round161-support-two-pass-frozen

Immutable full-result SHA256: 915d62a50575642e0051957359e85ba5455473d85d1360989b970a57134a3f91

STATEMENT
Let e=2^(-16), p=4/5, A=diag(1-e,e), and B=H A H, where H=2^(-1/2)[[1,1],[1,-1]]. On C^2 direct-sum C^2 define rho=diag(pA,(1-p)B) and sigma=diag((1-p)B,pA). These are faithful noncommuting density operators. Their fixed-single-copy measured Chernoff exponent satisfies C_M(rho,sigma)<log(5/4)+454/1000. A fixed two-outcome first instrument extracting the direct-sum label, followed by the explicit admissible two-pass policy described below, has liminf error exponent greater than log(5/4)+457/1000. Consequently underline-xi_LOCC and underline-xi_SEP are both strictly larger than C_M, by more than 3/1000. In particular Eq. (4) is false. The policy retains all n conditional qubits after its first pass and visits each exactly once thereafter.

PROOF
All logarithms are natural. We first prove the arbitrary-POVM bound used in the certificate, then describe and analyze the protocol.

1. An elementary measured-coefficient bound. For density matrices X,Y>0, 1/2<=alpha<1, beta=(1-alpha)/alpha, and any finite POVM {M_i}, write x_i=Tr X M_i and y_i=Tr Y M_i. Then
sum_i x_i^alpha y_i^(1-alpha) >= (Tr X Y^beta)^alpha.                                      (A)
Zero effects may be omitted. Scalar minimization gives the left side as the infimum over t_i>0 of alpha sum_i x_i t_i^(-beta)+(1-alpha)sum_i y_i t_i. Set G=sum_i t_i M_i. Operator Jensen for the operator-convex function t^(-beta) gives sum_i t_i^(-beta)M_i>=G^(-beta). Put v=Tr YG. Since G^(1/2)YG^(1/2)<=vI, Y<=vG^(-1). Operator monotonicity of t^beta gives Y^beta<=v^beta G^(-beta). Therefore each expression being minimized is at least alpha c v^(-beta)+(1-alpha)v, where c=Tr X Y^beta. Minimizing this scalar expression over v>0 gives c^alpha, proving (A). The operator properties used here hold for 0<=beta<=1: they follow from the positive integral representations of t^beta and t^(-beta) in terms of resolvents; Jensen for the inverse follows from the Schur complement. Thus no assertion about an optimal POVM or a saddle point is needed.

2. Applying (A) to the four-dimensional pair. Refining any POVM by the two direct-sum blocks can only decrease its coefficient, by scalar Holder applied to the two contributions to each original outcome. The refined POVM restricts to a POVM on each block. The Hadamard unitary interchanges A and B, and
Tr A B^beta=Tr B A^beta=((1-e)^beta+e^beta)/2.
For any Chernoff parameter s, put alpha=max(s,1-s). For 0<s<1, apply (A), interchanging X,Y when necessary. Every original POVM coefficient is at least
[p^alpha(1-p)^(1-alpha)+p^(1-alpha)(1-p)^alpha] [((1-e)^beta+e^beta)/2]^alpha.
For p=4/5 the bracket in front is (4/5)cosh((2alpha-1)log 2). Define, for 1/2<=alpha<=1,
h(alpha)=log cosh((2alpha-1)log 2)-alpha log 2+alpha log(1+2^(-16(1-alpha)/alpha)).
Because (1-e)^beta+e^beta >= (1-e)^beta(1+e^beta), the logarithm of the displayed coefficient is at least
log(4/5)+h(alpha)+(1-alpha)log(1-e).
We now establish uniformly that the last two terms exceed -454/1000.

Let l=log 2 and x=2^(-16/3). At alpha=3/4,
h(3/4)=log 3-(9/4)l+(3/4)log(1+x),
h'(3/4)=-l/3+log(1+x)+(64l/3)x/(1+x).
The rational estimates
6931/10000<log 2<6932/10000,
log 3>10986/10000,
248/10000<x<249/10000,
and x-x^2/2<=log(1+x)<=x imply
h(3/4)>-443/1000 and 0<h'(3/4)<77/500.
The last term in h is a convex perspective of log(1+exp(u)). The first term has second derivative 4l^2 sech^2((2alpha-1)l), which is at least (64/25)l^2>6/5 throughout [1/2,1]. Hence strong convexity and completing the square give
h(alpha)>-443/1000-(77/500)^2/(12/5)>-453/1000.
Also log(1-e)>=-e/(1-e)=-1/65535, so (1-alpha)log(1-e)>=-1/131070. In fact the preceding completed-square lower bound minus 1/131070 is greater than -454/1000. Thus every coefficient for every finite POVM and every interior s is bounded below by a common constant strictly greater than (4/5)exp(-454/1000). At s=0,1 the coefficient is 1. Taking the infimum over measurements and s proves
C_M<log(5/4)+454/1000.
All strict margins above are checked using rational arithmetic in the accompanying certificate; logarithm brackets follow from explicitly bounded positive series.

