is not EB. The pair ((Phi_1,Phi_2)) need not be trace
preserving. The next proposition switches two coordinating blocks and
-uses an absorbing flag, following Christandl--M\"uller-Hermes--Wolf~%
-\cite[Sec.~IV.A]{CMHW2019} and Filippov~%
-\cite[Prop.~3,~Eq.~(15)]{Filippov2021}.
+uses an absorbing flag, following Christandl--M"uller-Hermes--Wolf~%
+cite[Sec.~IV.A]{CMHW2019} and Filippov~%
+cite[Prop.~3,~Eq.~(15)]{Filippov2021}.
[Channel completion]
-Let \(\Psi_1,\Psi_2:\Mat m\to\Mat m\) be PPT maps. There are explicit
-\(c_1,c_2>0\), a trace-preserving PPT channel
-\(Θ:\Mat{2m+1}\to\Mat{2m+1}\), and a real coordinate isometry
-\(W_0:\C^m\to\C^{2m+1}\) such that
+Let (Psi_1,Psi_2:Mat mtoMat m) be PPT maps. There are explicit
+(c_1,c_2>0), a trace-preserving PPT channel
+(Θ:Mat{2m+1}toMat{2m+1}), and a real coordinate isometry
+(W_0:C^mtoC^{2m+1}) such that
-(W_0^* ⊗ W_0^*)J(Θ\circΘ)(W_0 ⊗ W_0)
-=c_1c_2J(\Psi_2\circ\Psi_1).
+(W_0^* ⊗ W_0^*)J(ΘcircΘ)(W_0 ⊗ W_0)
+=c_1c_2J(Psi_2circPsi_1).
-If \(J(\Psi_2\circ\Psi_1)\) is nonseparable, \(Θ\circΘ\) is
+If (J(Psi_2circPsi_1)) is nonseparable, (ΘcircΘ) is
not EB.
Proof.
Set
-\(A_i=\Psi_i^\dagger(I_m)\), \(c_i=1/(1+\tr A_i)\),
-\(F_i=I_m-c_iA_i\). Adjoints of CP maps are CP, so \(A_i\succeq0\).
-Since \(A_i\preceq(\tr A_i)I_m\), we get
-\(F_i\succeq(1-c_i\tr A_i)I_m=c_iI_m>0\). The adjoint identity gives
-\(\tr\Psi_i(B)=\tr(A_iB)\), whose complement wrt \(c_i\) is
-\(\tr(F_iB)\).
-
-Decompose \(\C^{2m+1}=\C^m_0 ⊕ \C^m_1 ⊕ \C e_\star\); \(W_0,W_1\)
-are the two coordinate inclusions. For \(X\in\Mat{2m+1}\) write
-\(X_{jj}=W_j^*XW_j\), \(x_\star=e_\star^*Xe_\star\),
-\(P_\star=e_\star e_\star^*\). Set
-
-Θ(X)&=c_2W_0\Psi_2(X_{11})W_0^*
-+c_1W_1\Psi_1(X_{00})W_1^*\notag\\
-&+(\tr(F_1X_{00})+\tr(F_2X_{11})+x_\star)P_\star.
+(A_i=Psi_i^dagger(I_m)), (c_i=1/(1+tr A_i)),
+(F_i=I_m-c_iA_i). Adjoints of CP maps are CP, so (A_isucceq0).
+Since (A_ipreceq(tr A_i)I_m), we get
+(F_isucceq(1-c_itr A_i)I_m=c_iI_m>0). The adjoint identity gives
+(trPsi_i(B)=tr(A_iB)), whose complement wrt (c_i) is
+(tr(F_iB)).
+
+Decompose (C^{2m+1}=C^m_0 ⊕ C^m_1 ⊕ C e_star); (W_0,W_1)
+are the two coordinate inclusions. For (XinMat{2m+1}) write
+(X_{jj}=W_j^*XW_j), (x_star=e_star^*Xe_star),
+(P_star=e_star e_star^*). Set
+
+Θ(X)&=c_2W_0Psi_2(X_{11})W_0^*
++c_1W_1Psi_1(X_{00})W_1^*notag
+&+(tr(F_1X_{00})+tr(F_2X_{11})+x_star)P_star.
The moving branches are CP compositions; the flag branches use
-\(F_i=\sum_r\lambda_{ir}v_{ir}v_{ir}^*\) and
-\(K_{ir}=\sqrt{\lambda_{ir}}e_\star v_{ir}^*W_{i-1}^*\), whose
-\(\sum_r K_{ir}XK_{ir}^*=\tr(F_iX_{i-1,i-1})P_\star\) is CP.
-Trace preservation: \(c_iA_i+F_i=I_m\) gives
-\(\trΘ(X)=\tr X_{00}+\tr X_{11}+x_\star=\tr X\).
-
-PPT: because \(W_0,W_1,e_\star\) are real, output transposition sends
-\(\Psi_i\) to \(T_m\circ\Psi_i\) in the moving branches and fixes the
-flag. The effects \(F_i\) may be complex; coordinate transpose does not
-conjugate the scalar multiplying \(P_\star\). Each transposed moving
+(F_i=sum_rlambda_{ir}v_{ir}v_{ir}^*) and
+(K_{ir}=sqrt{lambda_{ir}}e_star v_{ir}^*W_{i-1}^*), whose
+(sum_r K_{ir}XK_{ir}^*=tr(F_iX_{i-1,i-1})P_star) is CP.
+Trace preservation: (c_iA_i+F_i=I_m) gives
+(trΘ(X)=tr X_{00}+tr X_{11}+x_star=tr X).
+
+PPT: because (W_0,W_1,e_star) are real, output transposition sends
+(Psi_i) to (T_mcircPsi_i) in the moving branches and fixes the
+flag. The effects (F_i) may be complex; coordinate transpose does not
+conjugate the scalar multiplying (P_star). Each transposed moving
branch is CP by hypothesis, and the flag is CP; so
-\(T_{2m+1}\circΘ\) is CP.
+(T_{2m+1}circΘ) is CP.
-Flag absorbing: \(Θ(P_\star)=P_\star\). Applying (eq:channel)
-twice to \(W_0BW_0^*\), for \(B\in\Mat m\),
-\[
+Flag absorbing: (Θ(P_star)=P_star). Applying (eq:channel)
+twice to (W_0BW_0^*), for (BinMat m),
+[
Θ^2(W_0BW_0^*)
-=c_1c_2W_0\Psi_2(\Psi_1(B))W_0^*
-+(\tr(F_1B)+c_1\tr(F_2\Psi_1(B)))P_\star.
-\]
+=c_1c_2W_0Psi_2(Psi_1(B))W_0^*
++(tr(F_1B)+c_1tr(F_2Psi_1(B)))P_star.
