diff 56bc728..f6bf760 zero_distillable_entanglement_v2.txt
diff --git a/zero_distillable_entanglement_v2.txt b/zero_distillable_entanglement_v2.txt
index 845a139..ffd5fa9 100644
--- a/zero_distillable_entanglement_v2.txt
+++ b/zero_distillable_entanglement_v2.txt
@@ -19,7 +19,7 @@ hypersetup{pdftitle={Entanglement with zero distillable secret key in local dime
-newcommand{Mat}[1]{M_{#1}(mathbb C)}bbC)
+newcommand{Mat}[1]{M_{#1}(C)}
@@ -135,7 +135,7 @@ There is an explicitly specified trace-preserving PPT channel
entanglement breaking.
-The dimension (21) is (2cdot10+1): the pair from Theorem~%
+The dimension (21) is (2·10+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
Filippov~[cite]. Section~sec:applications presents the
@@ -250,17 +250,17 @@ a product vector (a_r ⊗ b_r). Since
nonzero. This is the range criterion of Horodecki~[cite].
-[The represented class (mathscr CscrC)]
+[The represented class (C]
A density operator (omega) on (C^a ⊗ C^b) belongs to
-(mathscr CscrC) if there exist positive integers (r,s), a vector
+(C 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
+(L:calL:Mat rtoMat a), (M:calM:Mat stoMat b) such
that
-omega=(mathcal LcalL ⊗ mathcal M)(calM)(chichi^*),
+omega=(L ⊗ M)(*),
-mathcal L,calL, mathcal LcalLcirc T_r, mathcal M,calM, mathcal McalMcirc T_s
+L, L T_r, M, M T_s
are CP.
Only (omega) is normalized. The vector (chi) may be unnormalized
@@ -268,17 +268,17 @@ and the maps may not preserve trace.
This is a representation property, not a physical claim. Lemma~%
-lem:ppt-stability shows that (mathcal L,calL,mathcal McalM) are also
+lem:ppt-stability shows that (L,M are also
PPT in the output-transpose convention. In particular (omega) itself
has positive partial transpose, since
((T_a ⊗ id_b)(omega)
-=((T_acircmathcal L)calL) ⊗ mathcal M)(calM)(chichi^*)succeq0).
+=((T_acircL) ⊗ M)(*)succeq0).
[Closure under tensor products]
-(mathscr CscrC) is closed under finite tensor products (after grouping
+(C is closed under finite tensor products (after grouping
all first-party systems and all second-party systems). In particular
-(omega^{ ⊗ n}inmathscr CscrC) for every (omegainmathscr CscrC)
+(omega^{ ⊗ n}inC for every (omegainC
and (nge1).
@@ -303,7 +303,7 @@ 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))inmathscr CscrC
+((Ω_bΩ_b^*)/(lambda))inC
on (C^a ⊗ C^c).
@@ -332,7 +332,7 @@ which is allowed.
The first step of the operational obstruction is a purely algebraic
-inequality about any state in (mathscr CscrC) and four arbitrary positive
+inequality about any state in (C 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
@@ -361,7 +361,7 @@ le(sum_knorm{a_k}^2)^{1/2}(sum_knorm{b_k}^2)^{1/2}
[Four-effect inequality]
-Let (omegainmathscr CscrC) act on (C^p ⊗ C^q). For arbitrary
+Let (omegainC 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
@@ -373,8 +373,8 @@ No normalization is assumed.
Proof. [Strategy]
-Represent (omega=(mathcal LcalL ⊗ mathcal M)(calM)(chichi^*)) with
-(mathcal L,calL,mathcal McalM) input-transpose CP. Pull the (X_i,Y_j) back
+Represent (omega=(L ⊗ M)(*)) with
+(L,M 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
@@ -386,10 +386,10 @@ terms on the RHS.
Proof.
Fix a representation
-(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.
+(omega=(L ⊗ M)(*)) with
+(L T_r), (M T_s) CP. Pull back:
+(A_i=L^ r),
+(B_j=M^ s). These are positive.