3. The fixed first instrument and the admissible policy. Write V_j:C^4->C^2 for extraction of direct-sum block j, j=0,1, and let E_j(X)=V_j X V_j^*. These CP maps have trace-preserving sum in the instrument sense. Independently apply this same instrument to all copies. Under rho, the labels have probabilities (p,1-p), with conditional qubits (A,B); under sigma they have probabilities (1-p,p), with conditional qubits (B,A).
Let k be the number of labels 0. If k>=n/2, each party measures its retained qubit in the eigenbasis of its rho-conditional state: the A basis at label 0 and the B basis at label 1. Otherwise each measures in the eigenbasis of its sigma-conditional state. Outcomes are ordered with the larger eigenvalue first. Finally perform the classical optimal equal-prior likelihood decision using all labels and outcomes. The visit order may be fixed. There is one final measurement per party and no subsequent quantum revisit. This is finite-round LOCC, with no joint input operation or entangled resource. It is permissible to use only the label count: this specifies a particular policy, not an assumption that all policies depend only on type.

4. Uniform asymptotic performance. Put P=(1-e,e) and Q=(1/2,1/2). In the k>=n/2 branch, conditional final outcomes are independent with laws P under rho and Q under sigma, irrespective of the positions of the labels. Set
q=sum_i P_i^(1/4) Q_i^(3/4)=2^(-3/4)((1-e)^(1/4)+1/16).
For a particular label string, its probabilities are a=p^k(1-p)^(n-k) and b=(1-p)^k p^(n-k). The sum over final outcomes of the minimum of the two full transcript probabilities is at most a^(1/4)b^(3/4)q^n, by min(u,v)<=u^(1/4)v^(3/4).
Writing t=k/n, the binomial bound binom(n,k)<=exp(n H(t)) shows
binom(n,k)a^(1/4)b^(3/4)<=exp(n f(t)),
where f(t)=H(t)+(1/4)[t log p+(1-t)log(1-p)]+(3/4)[t log(1-p)+(1-t)log p].
For t>=1/2, f'(t)=log((1-t)/t)-(1/2)log(p/(1-p))<0, and f(1/2)=log(2 sqrt(p(1-p)))=log(4/5). Thus this entire type contributes at most ((4/5)q)^n.
For k<n/2, the final laws are Q under rho and P under sigma. Using min(u,v)<=u^(3/4)v^(1/4) gives the same residual coefficient q. The corresponding type function is f(1-t), giving the same bound. Summing over all n+1 types and including the equal-prior factor proves, for every n,
p_e(policy,n)<=(n+1)((4/5)q)^n/2.
Consequently its lower exponent is at least E=log(5/4)-log q. Since (1-e)^(1/4)<1,
-log q>(3/4)log 2-log(17/16)> (3/4)(6931/10000)-1/16>457/1000.
This proves the stated strict gap. Optimal LOCC error is no larger than the policy error, and every LOCC test is SEP, so the same lower-exponent lower bound holds for both original classes.

Finally A and B are positive definite and [A,B] is nonzero because e is neither 0 nor 1/2. The two blocks of [rho,sigma] are nonzero multiples of [A,B] and [B,A]. Thus the example satisfies exactly the faithful, noncommuting assumptions of the original question.

REMAINING GAP
The exact optimal unrestricted LOCC and SEP exponents, their possible equality with each other, and existence of their limits for this pair and general pairs are not determined. The universal equality with C_M is disproved.

CHECKER
from fractions import Fraction as F

def log_bounds(t, N):
    # log((1+t)/(1-t)); positive series and geometric tail bound.
    lo = 2*sum((t**(2*k+1)/F(2*k+1) for k in range(N)), F(0))
    hi = lo + 2*t**(2*N+1)/(F(2*N+1)*(1-t*t))
    return lo, hi

l2,u2=F(6931,10000),F(6932,10000)
l3=F(10986,10000)
a,b=log_bounds(F(1,3),12)
assert l2<a<b<u2
assert log_bounds(F(1,2),16)[0]>l3
xl,xu=F(248,10000),F(249,10000)
e=F(1,65536)
assert xl**3<e<xu**3
logx_lower=xl-xu*xu/2
h0_lower=l3-F(9,4)*u2+F(3,4)*logx_lower
assert h0_lower>-F(443,1000)
d_lower=-u2/3+logx_lower+F(64,3)*l2*xl/(1+xu)
d_upper=-l2/3+xu+F(64,3)*u2*xu/(1+xl)
assert 0<d_lower<d_upper<F(77,500)
assert F(64,25)*l2*l2>F(6,5)
h_lower=-F(443,1000)-F(77,500)**2/F(12,5)
assert h_lower>-F(453,1000)
assert h_lower-F(1,131070)>-F(454,1000)
policy_lower=F(3,4)*l2-F(1,16)
assert policy_lower>F(457,1000)
assert policy_lower-F(454,1000)>F(3,1000)
print('All rational bounds verified; certified exponent gap > 3/1000.')

EXPECTED CHECKER OUTPUT
All rational bounds verified; certified exponent gap > 3/1000.

Coordinator assessment: reports/authorization-history/20261016T084317Z-through161-chernoff-support/preflight161/round161-support-two-pass-frozen-outcome.json