+]
Compression:
-\(W_0^*Θ^2(W_0BW_0^*)W_0=c_1c_2\Psi_2(\Psi_1(B))\). Coordinate
+(W_0^*Θ^2(W_0BW_0^*)W_0=c_1c_2Psi_2(Psi_1(B))). Coordinate
matrix units give (eq:choi-corner). If the compressed positive
multiple were separable, then because compression takes
-\(A ⊗ B\mapsto(W_0^*AW_0) ⊗ (W_0^*BW_0)\) on products,
-\(J(\Psi_2\circ\Psi_1)\) would be separable. Contradiction if it is not.
+(A ⊗ Bmapsto(W_0^*AW_0) ⊗ (W_0^*BW_0)) on products,
+(J(Psi_2circPsi_1)) would be separable. Contradiction if it is not.
Proof. [Proof of Theorem~thm:main-channel]
-Apply Proposition~prop:channel-completion with \(m=10\),
-\(\Psi_i=\Phi_i\). The composed Choi matrix \(J(\Phi_2\circ\Phi_1)\) is
+Apply Proposition~prop:channel-completion with (m=10),
+(Psi_i=Phi_i). The composed Choi matrix (J(Phi_2circPhi_1)) is
nonseparable (Theorem~thm:main-pair). Formula (eq:channel)
-produces a trace-preserving PPT channel on \(\Mat{21}\); its square's
+produces a trace-preserving PPT channel on (Mat{21}); its square's
Choi matrix has the nonseparable corner (eq:choi-corner), hence is
not EB.
For the exact positive evaluation used in Theorem~thm:main-pair,
Theorem~thm:main-pair's argument shows
-\[
-S(P_{e_{00}})=R(P_{e_{00}})=36^2I_6,\quad
+[
+S(P_{e_{00}})=R(P_{e_{00}})=36^2I_6,quad
(e_{00} ⊗ e_{00})^*Z(e_{00} ⊗ e_{00})
-=6\cdot36^4=10{,}077{,}696>0.
-\]
+=6cdot36^4=10{,}077{,}696>0.
+]
# An alternative construction from projection overlaps
The dimension-ten construction is done. This section builds another
-entangled member of \(\mathscr C\), in much larger dimension, by a
+entangled member of (mathscr CscrC), in much larger dimension, by a
different mechanism. Real unit vectors and projective measurements
produce a large sum of correlations. Zero trace overlaps among the
projections restrict the supports of product vectors in any hypothetical
separable decomposition. These two constraints force a contradiction.
-Two preliminaries. First, for self-adjoint projections \(E,F\),
+Two preliminaries. First, for self-adjoint projections (E,F),
-\tr(EF)=\tr(FEF)=\tr((EF)^*EF)\ge0.
+tr(EF)=tr(FEF)=tr((EF)^*EF)ge0.
-So \(\tr(EF)=0\) iff \(EF=0\). No commutation required.
+So (tr(EF)=0) iff (EF=0). No commutation required.
Second, each block of a Choi matrix can be tested by pairing against
matrix units. This will be the identity we test against in Section~%
@@ -1732,154 +1585,154 @@ sec:projection's proof.
[Rectangular projection criterion]
-Let \(Q,K,D\) be positive integers, \(\mathcal Q\) a set of size \(Q\).
-Take unit vectors \(u_q\in\mathbb R^K\) for \(q\in\mathcal Q\), with
-\(c_{qr}=u_q^{\T}u_r\). For each \(q\), \(\{P_i:i\in\mathcal I_q\}\) is
-a finite projective measurement on \(\C^D\); zero projections allowed.
-Take the \(\mathcal I_q\) disjoint, \(\mathcal I=⊔_q\mathcal I_q\),
-\(n=|\mathcal I|\). Suppose:
+Let (Q,K,D) be positive integers, (mathcal QcalQ) a set of size (Q).
+Take unit vectors (u_qinmathbb R^KbbR^K) for (qinmathcal QcalQ), with
+(c_{qr}=u_q^{T}u_r). For each (q), ({P_i:iinmathcal I_qcalI_q}) is
+a finite projective measurement on (C^D); zero projections allowed.
+Take the (mathcal I_qcalI_q) disjoint, (mathcal I=⊔_qcalI=⊔_qmathcal I_qcalI_q),
+(n=|mathcal I|calI|). Suppose:
-\item \(P_iP_j=P_jP_i\) whenever \(c_{qr}\ne0\);
-\item there is a real \(\kappa<Q/K\) such that every
- \(J⊆\mathcal I\) with \(\tr(P_iP_j)>0\) for all distinct
- \(i,j\in J\) has \(|J|\le\kappa\).
+item (P_iP_j=P_jP_i) whenever (c_{qr}ne0);
+item there is a real (kappa<Q/K) such that every
+ (J⊆mathcal IcalI) with (tr(P_iP_j)>0) for all distinct
+ (i,jin J) has (|J|lekappa).
-Define \(\Psi:\Mat n\to\Mat D\) on matrix units by
+Define (Psi:Mat ntoMat D) on matrix units by
-H_{ij}=c_{qr}P_iP_j,\quad \Psi(E_{ij})=H_{ij}
-\quad(i\in\mathcal I_q,\ j\in\mathcal I_r).
+H_{ij}=c_{qr}P_iP_j,quad Psi(E_{ij})=H_{ij}
+quad(iinmathcal I_q,calI_q, jinmathcal I_r).calI_r).
-Then \(\Psi,\Psi^\dagger\) are PPT, and
+Then (Psi,Psi^dagger) are PPT, and
-C:=J(\Psi^\dagger\circ\Psi) is nonseparable over \C^n ⊗ \C^n,
-\qquad \tr C=Q^2D.
+C:=J(Psi^daggercircPsi) is nonseparable over C^n ⊗ C^n,
+qquad tr C=Q^2D.
Proof. [Strategy]
-The block matrix \(H=J(\Psi)\succeq0\) since it is a sum of rank-one
-Psd block columns \(u_q ⊗ P_iw_i\). Block symmetry plus reality
-gives PPT. Then compress \(C\) to the diagonal tensor subspace; a
-vector-Gram bound gives \(\mathbf1^*B\mathbf1\ge(D/K)\tr B\). A separable
-decomposition would give the reverse bound \(\le\kappa\tr B\), and
-\(\kappa<Q/K\) closes the contradiction.
+The block matrix (H=J(Psi)succeq0) since it is a sum of rank-one
+Psd block columns (u_q ⊗ P_iw_i). Block symmetry plus reality
+gives PPT. Then compress (C) to the diagonal tensor subspace; a
+vector-Gram bound gives (mathbf1^*Bbf1^*Bmathbf1bf1ge(D/K)tr B). A separable
+decomposition would give the reverse bound (lekappatr B), and
+(kappa<Q/K) closes the contradiction.
Proof.
-Regard \(u_q\) as a vector in \(\C^K\). For \(w_i\in\C^D\),
+Regard (u_q) as a vector in (C^K). For (w_iinC^D),
-\sum_{i,j}< w_i,H_{ij}w_j>
-=\norm{\sum_{q}\sum_{i\in\mathcal I_q}u_q ⊗ P_iw_i}^2\ge0.
+sum_{i,j}< w_i,H_{ij}w_j>
+=norm{sum_{q}sum_{iinmathcal I_q}calI_qu_q ⊗ P_iw_i}^2ge0.