Normalization on the support.:
Let (H=A_0+A_1) and (Pi) be the orthogonal projection onto
@@ -401,14 +401,14 @@ 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 LcalL) by (mathcal LcalLcircAd_S) and (chi) by
+(L by (L 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
+(L'calL'circ T_r=(L T_r)circAd_{overlineoverline S}S) is
CP. Suppress the primes.
Two block estimates.:
-Let (C=mathcal LcalL ⊗ mathcal McalM). For vectors
+Let (C=L ⊗ M 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
@@ -465,7 +465,7 @@ Off-diagonal: (Z(v,w)) is bounded by (eq:reversed-block-bound) and
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 CscrC). Its lack of normalization is exactly what lets us apply it
+(C Its lack of normalization is exactly what lets us apply it
at each complete public transcript.
# Complete public transcripts
@@ -492,15 +492,15 @@ Suppose the protocol is finite (only finitely many steps, all classical
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.
+(A_i^t on her input; Bob's into (B_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^tcalA_i^t ⊗ mathcal B_j^tcalB_j^t). Public-source weights are
+(A_i^t ⊗ B_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)^calA_i^t)^dagger(I)) and
-(Y_j(t)=(mathcal B_j^t)^calB_j^t)^dagger(I)). These are positive matrices, and
+Set (X_i(t)=(A_i^t)^ and
+(Y_j(t)=(B_j^t)^ 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
[
@@ -547,13 +547,13 @@ 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 NbbN)-indexed sequence in standard Borel coordinate spaces. Local
+(N-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.
Public steps and the complete record. Public discussion is
-correspondingly finite or (mathbb NbbN)-indexed. Each message is
+correspondingly finite or (N-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
@@ -664,18 +664,18 @@ finite-prefix laws. Standard-Borel kernel sampling is standard
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=calF_m=sigma(T_m)).
+function of one local tape and (T). Let (F_m=
A step sent by Alice makes the new record measurable with respect to
-(sigma(U)veemathcal F_mcalF_m); Bob's symmetric. A public-source step
+(sigma(U)veeF_m; Bob's symmetric. A public-source step
has the sampling kernel as its regular conditional law. Induction shows
-(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]).
+(Uperp VmidF_m for every (m), and passing to
+(F_∞=calF_∞=sigma(T)) in (L^1) gives
+(E[u(U)v(V)bbE[u(U)v(V)mid T]=E[u(U)bbE[u(U)mid T]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
testing against countable generating algebras, we get
-(mathsf P_sfP_{U,Vmid T=t}=mathsf P_sfP_{Umid T=t} ⊗ mathsf P_sfP_{Vmid T=t})
+(P_{U,Vmid T=t}=P_{Umid T=t} ⊗ P_{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).
@@ -722,7 +722,7 @@ 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(,·,)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
@@ -749,24 +749,24 @@ null set if necessary. Transposition preserves positivity, giving
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:calR:mathcal T(calT(widetilde E)tomathcal T(E)calT(E)) mapping
+(R:calR:T( E)toT(E) mapping
(|psi>) to (|Psi_R>). If Alice and Bob run the same
admitted protocol and Eve retains her purification and full transcript,
-(mathcal RcalR ⊗ id_T) recovers the canonical output of
-Lemma~lem:transcript. Moreover (mathcal RcalR) maps every ideal
+(R ⊗ id_T) recovers the canonical output of
+Lemma~lem:transcript. Moreover (R maps every ideal
secret-bit law with Eve in (widetilde ET) to one with Eve in (ET).
Proof.
-Let (mathcal S=calS=supp(R^T)⊆ E). Schmidt decomposition gives
-an isometry (V:mathcal ScalStowidetilde E) with
+Let (S=⊆ E). Schmidt decomposition gives
+an isometry (V:S 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.
+R(*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 RcalR) is a channel. Its
+fixed state. Trace is preserved, so (R 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
@@ -775,7 +775,7 @@ the formula extends to trace-class inputs on larger (widetilde E).
An ideal bit has form
(frac12sum_{i}|i,i>< i,i| ⊗ sigma); conjugating by
-(mathcal RcalR ⊗ id_T) replaces (sigma) by a normalized
+(R ⊗ 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.
@@ -784,7 +784,7 @@ bound from the canonical to the general case.
