diff b1f9bc4..56bc728 zero_distillable_entanglement_v2.txt
diff --git a/zero_distillable_entanglement_v2.txt b/zero_distillable_entanglement_v2.txt
index 9d8cce0..845a139 100644
--- a/zero_distillable_entanglement_v2.txt
+++ b/zero_distillable_entanglement_v2.txt
@@ -156,7 +156,7 @@ Our Choi matrix convention is
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.
+ Ω_a=sum_{i=1}^a e_i ⊗ e_i.
So (norm{Ω_a}^2=a), and the first tensor factor is the input
reference. This fixes every dimensional factor in the transfer formula
@@ -203,9 +203,8 @@ Products follow from (T_{b_1b_2}=T_{b_1} ⊗ T_{b_2}).
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 (kge1) and (Wsucceq0) in (Mat k ⊗ Mat a),
+ (J(F)) is separable;
+ for every (kge1) and (Wsucceq0) in (Mat k ⊗ Mat a),
((id_k ⊗ F)(W)) is separable.
Also (J(T_bcirc F)=(id_a ⊗ T_b)J(F)) and
@@ -260,7 +259,7 @@ A density operator (omega) on (C^a ⊗ C^b) belongs to
that
omega=(mathcal LcalL ⊗ mathcal M)(calM)(chichi^*),
-qquad
+
mathcal L,calL, mathcal LcalLcirc T_r, mathcal M,calM, mathcal McalMcirc T_s
are CP.
@@ -441,8 +440,8 @@ onto the eigenvectors where the (A_0)-eigenvalue is at least the
(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),
+p_{11}(v)le p_{01}(v), p_{00}(w)le p_{10}(w).
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
@@ -453,7 +452,7 @@ Diagonal terms: (eq:direct-block-bound) with (eq:spectral-weights)
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
@@ -487,7 +486,7 @@ 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 Econg A ⊗ B.
+ Econg A ⊗ B.
Suppose the protocol is finite (only finitely many steps, all classical
records finite) and produces bits (i,jin{0,1}) almost surely on
@@ -528,7 +527,6 @@ consists of local quantum instruments, local memories and randomness, and
authenticated two-way public classical communication. Data and joint
laws satisfy:
-item
Initial resources. A recorded standard Borel public variable
(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.
@@ -536,14 +534,14 @@ 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
+
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
+
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
@@ -553,7 +551,7 @@ outcomes and public records, is a finite sequence or an
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
+
Public steps and the complete record. Public discussion is
correspondingly finite or (mathbb NbbN)-indexed. Each message is
produced by its sender's local process from that sender's private
@@ -565,14 +563,14 @@ registers. Public choices of speaker and stopping decisions use these
same two rules. (T) records (C_0), all public messages, draws, speaker
choices, and stopping decisions. Private outcomes and quantum memories
stay home.
-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
+
Terminal outputs. A terminal classical output may be a measurable
function of that laboratory's instrument history and received (T). A
terminal quantum readout is supplied with its incoming memory (via one
@@ -585,7 +583,7 @@ 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
+
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
@@ -639,8 +637,8 @@ Set (a=dim A), (b=dim B), and attach untouched references
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_a>=a^{-1/2}sum_{r=1}^a |r,r>,
|Phi_b>=b^{-1/2}sum_{s=1}^b |s,s>.
Assign bit zero in a laboratory if its prescribed bit is never produced,
@@ -700,8 +698,8 @@ 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_jtr_{B_0}W_{ij}(t),quad
-V_j(t)=sum_itr_{A_0}W_{ij}(t),quad
+U_i(t)=sum_jtr_{B_0}W_{ij}(t),
+V_j(t)=sum_itr_{A_0}W_{ij}(t),
sum_itr U_i(t)=sum_jtr V_j(t)=1.
]
The conditional POVM statistics and their factorized form give
@@ -917,7 +915,7 @@ the common transcript readout, and define
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 qle a_0+a_1=eta-e.
+eta=e+a_0+a_1, qle a_0+a_1=eta-e.
Trace conservation gives:
since actual equal-bit branches have total trace (1-e) while ideal
@@ -934,7 +932,7 @@ last give
(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)lemathfrak ffrakfle√(e(1-e))+e,qquad elefraceta2.