-So \(H\succeq0\); Lemma~lem:choi-separable gives \(\Psi\) CP.
+So (Hsucceq0); Lemma~lem:choi-separable gives (Psi) CP.
-Block symmetry: since \(c_{qr}=c_{rq}\) is real and hypothesis 1 makes
-\(P_i,P_j\) commute when \(c_{qr}\ne0\), \(H_{ij}=H_{ji}=H_{ij}^*\).
+Block symmetry: since (c_{qr}=c_{rq}) is real and hypothesis 1 makes
+(P_i,P_j) commute when (c_{qr}ne0), (H_{ij}=H_{ji}=H_{ij}^*).
The two partial transposes:
-H^{\Gamma_1}=H,\qquad H^{\Gamma_2}=H^{\T}\succeq0,
+H^{Gamma_1}=H,qquad H^{Gamma_2}=H^{T}succeq0,
-where \(H^{\T}\succeq0\) because \(H=R^*R⇒ H^{\T}=(\bar R)^*\bar R\).
-Lemma~lem:choi-separable then gives \(T_D\circ\Psi\) CP; so
-\(\Psi\) is PPT.
+where (H^{T}succeq0) because (H=R^*R⇒ H^{T}=(barbar R)^*R)^*barbar RR).
+Lemma~lem:choi-separable then gives (T_DcircPsi) CP; so
+(Psi) is PPT.
-Also \(\Psi^\dagger\) is PPT and CP (Lemma~lem:ppt-stability).
+Also (Psi^dagger) is PPT and CP (Lemma~lem:ppt-stability).
Explicit form:
-[\Psi^\dagger(A)]_{ij}=\tr(H_{ij}^*A);
+[Psi^dagger(A)]_{ij}=tr(H_{ij}^*A);
-block symmetry gives \(T_n\circ\Psi^\dagger=\Psi^\dagger\).
+block symmetry gives (T_ncircPsi^dagger=Psi^dagger).
Compression to diagonal.:
-\(C=\Psi^\dagger\circ\Psi\succeq0\). Its fully indexed entries:
+(C=Psi^daggercircPsisucceq0). Its fully indexed entries:
-C_{(i,a),(j,b)}=\tr(H_{ab}^*H_{ij}).
+C_{(i,a),(j,b)}=tr(H_{ab}^*H_{ij}).
-Since \(c_{qq}=1\), \(P_i^2=P_i\), we get \(H_{ii}=P_i\). The diagonal
-entries and the compression \(B\) to
-\(span\{e_i ⊗ e_i\}\) compute as
+Since (c_{qq}=1), (P_i^2=P_i), we get (H_{ii}=P_i). The diagonal
+entries and the compression (B) to
+(span{e_i ⊗ e_i}) compute as
-C_{(i,j),(i,j)}&=\tr(P_iP_j),\\
-B_{ij}:=C_{(i,i),(j,j)}&=c_{qr}^2\tr(P_iP_j).
+C_{(i,j),(i,j)}&=tr(P_iP_j),
+B_{ij}:=C_{(i,i),(j,j)}&=c_{qr}^2tr(P_iP_j).
Completeness of each measurement gives
-\(\sum_{i\in\mathcal I_q}\sum_{j\in\mathcal I_r}\tr(P_iP_j)=D\). Summing
-over \(q,r\): \(\tr C=Q^2D\). Also \(\tr B=QD\), and
+(sum_{iinmathcal I_q}calI_qsum_{jinmathcal I_r}calI_rtr(P_iP_j)=D). Summing
+over (q,r): (tr C=Q^2D). Also (tr B=QD), and
-\mathbf1^*B\mathbf1=D\sum_{q,r}c_{qr}^2\ge(D Q^2)/(K).
+mathbf1^*Bbf1^*Bmathbf1=Dbf1=Dsum_{q,r}c_{qr}^2ge(D Q^2)/(K).
-Indeed \(M=\sum_q u_qu_q^{\T}\) has \(\tr M=Q\) and
-\(\tr(M^2)=\sum_{q,r}c_{qr}^2\); Cauchy--Schwarz on its \(K\) eigenvalues
-gives \(\tr M^2\ge Q^2/K\).
+Indeed (M=sum_q u_qu_q^{T}) has (tr M=Q) and
+(tr(M^2)=sum_{q,r}c_{qr}^2); Cauchy--Schwarz on its (K) eigenvalues
+gives (tr M^2ge Q^2/K).
Separability bound.:
-Suppose \(C\) separable over \(\C^n ⊗ \C^n\). Spectral decomposition
-of local factors yields \(C=\sum_\ell(x^{(\ell)} ⊗ y^{(\ell)})(...)^*\).
-Whenever \(\tr(P_iP_j)=0\), the diagonal entry
-\(0=C_{(i,j),(i,j)}=\sum_\ell|x_i^{(\ell)}y_j^{(\ell)}|^2\) forces each
-term zero, so for one \(z_i=x_iy_i\), \(\supp z\) has no pair with
-\(\tr(P_iP_j)=0\). Thus \(|\supp z|\le\kappa\).
-
-Compressing to the diagonal subspace gives \(B=\sum_\ell z^{(\ell)}(z^{(\ell)})^*\).
+Suppose (C) separable over (C^n ⊗ C^n). Spectral decomposition
+of local factors yields (C=sum_ell(x^{(ell)} ⊗ y^{(ell)})(...)^*).
+Whenever (tr(P_iP_j)=0), the diagonal entry
+(0=C_{(i,j),(i,j)}=sum_ell|x_i^{(ell)}y_j^{(ell)}|^2) forces each
+term zero, so for one (z_i=x_iy_i), (supp z) has no pair with
+(tr(P_iP_j)=0). Thus (|supp z|lekappa).
+
+Compressing to the diagonal subspace gives (B=sum_ell z^{(ell)}(z^{(ell)})^*).
Complex Cauchy--Schwarz on supports:
-\mathbf1^*B\mathbf1=\sum_\ell|\sum_i z_i^{(\ell)}|^2
-\le\kappa\sum_\ell\norm{z^{(\ell)}}^2=\kappa\tr B
-<\frac QK\tr B=(DQ^2)/(K),
+mathbf1^*Bbf1^*Bmathbf1=bf1=sum_ell|sum_i z_i^{(ell)}|^2
+lekappasum_ellnorm{z^{(ell)}}^2=kappatr B
+<frac QKtr B=(DQ^2)/(K),
-contradicting (eq:projection-gram-bound). Hence \(C\)
+contradicting (eq:projection-gram-bound). Hence (C)
nonseparable.
[Same local channel for real projections]
Under the hypotheses of Proposition~prop:projection-criterion,
-suppose every \(P_i\) is real symmetric. Put \(Λ=\Psi^\dagger\).
+suppose every (P_i) is real symmetric. Put (Λ=Psi^dagger).
Then
-Λ\circ T_D=Λ=T_n\circΛ,
-\qquad \trΛ(A)=Q\,\tr A.