The coherence inequality applies at each transcript. Take
-(mathcal ScalS) = the trace-norm distance from the actual transcript joint
+(S = 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
@@ -855,7 +855,7 @@ The transpose identity follows from functional calculus.
[Uniform obstruction to a secret bit]
-Let (omegainmathscr CscrC). For every integer (nge1), let an
+Let (omegainC 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
@@ -870,7 +870,7 @@ Thus the traditional trace distance is at least (1/10), and
Proof.
Group Alice's and Bob's factors, and set (R=omega^{ ⊗ n}). Lemma~%
-lem:tensor-closure gives (Rinmathscr CscrC). Apply Eve's classical
+lem:tensor-closure gives (RinC 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).
@@ -949,7 +949,7 @@ all sufficiently large (n), and (K_D(omega)=0).
[Conditioning, aborts, completed-output limits]
-Consider inputs (omega^{ ⊗ n}) with (omegainmathscr CscrC) and
+Consider inputs (omega^{ ⊗ n}) with (omegainC and
(nge1), and admitted protocols whose normalized completed outputs
retain an arbitrary initial purification and the full transcript.
If (A) is a measurable transcript event with probability
@@ -961,11 +961,11 @@ retain an arbitrary initial purification and the full transcript.
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 CscrC).
+ bits. Such a sequence is impossible for inputs in (C
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_kcalR_k). If
+ a common (F) with embeddings (R_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).
@@ -983,17 +983,17 @@ and root fidelity scale by the same factor. The proof of
(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)
+channels (Q 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
+(norm{Q(-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_kcalR_k) transfers ideal bits:
-(mathcal R_k:calR_k:sigmamapstosigma_{mathcal R_k}calR_k). Contraction and the
+(3) Recovery channel (R_k transfers ideal bits:
+(R_k:calR_k:sigma|->sigma_{R_k} Contraction and the
per-protocol gap give
-(norm{widehattau_k-gamma_sigma}_1genorm{tau_k-gamma_{mathcal R_k(calR_k(sigma)}}_1ge1/5).
+(norm{widehattau_k-gamma_sigma}_1genorm{tau_k-gamma_{R_k(}}_1ge1/5).
Then
(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
@@ -1021,11 +1021,11 @@ projective direction ([x]) is the complex line through (xne0).
[Tensor criterion]
Let (L:Mat 4toMat 4) be PPT. Suppose
-(mathcal X=calX={x_1,...,x_{20}}subsetC^4∖{0}) represents
+(X={x_1,...,x_{20}}subsetC^4∖{0}) represents
twenty distinct projective directions with:
- (rank L(P_x)<4) for every (xinmathcal XcalX), and every nonzero
+ (rank L(P_x)<4) for every (xinX, and every nonzero
homogeneous quadratic in (C[x_0,x_1,x_2,x_3]) vanishes on at
- most nine members of (mathcal XcalX);
+ most nine members of (X;
(L(P_y)) is positive definite for some (yinC^4).
Then there are explicitly constructed PPT maps
@@ -1060,7 +1060,7 @@ 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 (hathat xx)'s are independent; equivalently the ten
-(hathat xx)'s span (C^{10}) for any ten of the (xinmathcal XcalX).
+(hathat xx)'s span (C^{10}) for any ten of the (xinX
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).
@@ -1072,7 +1072,7 @@ 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 (wmapsto Koverlineoverline ww) is the
+so (K=K^*=K^{-1}). The conjugation (w|-> Koverlineoverline ww) is the
complementary-pair exterior dual used by Schliemann, Loss, and
MacDonald~[cite].
@@ -1123,16 +1123,16 @@ conjugate-linear in the first argument, the definition of (K) gives
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).
+(xinX (including (dim W_x<2), when both are zero).
Hence
[
(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
+for (xinX Positive (Z) with (v^*Zv=0) implies
(Zv=0), so
-hathat xx ⊗ hathat xxinker Zqquad(xinmathcal X).calX).
+hathat xx ⊗ hathat xxinker Z (xinX).
For the second hypothesis' input (y), (S(P_{hathat y}y)=∧^2L(P_y))
@@ -1143,10 +1143,10 @@ is positive definite, as is (R(P_{hathat y}y)=KS(P_{hathat y}y)^T K^*), so
Excluding product vectors from the range.