+(1-eta)/(2)lemathfrak ffrakfle√(e(1-e))+e, elefraceta2.
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).
@@ -954,18 +952,17 @@ all sufficiently large (n), and (K_D(omega)=0).
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
+ 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
+ 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 CscrC).
-item Let (tau_k) be completed normalized bit outputs to which the
+ 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
@@ -1026,11 +1023,10 @@ projective direction ([x]) is the complex line through (xne0).
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 (xinmathcal XcalX), and every nonzero
+ (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).
+ (L(P_y)) is positive definite for some (yinC^4).
Then there are explicitly constructed PPT maps
(Phi_1,Phi_2:Mat{10}toMat{10}) with
@@ -1048,9 +1044,9 @@ 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),
+U e_{ij}&=(e_i ⊗ e_j+e_j ⊗ e_i)/(sqrt2)(i<j),
notag
-V e_{ij}&=(e_i ⊗ e_j-e_j ⊗ e_i)/(sqrt2)quad(i<j).
+V e_{ij}&=(e_i ⊗ e_j-e_j ⊗ e_i)/(sqrt2)(i<j).
Their ranges are the symmetric and antisymmetric tensor subspaces. Put
@@ -1082,7 +1078,7 @@ MacDonald~[cite].
Define (S,R:Mat{10}toMat 6) by
-S(A)=V^*(L ⊗ L)(UAU^*)V,qquad
+S(A)=V^*(L ⊗ L)(UAU^*)V,
R(A)=K,S(A)^T,K^*.
Embeddings and compressions are CP; (L) PPT gives (L ⊗ L) PPT;
@@ -1096,7 +1092,7 @@ The rectangular maps give an endomorphism
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_1(A)=E,S(A^T),E^*,
Phi_2(B)=R^dagger(E^* B E).
The stability lemma gives (Phi_1,Phi_2) PPT. Since (E^*E=I_6),
@@ -1199,7 +1195,7 @@ L(E_{ij})=M_i^{T}M_jqquad(0le i,j<4).
(L) is PPT. For every (xinC^4),
-L(P_x)=M(x)^{T}M(x)bar=M(x)^*M(x),qquad
+L(P_x)=M(x)^{T}M(x)bar=M(x)^*M(x),
rank L(P_x)=rank M(x).
@@ -1239,13 +1235,12 @@ product rays (x ⊗ y) there are different from the twenty
Proof. [Proof of Proposition~prop:twenty-directions]
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
+[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 (xne0,yne0).)
-item[B] A (20times20) multiplication operator (N) on that span
+[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),
@@ -1255,7 +1250,7 @@ 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
+[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
@@ -1364,7 +1359,7 @@ 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=e_{alpha+(1,0,0)} (for |alpha|le2),
Ne_alpha=textstylesum_{|beta|le3}C_{alpha+(1,0,0),beta}e_beta
(for |alpha|=3).
]
@@ -1376,13 +1371,13 @@ twenty-one characteristic coefficients and the modular gcd degrees
Exact modular data (see Section~app:arithmetic for recorded values):
{ccll}
-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.
@@ -1406,7 +1401,7 @@ Irreducibility makes (N) have no proper invariant subspace, so
polynomial of (N) equals (f), the subalgebra generated by (t_1) is
already all of (mathcal AcalA):
-mathcal AcalAsimeqQ[z]/(f),qquad zmapsto t_1.
+mathcal AcalAsimeqQ[z]/(f), zmapsto 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)),
@@ -1558,7 +1553,7 @@ 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,
(e_{00} ⊗ e_{00})^*Z(e_{00} ⊗ e_{00})
=6cdot36^4=10{,}077{,}696>0.
]
@@ -1591,21 +1586,20 @@ Take unit vectors (u_qinmathbb R^KbbR^K) for (qinmathcal QcalQ), with
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
+ (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
(i,jin J) has (|J|lekappa).
Define (Psi:Mat ntoMat D) on matrix units by
-H_{ij}=c_{qr}P_iP_j,quad Psi(E_{ij})=H_{ij}
-quad(iinmathcal I_q,calI_q, jinmathcal I_r).calI_r).
+H_{ij}=c_{qr}P_iP_j, Psi(E_{ij})=H_{ij}
+(iinmathcal I_q,calI_q, jinmathcal I_r).calI_r).