+Λcirc T_D=Λ=T_ncircΛ,
+qquad trΛ(A)=Q,tr A.
-So \(Λ/Q:\Mat D\to\Mat n\) is a trace-preserving PPT channel.
-With \(\Phi_D=Ω_D/\sqrt D\) the normalized maximally entangled
+So (Λ/Q:Mat DtoMat n) is a trace-preserving PPT channel.
+With (Phi_D=Ω_D/sqrt D) the normalized maximally entangled
vector,
(C)/(Q^2D)=
-((Λ)/(Q) ⊗ (Λ)/(Q))(\Phi_D\Phi_D^*)
+((Λ)/(Q) ⊗ (Λ)/(Q))(Phi_DPhi_D^*)
-is an entangled density operator in \(\mathscr C\).
+is an entangled density operator in (mathscr CscrC).
Proof.
-Reality + block symmetry gives \(H_{ij}^{\T}=H_{ij}\). Equation~%
+Reality + block symmetry gives (H_{ij}^{T}=H_{ij}). Equation~%
(eq:projection-adjoint) gives both transpose invariances. Trace:
-\(\trΛ(A)=\sum_i\tr(H_{ii}A)=Q\tr A\). So \(Λ/Q\) is a
+(trΛ(A)=sum_itr(H_{ii}A)=Qtr A). So (Λ/Q) is a
trace-preserving PPT channel. Lemma~lem:choi-transfer with
-\(\Psi,Λ\) gives \(\Psi^\sharp=T_n\circΛ\circ T_D=Λ\)
+(Psi,Λ) gives (Psi^sharp=T_ncircΛcirc T_D=Λ)
by invariances, so
-\(G:=(Λ ⊗ Λ)(Ω_DΩ_D^*)=C\). Divide by
-\(Q^2D\). Nonseparability and trace \(Q^2D\) give the claim.
+(G:=(Λ ⊗ Λ)(Ω_DΩ_D^*)=C). Divide by
+(Q^2D). Nonseparability and trace (Q^2D) give the claim.
[Square embedding]
Under Proposition~prop:projection-criterion's hypotheses with
-\(n\ge D\), let \(U:\C^D\to\C^n\) be the real coordinate isometry onto
-the first \(D\) coordinates. With
+(nge D), let (U:C^DtoC^n) be the real coordinate isometry onto
+the first (D) coordinates. With
-\Xi_1=\Ad_U\circ\Psi,\qquad
-\Xi_2=\Psi^\dagger\circ\Ad_{U^*}=\Xi_1^\dagger,
+Xi_1=Ad_UcircPsi,qquad
+Xi_2=Psi^daggercircAd_{U^*}=Xi_1^dagger,
-the maps \(\Xi_1,\Xi_2:\Mat n\to\Mat n\) are PPT and
-\(\Xi_2\circ\Xi_1=\Psi^\dagger\circ\Psi\). Their composition is not EB.
+the maps (Xi_1,Xi_2:Mat ntoMat n) are PPT and
+(Xi_2circXi_1=Psi^daggercircPsi). Their composition is not EB.
Proof.
-Embedding and compression by \(U\) are CP; the composite is
-\(\Psi^\dagger\circ\Psi\) with Choi \(C\) nonseparable. The pair inherits
-PPT from \Ssec:projection's invariances; the composite Choi matrix
-is the nonseparable \(C\). Lemma~lem:choi-separable gives non-EB.
+Embedding and compression by (U) are CP; the composite is
+(Psi^daggercircPsi) with Choi (C) nonseparable. The pair inherits
+PPT from Ssec:projection's invariances; the composite Choi matrix
+is the nonseparable (C). Lemma~lem:choi-separable gives non-EB.
# A finite combinatorial realization
@@ -1895,122 +1748,122 @@ orthogonal eight-set into a forbidden positive-overlap choice.
## Vectors with forced orthogonality
-Set \(p=257\), \(k=4p=1028\), \([k]=\{1,...,k\}\). Define
+Set (p=257), (k=4p=1028), ([k]={1,...,k}). Define
-\mathcal U=\{A⊆[k]:|A|=2p,\ 1\in A\},\quad
-b=|\mathcal U|=\binom{1027}{513}=\frac12\binom{4p}{2p},
+mathcal U=calU={A⊆[k]:|A|=2p, 1in A},quad
+b=|mathcal U|=calU|=binom{1027}{513}=frac12binom{4p}{2p},
-and \(s_A=k^{-1/2}(2\mathbf1_{\{h\in A\}}-1)_{h=1}^k\in\mathbb R^k\).
-Each \(s_A\) is a unit vector; by counting agreeing entries,
+and (s_A=k^{-1/2}(2mathbf1_bf1_{{hin A}}-1)_{h=1}^kinmathbb R^kbbR^k).
+Each (s_A) is a unit vector; by counting agreeing entries,
-s_A^{\T}s_B=(k-2|A△ B| )/(k)
-=(|A\cap B|)/(p)-1.
+s_A^{T}s_B=(k-2|A△ B| )/(k)
+=(|Acap B|)/(p)-1.
-So \(s_A\perp s_B\) iff \(|A\cap B|=p\).
+So (s_Aperp s_B) iff (|Acap B|=p).
-Put \(a_0=\sum_{j=0}^{p-1}\binom{k}{j}=\sum_{j=0}^{256}\binom{1028}{j}\)
-and \(\delta=a_0/b\). Frankl--R\"odl~\cite[Thm.~1.11]{FR1987} forces
+Put (a_0=sum_{j=0}^{p-1}binom{k}{j}=sum_{j=0}^{256}binom{1028}{j})
+and (delta=a_0/b). Frankl--R"odl~cite[Thm.~1.11]{FR1987} forces
orthogonal families in subsets, but here we get a precise bound by
elementary multilinear methods.
[Restricted intersections]
-If \(\mathcal F⊆\mathcal U\) has no two distinct sets with
-orthogonal associated vectors, then \(|\mathcal F|\le a_0=\delta b\).
+If (mathcal F⊆calF⊆mathcal UcalU) has no two distinct sets with
+orthogonal associated vectors, then (|mathcal F|calF|le a_0=delta b).
Proof.
-For distinct \(A,B\in\mathcal F\), common element \(1\) and equal sizes
-give \(1\le|A\cap B|\le2p-1\); orthogonality is excluded, so
-\(|A\cap B|\ne p\) and \(|A\cap B|\not\equiv0\pmod p\); also
-\(|A\cap A|=2p\equiv0\pmod p\). Over \(\F_p\) define
-\(f_A(x)=\prod_{h\in A}(1-x_h^{p-1})=1-(\sum_{h\in A}x_h)^{p-1}\)
-in the identity \(a^{p-1}=1\) for \(a\in\F_p^\times\). On indicators:
-\(f_A(\mathbf1_B)=\delta_{AB}\).
-Replace every positive power \(x_h^{d}\to x_h\) for \(d>1\); evaluations
-on Boolean vectors are unchanged and degrees drop to \(\le p-1\).