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)p_v(x)=0) at every (xinX, 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
+vanishes on at most nine of (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.
@@ -1187,7 +1187,7 @@ M_3&=
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_jqquad(0le i,j<4).
+L(E_{ij})=M_i^{T}M_j (0le i,j<4).
@@ -1220,7 +1220,7 @@ the rank identity. Finally (M_0^{T}M_0=36I_4).
[Twenty directions]
-The projective directions ([x]inmathbb P^3(bbP^3(C)) with
+The projective directions ([x]inP^3( 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.
@@ -1251,7 +1251,7 @@ local Frobenius lemma then gives Galois elements of cycle types
transitive; a transposition (seventeenth power of ((1,2,17))) plus
transitivity on ordered pairs gives (S_{20}).
[C] With (S_{20}) acting, the evaluation matrix
-(mathcal E_calE_{j,alpha}=(t^{(j)})^alpha) for (|alpha|le2) has
+(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
ten of the twenty triples, the rank-ten evaluation matrix would force
@@ -1271,7 +1271,7 @@ Any six complex bilinear forms on (C^4timesC^4) have a common zero
Proof.
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
+(R=[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
@@ -1324,18 +1324,18 @@ 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
+(O=[alpha]congZ_p[X]/(g)) where (alpha) is a root;
+(O is a free rank-(s) (Z_p)-module with
+(O/pcalO/pO~=F_{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 OcalO).
+polynomial lifts uniquely to a root in (O
Since (overlineoverline FF) is squarefree with roots in (F_{p^s}), we get
-(deg F) distinct roots in (mathcal OcalO). Finite fields are perfect,
+(deg F) distinct roots in (O Finite fields are perfect,
so (overlinealpha^p) is a simple root of (overlineoverline gg) and lifts to
-some (betainmathcal OcalO). Sending (alphamapstobeta) gives a
+some (betainO Sending (alpha|->beta) 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)
+Finiteness makes it an automorphism, and (sigma(O)= 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
@@ -1346,7 +1346,7 @@ giving the required rational Galois element.
## The multiplication operator and (f(z))
-Let (mathcal A=calA=Q[t_1,t_2,t_3]/I), (I) generated by the fifteen
+Let (A=[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
@@ -1354,7 +1354,7 @@ 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)
+(H) is invertible. Define (C=-H^{-1}D); in (A, (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):
@@ -1391,27 +1391,27 @@ transitivity; a transposition from the 17th power of ((1,2,17)) plus
## The affine quotient is a field
-The (w=Cv) relations force (mathcal AcalA) to be spanned by the twenty
+The (w=Cv) relations force (A 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:Q^{20}toA with (e_alpha|-> 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
+(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 AcalA):
+already all of (A:
-mathcal AcalAsimeqQ[z]/(f), zmapsto t_1.
+A~=Q[z]/(f), z|-> t_1.
-In particular (mathcal AcalA) is a field, and there are
+In particular (A 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:
-C[t_1,t_2,t_3]/IC[t_1,t_2,t_3]simeqC[z]/(f)
-simeqprod_{j=1}^{20}C,
+C[t_1,t_2,t_3]/IC[t_1,t_2,t_3]~=C[z]/(f)
+~=prod_{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
@@ -1419,16 +1419,16 @@ reduced.
## Quadratic evaluations via Galois
-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
+Let (K be a splitting field of (f),
+(G=Gal(K/bbK/Q)). The evaluation matrix
+(E_{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
+(K 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}).
+(sigmaE_{j,alpha}=E_{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)
+submatrix. Hence (E has column rank ten over (K
and over (C).
A nonzero homogeneous quadratic (q) dehomogenizes to a nonzero
@@ -1463,7 +1463,7 @@ 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.
+inC.
]
Theorem~thm:zero-key and Proposition~%
prop:operational-extensions give (eq:main-gap) and
@@ -1537,7 +1537,7 @@ Compression:
(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 ⊗ Bmapsto(W_0^*AW_0) ⊗ (W_0^*BW_0)) on products,
+(A ⊗ B|->(W_0^*AW_0) ⊗ (W_0^*BW_0)) on products,
(J(Psi_2circPsi_1)) would be separable. Contradiction if it is not.