Then (Psi,Psi^dagger) are PPT, and
C:=J(Psi^daggercircPsi) is nonseparable over C^n ⊗ C^n,
-qquad tr C=Q^2D.
+ tr C=Q^2D.
@@ -1630,7 +1624,7 @@ 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, H^{Gamma_2}=H^{T}succeq0,
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
@@ -1692,7 +1686,7 @@ suppose every (P_i) is real symmetric. Put (Λ=Psi^dagger).
Then
Λcirc T_D=Λ=T_ncircΛ,
-qquad trΛ(A)=Q,tr A.
+ trΛ(A)=Q,tr A.
So (Λ/Q:Mat DtoMat n) is a trace-preserving PPT channel.
With (Phi_D=Ω_D/sqrt D) the normalized maximally entangled
@@ -1721,7 +1715,7 @@ Under Proposition~prop:projection-criterion's hypotheses with
(nge D), let (U:C^DtoC^n) be the real coordinate isometry onto
the first (D) coordinates. With
-Xi_1=Ad_UcircPsi,qquad
+Xi_1=Ad_UcircPsi,
Xi_2=Psi^daggercircAd_{U^*}=Xi_1^dagger,
the maps (Xi_1,Xi_2:Mat ntoMat n) are PPT and
@@ -1750,7 +1744,7 @@ 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},quad
+mathcal U=calU={A⊆[k]:|A|=2p, 1in A},
b=|mathcal 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).
@@ -1815,7 +1809,7 @@ Summing gives
Set questions:
-mathcal Q=calQ=mathcal U^3,calU^3,quad Q=b^3,quad K=k^3=1028^3=1{,}086{,}373{,}952,
+mathcal Q=calQ=mathcal U^3,calU^3, 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)
@@ -1891,14 +1885,14 @@ 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},quad
-L=|mathcal S|=calS|=binom Q8,quad
-mathcal H=⊗_calH=⊗_{Sinmathcal S}calSC^4congC^D,quad D=4^L.
+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.
For each (qinmathcal QcalQ) and outcome string
(lambdain{-1,+1}^{2timesmathcal S}calS),
-mathcal I_q=calI_q={q}timesmathcal A^calA^{mathcal S}calS,quad
+mathcal I_q=calI_q={q}timesmathcal A^calA^{mathcal S}calS,
P_{q,lambda}=⊗_{Sinmathcal S}calSh_{S,q}(lambda_S).
where (h_{S,q}) is the measurement above (or trivial). These are real
@@ -1937,8 +1931,7 @@ Take the explicit data:
(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
+ (Λ/Q) is a trace-preserving PPT channel, and
rho_{comb}
=(Λ/Q ⊗ Λ/Q)(Phi_DPhi_D^*)
@@ -1947,12 +1940,12 @@ rho_{comb}
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,
+ 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
+ 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
nonseparable corner
@@ -1989,14 +1982,13 @@ 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:
-
-item an assertion of the block-symmetry identity
+ 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
+ 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
+ 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).
@@ -2030,21 +2022,20 @@ 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
+ 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
+ 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
+ (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)=20ne0)
-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))
+41 & 0,...,0, 20 (only d_{41}(20)=20ne0)
+131 & 1,,3,,0,,0,...,0, 18 (at k=1,2,17)
+139 & 1,,0,...,0, 20 (at k=1,19)
]
and the residues of (det H) are (40, 23, 9) modulo (41,131,139),
@@ -2086,7 +2077,7 @@ 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 zero-overlap pairs & 1 & 3 & 5 & 7 & 9
# of selections & 144 & 1344 & 2016 & 576 & 16
]
@@ -2107,89 +2098,89 @@ objects are specified and analyzed by the finite formulas in the text.
{99}
-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}
+
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,
``When do composed maps become entanglement breaking?''
Ann. Henri Poincar'e 20 (2019), 2295--2322.
-bibitem{DevetakWinter05}
+
I.~Devetak and A.~Winter, ``Distillation of secret key and entanglement
from quantum states,'' Proc. Roy. Soc. A 461 (2005),
207--235.
-bibitem{Filippov2021}
+
S.~N.~Filippov, ``Capacity of trace decreasing quantum operations
and superadditivity of coherent information for a generalized erasure
channel,'' J. Phys. A 54 (2021), 255301.
-bibitem{FR1987}
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