+For distinct (A,Binmathcal FcalF), common element (1) and equal sizes
+give (1le|Acap B|le2p-1); orthogonality is excluded, so
+(|Acap B|ne p) and (|Acap B|notequiv0pmod p); also
+(|Acap A|=2pequiv0pmod p). Over (F_p) define
+(f_A(x)=prod_{hin A}(1-x_h^{p-1})=1-(sum_{hin A}x_h)^{p-1})
+in the identity (a^{p-1}=1) for (ainF_p^times). On indicators:
+(f_A(mathbf1_B)=bf1_B)=delta_{AB}).
+Replace every positive power (x_h^{d}to x_h) for (d>1); evaluations
+on Boolean vectors are unchanged and degrees drop to (le p-1).
Independence follows by evaluating any linear relation at each
-\(\mathbf1_B\). The multilinear polynomials of degree \(\le p-1\) have
-basis \(\{x^\alpha:|\alpha|\le p-1\}\), size \(a_0\). So
-\(|\mathcal F|\le a_0\).
+(mathbf1_Bbf1_B). The multilinear polynomials of degree (le p-1) have
+basis ({x^alpha:|alpha|le p-1}), size (a_0). So
+(|mathcal F|calF|le a_0).
[Fiber amplification]
-For \(t=1,2,3\), to \((A_1,...,A_t)\in\mathcal U^t\) associate the
-unit tensor \(s_{A_1} ⊗ ... ⊗ s_{A_t}\). If
-\(\mathcal R⊆\mathcal U^t\) contains no collection of \(2^t\)
+For (t=1,2,3), to ((A_1,...,A_t)inmathcal U^tcalU^t) associate the
+unit tensor (s_{A_1} ⊗ ... ⊗ s_{A_t}). If
+(mathcal R⊆calR⊆mathcal U^tcalU^t) contains no collection of (2^t)
distinct tuples whose associated tensors are pairwise orthogonal, then
-\(|\mathcal R|\le t\delta b^t\).
+(|mathcal R|calR|le tdelta b^t).
Proof.
-\(t=1\) is Lemma~lem:restricted-intersections. For
-\(t=2,3\), set
-\(\mathcal R_A=\{(A_2,...,A_t):(A,A_2,...,A_t)\in\mathcal R\}\).
-Let \(\mathcal G\) be those \(A\) for which \(\mathcal R_A\) contains
-\(2^{t-1}\) pairwise orthogonal tuples. Two \(A,B\in\mathcal G\) with
-\(s_A\perp s_B\) and disjoint fibers would produce \(2^t\) forbidden
-tuples; so \(\mathcal G\) has no orthogonal pair, forcing
-\(|\mathcal G|\le a_0=\delta b\). Fibers in \(\mathcal G\) have
-\(\le b^{t-1}\) elements; other fibers have \(\le(t-1)\delta b^{t-1}\).
+(t=1) is Lemma~lem:restricted-intersections. For
+(t=2,3), set
+(mathcal R_A=calR_A={(A_2,...,A_t):(A,A_2,...,A_t)inmathcal RcalR}).
+Let (mathcal GcalG) be those (A) for which (mathcal R_AcalR_A) contains
+(2^{t-1}) pairwise orthogonal tuples. Two (A,Binmathcal GcalG) with
+(s_Aperp s_B) and disjoint fibers would produce (2^t) forbidden
+tuples; so (mathcal GcalG) has no orthogonal pair, forcing
+(|mathcal G|calG|le a_0=delta b). Fibers in (mathcal GcalG) have
+(le b^{t-1}) elements; other fibers have (le(t-1)delta b^{t-1}).
Summing gives
-\[
-|\mathcal R|\le(\delta b)b^{t-1}+b(t-1)\delta b^{t-1}=t\delta b^t.
-\]
+[
+|mathcal R|calR|le(delta b)b^{t-1}+b(t-1)delta b^{t-1}=tdelta b^t.
+]
Set questions:
-\mathcal Q=\mathcal U^3,\quad Q=b^3,\quad K=k^3=1028^3=1{,}086{,}373{,}952,
+mathcal Q=calQ=mathcal U^3,calU^3,quad Q=b^3,quad K=k^3=1028^3=1{,}086{,}373{,}952,
-\(u_q=s_{A_1} ⊗ s_{A_2} ⊗ s_{A_3}\) for
-\(q=(A_1,A_2,A_3)\). An orthogonal eight-set is \(S⊆\mathcal Q\)
-of size eight with pairwise orthogonal \(u_q\)'s. Lemma~%
-lem:fiber-amplification at \(t=3\) gives
+(u_q=s_{A_1} ⊗ s_{A_2} ⊗ s_{A_3}) for
+(q=(A_1,A_2,A_3)). An orthogonal eight-set is (S⊆mathcal QcalQ)
+of size eight with pairwise orthogonal (u_q)'s. Lemma~%
+lem:fiber-amplification at (t=3) gives
-|\mathcal R|\le\kappa:=3\delta Q=3a_0b^2
+|mathcal R|calR|lekappa:=3delta Q=3a_0b^2
-for any \(\mathcal R⊆\mathcal Q\) with no orthogonal eight-set.
+for any (mathcal R⊆calR⊆mathcal QcalQ) with no orthogonal eight-set.
The analytic bound: central binomial
-\(b=\frac12\binom{4p}{2p}\ge 16^p/(2(4p+1))\), and
-\(a_0\le p\binom{4p}{p}\le p(4/3)^{4p}3^p < p\,12^p\). So
-\(\delta < 2p(4p+1)(3/4)^p<2^{-65}\). With \(2p(4p+1)<2^{20}\),
-\(p\le85\cdot3+2\), \((3/4)^3<1/2\):
-\(3\delta<2^{-33}<1/K\), giving
+(b=frac12binom{4p}{2p}ge 16^p/(2(4p+1))), and
+(a_0le pbinom{4p}{p}le p(4/3)^{4p}3^p < p,12^p). So
+(delta < 2p(4p+1)(3/4)^p<2^{-65}). With (2p(4p+1)<2^{20}),
+(ple85cdot3+2), ((3/4)^3<1/2):
+(3delta<2^{-33}<1/K), giving
-\kappa=3a_0b^2<\frac QK.
+kappa=3a_0b^2<frac QK.
## Six parity measurements (Mermin--Peres)
-On \(\C^2\):
-\(X=0&1\\1&0\),
-\(Z=1&0\\0&-1\),
-\(Y=0&-i\\i&0\).
+On (C^2):
+(X=0&11&0),
+(Z=1&00&-1),
+(Y=0&-ii&0).