@@ -1555,14 +1555,14 @@ Theorem~thm:main-pair's argument shows
[
S(P_{e_{00}})=R(P_{e_{00}})=36^2I_6,
(e_{00} ⊗ e_{00})^*Z(e_{00} ⊗ e_{00})
-=6cdot36^4=10{,}077{,}696>0.
+=6·36^4=10{,}077{,}696>0.
]
# An alternative construction from projection overlaps
The dimension-ten construction is done. This section builds another
-entangled member of (mathscr CscrC), in much larger dimension, by a
+entangled member of (C, 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
@@ -1580,21 +1580,21 @@ sec:projection's proof.
[Rectangular projection criterion]
-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
+Let (Q,K,D) be positive integers, (Q a set of size (Q).
+Take unit vectors (u_qinR^K for (qinQ, with
+(c_{qr}=u_q^{T}u_r). For each (q), ({P_i:iinI_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:
+Take the (I_q disjoint, (I=⊔_qcalI=⊔_qI_q,
+(n=|I|calI|). Suppose:
(P_iP_j=P_jP_i) whenever (c_{qr}ne0);
there is a real (kappa<Q/K) such that every
- (J⊆mathcal IcalI) with (tr(P_iP_j)>0) for all distinct
+ (J⊆I with (tr(P_iP_j)>0) for all distinct
(i,jin J) has (|J|lekappa).
Define (Psi:Mat ntoMat D) on matrix units by
H_{ij}=c_{qr}P_iP_j, Psi(E_{ij})=H_{ij}
-(iinmathcal I_q,calI_q, jinmathcal I_r).calI_r).
+(iinI_q, jinI_r).
Then (Psi,Psi^dagger) are PPT, and
@@ -1616,7 +1616,7 @@ Proof.
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_{iinmathcal I_q}calI_qu_q ⊗ P_iw_i}^2ge0.
+=norm{sum_{q}sum_{iinI_q} ⊗ P_iw_i}^2ge0.
So (Hsucceq0); Lemma~lem:choi-separable gives (Psi) CP.
@@ -1650,7 +1650,7 @@ 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_{iinmathcal I_q}calI_qsum_{jinmathcal I_r}calI_rtr(P_iP_j)=D). Summing
+(sum_{iinI_q}{jinI_r} Summing
over (q,r): (tr C=Q^2D). Also (tr B=QD), and
mathbf1^*Bbf1^*Bmathbf1=Dbf1=Dsum_{q,r}c_{qr}^2ge(D Q^2)/(K).
@@ -1695,7 +1695,7 @@ vector,
(C)/(Q^2D)=
((Λ)/(Q) ⊗ (Λ)/(Q))(Phi_DPhi_D^*)
-is an entangled density operator in (mathscr CscrC).
+is an entangled density operator in (C
Proof.
@@ -1744,10 +1744,10 @@ orthogonal eight-set into a forbidden positive-overlap choice.
Set (p=257), (k=4p=1028), ([k]={1,...,k}). Define
-mathcal U=calU={A⊆[k]:|A|=2p, 1in A},
-b=|mathcal U|=calU|=binom{1027}{513}=frac12binom{4p}{2p},
+U={A⊆[k]:|A|=2p, 1in A},
+b=|U|=calU|=binom{1027}{513}=frac12binom{4p}{2p},
-and (s_A=k^{-1/2}(2mathbf1_bf1_{{hin A}}-1)_{h=1}^kinmathbb R^kbbR^k).
+and (s_A=k^{-1/2}(2mathbf1_bf1_{{hin A}}-1)_{h=1}^kinR^k
Each (s_A) is a unit vector; by counting agreeing entries,
s_A^{T}s_B=(k-2|A△ B| )/(k)
@@ -1762,12 +1762,12 @@ elementary multilinear methods.
[Restricted intersections]
-If (mathcal F⊆calF⊆mathcal UcalU) has no two distinct sets with
-orthogonal associated vectors, then (|mathcal F|calF|le a_0=delta b).
+If (F⊆calF⊆U has no two distinct sets with
+orthogonal associated vectors, then (|F|calF|le a_0=delta b).