The Mermin--Peres square~[cite]:
{ccc|c}
-X ⊗ I_2&I_2 ⊗ X&X ⊗ X&+I_4\\
-I_2 ⊗ Z&Z ⊗ I_2&Z ⊗ Z&+I_4\\
-X ⊗ Z&Z ⊗ X&Y ⊗ Y&+I_4\\\hline
+X ⊗ I_2&I_2 ⊗ X&X ⊗ X&+I_4
+I_2 ⊗ Z&Z ⊗ I_2&Z ⊗ Z&+I_4
+X ⊗ Z&Z ⊗ X&Y ⊗ Y&+I_4hline
+I_4&+I_4&-I_4&
Each entry is a real symmetric involution. Order the six contexts as
-the three rows and then three columns. In context \(t\), the three
-observables \((O_1,O_2,O_3)\) satisfy \(O_1O_2=O_2O_1\), \(O_1,O_2,O_3\)
+the three rows and then three columns. In context (t), the three
+observables ((O_1,O_2,O_3)) satisfy (O_1O_2=O_2O_1), (O_1,O_2,O_3)
commute, and
-\(O_3=e_tO_1O_2\) with
-\((e_1,...,e_6)=(1,1,1,1,1,-1)\). Signs from \(XZ=-iY\) and \(ZX=iY\).
+(O_3=e_tO_1O_2) with
+((e_1,...,e_6)=(1,1,1,1,1,-1)). Signs from (XZ=-iY) and (ZX=iY).
-For \((\alpha,\beta)\in\{-1,+1\}^2\), set
+For ((alpha,beta)in{-1,+1}^2), set
-h_t(\alpha,\beta)=\frac14(I_4+\alpha O_1)(I_4+\beta O_2).
+h_t(alpha,beta)=frac14(I_4+alpha O_1)(I_4+beta O_2).
-These are self-adjoint projections with sums \(I_4\), ranges carrying
-eigensigns \((\alpha,\beta,e_t\alpha\beta)\).
+These are self-adjoint projections with sums (I_4), ranges carrying
+eigensigns ((alpha,beta,e_talphabeta)).
[Parity obstruction]
@@ -2021,113 +1874,113 @@ overlap.
Proof.
-If \(E\) (row) and \(F\) (column) both select an eigensign \(a,b\)
-for their shared \(O\), then \(aEF=EOF=bEF\); positive overlap forces
-\(a=b\). So a hypothetical positive-overlap choice assigns nine
+If (E) (row) and (F) (column) both select an eigensign (a,b)
+for their shared (O), then (aEF=EOF=bEF); positive overlap forces
+(a=b). So a hypothetical positive-overlap choice assigns nine
consistent signs to the square. The product over each row's three
-signs is \(e_t\): \(+1\) for rows; the column constraint \(-1\) at the
-last column contradicts the row product \(+1\).
+signs is (e_t): (+1) for rows; the column constraint (-1) at the
+last column contradicts the row product (+1).
## Assembling the tensor product
-For each 8-subset \(S⊆\mathcal Q\), a separate \(\C^4\) factor.
+For each 8-subset (S⊆mathcal QcalQ), a separate (C^4) factor.
On an orthogonal eight-set, the six smallest-index elements receive the
six parity measurements; the remaining two get a trivial one-outcome
-measurement with the other outcomes zero. Nonorthogonal \(S\) and all
+measurement with the other outcomes zero. Nonorthogonal (S) and all
non-membership questions get trivial measurements.
-\mathcal S=\{S⊆\mathcal Q:|S|=8\},\quad
-L=|\mathcal S|=\binom Q8,\quad
-\mathcal H=⊗_{S\in\mathcal S}\C^4\cong\C^D,\quad D=4^L.
+mathcal S=calS={S⊆mathcal Q:|S|=8calQ:|S|=8},quad
+L=|mathcal S|=calS|=binom Q8,quad
+mathcal H=⊗_calH=⊗_{Sinmathcal S}calSC^4congC^D,quad D=4^L.
-For each \(q\in\mathcal Q\) and outcome string
-\(\lambda\in\{-1,+1\}^{2\times\mathcal S}\),
+For each (qinmathcal QcalQ) and outcome string
+(lambdain{-1,+1}^{2timesmathcal S}calS),
-\mathcal I_q=\{q\}\times\mathcal A^{\mathcal S},\quad
-P_{q,\lambda}=⊗_{S\in\mathcal S}h_{S,q}(\lambda_S).
+mathcal I_q=calI_q={q}timesmathcal A^calA^{mathcal S}calS,quad
+P_{q,lambda}=⊗_{Sinmathcal S}calSh_{S,q}(lambda_S).
-where \(h_{S,q}\) is the measurement above (or trivial). These are real
-symmetric projections summing to \(I_D\) for each fixed \(q\).
-Keeping every outcome string gives exactly \(D=4^L\) labels per
-question, so \(n=|⊔_q\mathcal I_q|=QD\).
+where (h_{S,q}) is the measurement above (or trivial). These are real
+symmetric projections summing to (I_D) for each fixed (q).
+Keeping every outcome string gives exactly (D=4^L) labels per
+question, so (n=|⊔_qmathcal I_q|=QDcalI_q|=QD).
-The \(u_q\) and \(P_{q,\lambda}\) satisfy Proposition~%
-prop:projection-criterion's two hypotheses with \(n=m=QD\) and
-\(\kappa=3a_0b^2<Q/K\).
+The (u_q) and (P_{q,lambda}) satisfy Proposition~%
+prop:projection-criterion's two hypotheses with (n=m=QD) and
+(kappa=3a_0b^2<Q/K).
Proof.
-Hypothesis 1: if \(q=r\) then projections are in one POVM and commute.
-If \(q\ne r\) with \(c_{qr}\ne0\) and both nontrivial on one factor,
+Hypothesis 1: if (q=r) then projections are in one POVM and commute.
+If (qne r) with (c_{qr}ne0) and both nontrivial on one factor,
they would be together in an orthogonal eight-set forcing
-\(c_{qr}=0\). Contradiction. So on each factor at least one selected
+(c_{qr}=0). Contradiction. So on each factor at least one selected
projection is trivial or zero, and they commute.
Hypothesis 2: labels per question differ only on one factor and have
zero product; labels for one question are one per question. The trace
overlap factors over the tensor factors. If the questions represented
-by a positive-overlap label set contained an orthogonal eight-set \(S\),
+by a positive-overlap label set contained an orthogonal eight-set (S),
the six selected projections would violate Lemma~%
-lem:parity-obstruction on factor \(S\). So
-\(|J|\le\kappa\). The strict inequality is (eq:combinatorial-strict-bound).
+lem:parity-obstruction on factor (S). So
+(|J|lekappa). The strict inequality is (eq:combinatorial-strict-bound).
[Combinatorial construction]
Take the explicit data:
-\(p=257\), \(k=1028\), \(b=\binom{1027}{513}\),
-\(a_0=\sum_{j=0}^{256}\binom{1028}{j}\), \(Q=b^3\),
-\(K=1086373952\), \(\kappa=3a_0b^2<Q/K\), \(L=\binom Q8\),
-\(D=4^L\), \(m=QD\). Let \(\Psi\) be
-from (eq:projection-map), \(Λ=\Psi^\dagger\),
-\(\Phi_D=Ω_D/\sqrt D\). Then:
-
-\item \(Λ/Q\) is a trace-preserving PPT channel, and
-
-\rho_{comb}
-=(Λ/Q ⊗ Λ/Q)(\Phi_D\Phi_D^*)
-=(J(\Psi^\dagger\circ\Psi))/(Q^2 D)
-
-is an entangled density operator on \(\C^m ⊗ \C^m\) in
-\(\mathscr C\), with \(K_D(\rho_{comb})=0\) and the uniform
-unhalved \(1/5\) gap.
-\item For \(U:\C^D\to\C^m\) the first-coordinate isometry,
-\(\Phi_1^{comb}=\Ad_U\circ\Psi\),
-\(\Phi_2^{comb}=Λ\circ\Ad_{U^*}\) are PPT with
-\(J(\Phi_2^{comb}\circ\Phi_1^{comb})=Q^2D\rho_{comb}\)
+(p=257), (k=1028), (b=binom{1027}{513}),
+(a_0=sum_{j=0}^{256}binom{1028}{j}), (Q=b^3),
+(K=1086373952), (kappa=3a_0b^2<Q/K), (L=binom Q8),
+(D=4^L), (m=QD). Let (Psi) be
+from (eq:projection-map), (Λ=Psi^dagger),
+(Phi_D=Ω_D/sqrt D). Then:
+
+item (Λ/Q) is a trace-preserving PPT channel, and
+
+rho_{comb}
+=(Λ/Q ⊗ Λ/Q)(Phi_DPhi_D^*)
+=(J(Psi^daggercircPsi))/(Q^2 D)
+
+is an entangled density operator on (C^m ⊗ C^m) in
+(mathscr CscrC), with (K_D(rho_{comb})=0) and the uniform
+unhalved (1/5) gap.
+item For (U:C^DtoC^m) the first-coordinate isometry,
+(Phi_1^{comb}=Ad_UcircPsi),
+(Phi_2^{comb}=ΛcircAd_{U^*}) are PPT with
+(J(Phi_2^{comb}circPhi_1^{comb})=Q^2Drho_{comb})
nonseparable, so the composition is not EB.
-\item The channel completion \(Θ_{comb}\) of
+item The channel completion (Θ_{comb}) of
Proposition~prop:channel-completion applied to this pair is a
-trace-preserving PPT channel on \(\Mat{2m+1}\) whose square has the
+trace-preserving PPT channel on (Mat{2m+1}) whose square has the
nonseparable corner
-\((W_0^* ⊗ W_0^*)J(Θ_{comb}^2)(W_0 ⊗ W_0)
-=c_1c_2Q^2D\,\rho_{comb}\).
+((W_0^* ⊗ W_0^*)J(Θ_{comb}^2)(W_0 ⊗ W_0)
+=c_1c_2Q^2D,rho_{comb}).
Proof.
Lemma~lem:combinatorial-measurements verifies the projection
-criterion. Its composite \(C=J(\Psi^\dagger\circ\Psi)\) is positive and
-nonseparable, \(\tr C=Q^2D\). Real symmetric projections give
+criterion. Its composite (C=J(Psi^daggercircPsi)) is positive and
+nonseparable, (tr C=Q^2D). Real symmetric projections give
Corollary~cor:real-channel; then Theorem~thm:zero-key gives
the operational conclusions. Corollary~cor:square-embedding gives
the square PPT pair, and Proposition~prop:channel-completion gives
the channel.
-All quantities are given by finite exact formulas. The \(s_A\) have
+All quantities are given by finite exact formulas. The (s_A) have
irrational coordinates, but the Gram entries of question vectors
-\(c_{qr}\) are rational; observables and projections are integral; so
-every block \(H_{ij},\Psi,Λ,\Phi_1^{comb},\Phi_2^{comb}\),
-and \(\rho_{comb}\) entry is rational. The channel completion
+(c_{qr}) are rational; observables and projections are integral; so
+every block (H_{ij},Psi,Λ,Phi_1^{comb},Phi_2^{comb}),
+and (rho_{comb}) entry is rational. The channel completion
in Proposition~prop:channel-completion has rational coefficients.
Appendix~app:supplementary-checks records the exact integer checks.
-\appendix
+appendix
# Exact arithmetic for the dimension-ten certificate
@@ -2135,18 +1988,18 @@ Appendix~app:supplementary-checks records the exact integer checks.
Everything in Section~sec:certificate that can be reduced to exact
finite arithmetic is verified by a single Python~3 program using only
the standard library and integer arithmetic. Its inputs are the matrices
-\(M_0,...,M_3\) of (eq:explicit-matrices). Its outputs are:
+(M_0,...,M_3) of (eq:explicit-matrices). Its outputs are:
-\item an assertion of the block-symmetry identity
- \(M_i^{\T}M_j=M_j^{\T}M_i\) for all \(0\le i,j<4\);
-\item the integer matrices \(D\in\Z^{15\times20}\) and
- \(H\in\Z^{15\times15}\) in the decomposition
- \((p_\rho(t))_\rho=Dv(t)+Hw(t)\), where the
+item an assertion of the block-symmetry identity
+ (M_i^{T}M_j=M_j^{T}M_i) for all (0le i,j<4);
+item the integer matrices (DinZ^{15times20}) and
+ (HinZ^{15times15}) in the decomposition
+ ((p_rho(t))_rho=Dv(t)+Hw(t)), where the
monomial order is total degree first, then lexicographic;
-\item for each of \(p=41,131,139\): the residue of \(\det H\); the
- twenty-one characteristic coefficients of the \(\F_p\)-reduction
- of \(f(z)=\det(zI_{20}-N)\); and the gcd degrees
- \(d_p(k)=\deg\gcd(\overline f, z^{p^k}-z)\) for \(k=1,...,20\).
+item for each of (p=41,131,139): the residue of (det H); the
+ twenty-one characteristic coefficients of the (F_p)-reduction
+ of (f(z)=det(zI_{20}-N)); and the gcd degrees
+ (d_p(k)=deggcd(overlineoverline f,f, z^{p^k}-z)) for (k=1,...,20).
The program runs with assertions enabled (no -O, no
-OO, PYTHONOPTIMIZE unset). From the paper directory:
@@ -2158,43 +2011,43 @@ an identical build copy supplies the typeset listing.
Details of the computation.:
The function checksym verifies block symmetry using only
-\(i<j\) and column indices \(a<b\). The cases \(i=j\) and \(a=b\)
+(i<j) and column indices (a<b). The cases (i=j) and (a=b)
vanish identically (equal matrix entries), while swapping either also
leaves the same value; so strict loops cover every equality. The array
-R is \([D\mid H]\); T is the reduction of \(N\)
+R is ([Dmid H]); T is the reduction of (N)
modulo the current prime. The program expands each determinant over the
-\(4!=24\) permutations, collects coefficients in that order, and performs Gaussian elimination on \(H\)
+(4!=24) permutations, collects coefficients in that order, and performs Gaussian elimination on (H)
modulo each prime.