Proof.
-For distinct (A,Binmathcal FcalF), common element (1) and equal sizes
+For distinct (A,BinF, 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
@@ -1779,52 +1779,52 @@ on Boolean vectors are unchanged and degrees drop to (le p-1).
Independence follows by evaluating any linear relation at each
(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).
+(|F|calF|le a_0).
[Fiber amplification]
-For (t=1,2,3), to ((A_1,...,A_t)inmathcal U^tcalU^t) associate the
+For (t=1,2,3), to ((A_1,...,A_t)inU^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)
+(R⊆calR⊆U^t contains no collection of (2^t)
distinct tuples whose associated tensors are pairwise orthogonal, then
-(|mathcal R|calR|le tdelta b^t).
+(|R|calR|le tdelta b^t).
Proof.
(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
+(R_A={(A_2,...,A_t):(A,A_2,...,A_t)inR}).
+Let (G be those (A) for which (R_A contains
+(2^{t-1}) pairwise orthogonal tuples. Two (A,BinG 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
+tuples; so (G has no orthogonal pair, forcing
+(|G|calG|le a_0=delta b). Fibers in (G have
(le b^{t-1}) elements; other fibers have (le(t-1)delta b^{t-1}).
Summing gives
[
-|mathcal R|calR|le(delta b)b^{t-1}+b(t-1)delta b^{t-1}=tdelta b^t.
+|R|calR|le(delta b)b^{t-1}+b(t-1)delta b^{t-1}=tdelta b^t.
]
Set questions:
-mathcal Q=calQ=mathcal U^3,calU^3, Q=b^3, K=k^3=1028^3=1{,}086{,}373{,}952,
+Q=, Q=b^3, 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 QcalQ)
+(q=(A_1,A_2,A_3)). An orthogonal eight-set is (S⊆Q
of size eight with pairwise orthogonal (u_q)'s. Lemma~%
lem:fiber-amplification at (t=3) gives
-|mathcal R|calR|lekappa:=3delta Q=3a_0b^2
+|R|calR|lekappa:=3delta Q=3a_0b^2
-for any (mathcal R⊆calR⊆mathcal QcalQ) with no orthogonal eight-set.
+for any (R⊆calR⊆Q with no orthogonal eight-set.
The analytic bound: central binomial
(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):
+(ple85·3+2), ((3/4)^3<1/2):
(3delta<2^{-33}<1/K), giving
kappa=3a_0b^2<frac QK.
@@ -1878,27 +1878,27 @@ last column contradicts the row product (+1).
## Assembling the tensor product
-For each 8-subset (S⊆mathcal QcalQ), a separate (C^4) factor.
+For each 8-subset (S⊆Q, 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
non-membership questions get trivial measurements.
-mathcal S=calS={S⊆mathcal Q:|S|=8calQ:|S|=8},
-L=|mathcal S|=calS|=binom Q8,
-mathcal H=⊗_calH=⊗_{Sinmathcal S}calSC^4congC^D, D=4^L.
+S={S⊆Q:|S|=8calQ:|S|=8},
+L=|S|=calS|=binom Q8,
+H=⊗_calH=⊗_{SinS}, D=4^L.
-For each (qinmathcal QcalQ) and outcome string
-(lambdain{-1,+1}^{2timesmathcal S}calS),
+For each (qinQ and outcome string
+(lambdain{-1,+1}^{2timesS},
-mathcal I_q=calI_q={q}timesmathcal A^calA^{mathcal S}calS,
-P_{q,lambda}=⊗_{Sinmathcal S}calSh_{S,q}(lambda_S).
+I_q={q}timesA^{S},
+P_{q,lambda}=⊗_{SinS}{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=|⊔_qmathcal I_q|=QDcalI_q|=QD).
+question, so (n=|⊔_qI_q|=QDcalI_q|=QD).
The (u_q) and (P_{q,lambda}) satisfy Proposition~%
@@ -1938,7 +1938,7 @@ rho_{comb}
=(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
+(C, with (K_D(rho_{comb})=0) and the uniform
unhalved (1/5) gap.
For (U:C^DtoC^m) the first-coordinate isometry,
(Phi_1^{comb}=Ad_UcircPsi),