-For \(f(z)=z^{20}+c_1z^{19}+...+c_{20}\), Newton's identities
-\(k c_k=-\sum_{i=0}^{k-1}c_i\tr(N^{k-i})\) determine \(c_k\) from the
-traces \(\tr(N^{k-i})\). Division by \(k\) is valid at each of
-\(41,131,139\) because \(p>20\). Repeated \(p\)-th powers of \(z\) mod
-\(\overline f\) compute the residues of \(z^{p^k}\), and the polynomial
-Euclidean algorithm gives \(d_p(k)\). All inverses are in the stated
+For (f(z)=z^{20}+c_1z^{19}+...+c_{20}), Newton's identities
+(k c_k=-sum_{i=0}^{k-1}c_itr(N^{k-i})) determine (c_k) from the
+traces (tr(N^{k-i})). Division by (k) is valid at each of
+(41,131,139) because (p>20). Repeated (p)-th powers of (z) mod
+(overlineoverline ff) compute the residues of (z^{p^k}), and the polynomial
+Euclidean algorithm gives (d_p(k)). All inverses are in the stated
prime field.
No floating-point approximation, no heuristic search, no general
elimination, and no Galois-group computation is performed. In
particular:
-\item The irreducibility of \(f\) over \(\Q\) follows from the irreducible
+item The irreducibility of (f) over (Q) follows from the irreducible
reduction modulo 41 together with Lemma~lem:monic-local-reduction.
-\item The Galois cycle types \((1,2,17)\) and \((1,19)\) follow from
- Lemma~lem:local-frobenius at \(p=131,139\).
-\item \(G=S_{20}\) follows from the transposition plus \(2\)-transitivity
+item The Galois cycle types ((1,2,17)) and ((1,19)) follow from
+ Lemma~lem:local-frobenius at (p=131,139).
+item (G=S_{20}) follows from the transposition plus (2)-transitivity
argument of Section~sec:certificate.
The exact gcd degrees recorded are:
-\[
+[
{c|c}
-p & (d_p(1),...,d_p(20))\\ \hline
-41 & 0,...,0,\ 20 \quad(only d_{41}(20)=20\ne0)\\
-131 & 1,\,3,\,0,\,0,...,0,\ 18 \quad(at k=1,2,17)\\
-139 & 1,\,0,...,0,\ 20 \quad(at k=1,19)
+p & (d_p(1),...,d_p(20)) hline
+41 & 0,...,0, 20 quad(only d_{41}(20)=20ne0)
+131 & 1,,3,,0,,0,...,0, 18 quad(at k=1,2,17)
+139 & 1,,0,...,0, 20 quad(at k=1,19)
-\]
-and the residues of \(\det H\) are \(40, 23, 9\) modulo \(41,131,139\),
+]
+and the residues of (det H) are (40, 23, 9) modulo (41,131,139),
respectively, each nonzero.
# Supplementary finite checks
@@ -2206,38 +2059,38 @@ programs verify its exact integer components. Neither is a premise of
the proof.
Combinatorics.:
-verification/combinatorics\_exact\_checks.py uses only the
+verification/combinatorics_exact_checks.py uses only the
Python standard library. It checks:
-(1) primality of \(257\) by trial division by the primes up to
- \(√(257)\), namely \(2,3,5,7,11,13\);
+(1) primality of (257) by trial division by the primes up to
+ (√(257)), namely (2,3,5,7,11,13);
(2) the intermediate binomial and power inequalities used to derive
- \(\delta<2^{-65}\);
-(3) \(3a_0K<b\) and \(3a_0b^2\ge3\).
-Its recorded output combinatorics\_exact\_checks.json stores
-the exact integers \(a_0\) (under key A) and \(b\), confirms
-\(K=1086373952\), and records the stronger interval
+ (delta<2^{-65});
+(3) (3a_0K<b) and (3a_0b^2ge3).
+Its recorded output combinatorics_exact_checks.json stores
+the exact integers (a_0) (under key A) and (b), confirms
+(K=1086373952), and records the stronger interval
-2^{-194}\le\delta<2^{-193}.
+2^{-194}ledelta<2^{-193}.
-The analytic proof requires only \(\delta<2^{-65}\).
+The analytic proof requires only (delta<2^{-65}).
Parity gadget.:
-verification/gadget\_exact\_check.py represents each local
-projection \(h_t(\alpha,\beta)\) by the integral matrix
-\(R=4h_t(\alpha,\beta)\). It checks observable relations, all operator
-identities, eigensigns for all \(24\) projections, and measurement
-completeness. It computes the \(36\) ordered context-pair overlap
-tables; the stored integer \(\tr(RR')\) is exactly \(16\tr(h_th_{t'})\).
-It then enumerates all \(4^6=4096\) selections of one outcome per
-context, counts zero overlaps among the \(15\) unordered pairs of
+verification/gadget_exact_check.py represents each local
+projection (h_t(alpha,beta)) by the integral matrix
+(R=4h_t(alpha,beta)). It checks observable relations, all operator
+identities, eigensigns for all (24) projections, and measurement
+completeness. It computes the (36) ordered context-pair overlap
+tables; the stored integer (tr(RR')) is exactly (16tr(h_th_{t'})).
+It then enumerates all (4^6=4096) selections of one outcome per
+context, counts zero overlaps among the (15) unordered pairs of
distinct contexts, and produces:
-\[
+[
{c|rrrrr}
-\# of zero-overlap pairs & 1 & 3 & 5 & 7 & 9\\ \hline
-\# of selections & 144 & 1344 & 2016 & 576 & 16
+# of zero-overlap pairs & 1 & 3 & 5 & 7 & 9 hline
+# of selections & 144 & 1344 & 2016 & 576 & 16
-\]
-The recorded output gadget\_exact\_check.json has no selection
+]
+The recorded output gadget_exact_check.json has no selection
with all overlaps strictly positive.
Running.:
@@ -2254,92 +2107,92 @@ objects are specified and analyzed by the finite formulas in the text.
{99}
-\bibitem{AnLee2026}
+bibitem{AnLee2026}
J.~An and S.~Lee, ``Beyond the Positive Partial Transpose Squared
Conjecture: The Qutrit Case,'' arXiv:2607.15947 [quant-ph] (2026).
-\bibitem{Choi1975}
+bibitem{Choi1975}
M.-D.~Choi, ``Completely positive linear maps on complex matrices,''
Linear Algebra Appl. 10 (1975), 285--290.
-\bibitem{CMHW2019}
-M.~Christandl, A.~M\"uller-Hermes, and M.~M.~Wolf,
+bibitem{CMHW2019}
+M.~Christandl, A.~M"uller-Hermes, and M.~M.~Wolf,
``When do composed maps become entanglement breaking?''
-Ann.\ Henri Poincar\'e 20 (2019), 2295--2322.
+Ann. Henri Poincar'e 20 (2019), 2295--2322.
-\bibitem{DevetakWinter05}
+bibitem{DevetakWinter05}
I.~Devetak and A.~Winter, ``Distillation of secret key and entanglement
-from quantum states,'' Proc.\ Roy.\ Soc.\ A 461 (2005),
+from quantum states,'' Proc. Roy. Soc. A 461 (2005),
207--235.
-\bibitem{Filippov2021}
+bibitem{Filippov2021}
S.~N.~Filippov, ``Capacity of trace decreasing quantum operations
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