Datta–Dorlas–Jozsa–Benatti: conditional subentropy and guessing

Conjecture 2. For a c-q state ρXB, the conditional subentropy Q(X|B) satisfies the following bound: Q(X|B) ≤ Hmin(X|B) = − ln pguess(X|B).

The supplied locator is arXiv:1310.1312v1, §2.5.1, Conjecture 2. Let p be any finite probability distribution and rho_x any finite-dimensional density states. Conditional subentropy is Q(recordState p rho) minus Q(mixtureState p rho), using the canonical states. The strengthened target is

$$ Q(X|B)\le1-p_{\rm guess}(X|B)\le-\log p_{\rm guess}(X|B). $$

The target is preregistered in issue #18957. It concerns arbitrary finite ensembles, including zero probabilities, singular states and mixed conditional states.

The key estimate holds against every labelled POVM:

$$ Q(\rho_{XB})-Q(\rho_B) \le1-\sum_xp_x\operatorname{Re}\operatorname{Tr}(M_x\rho_x). $$

The proof uses convexity of the actual canonical subentropy pinching gap, not an assumed abstract coherence functional. The measurement outcome label is retained through its positive Gram piece and pure-state decomposition.

The success-probability set consists of exactly

$$ \left{\sum_xp_x\operatorname{Re}\operatorname{Tr}(M_x\rho_x): M_x\succeq0,\ \sum_xM_x=I\right}, $$

and p_guess is its real supremum. A deterministic guess at a positive-probability label proves the set nonempty and p_guess>0. Positivity and normalization of the POVM give success probabilities in [0,1], so the set is bounded above and p_guess<=1.

Let Y_x=p_x rho_x and V(w)=sum_x sqrt(Y_x) w_x. The two Gram identities VV*=rho_B and Delta(VV)=rho_XB give Q(X|B)=C_Delta(VV). For each POVM form G_x=V* M_x V. These positive matrices sum to V*V. Diagonalize each G_x separately as sum_j w_(x,j) |v_(x,j)><v_(x,j)|, with normalized vectors and nonnegative weights of total sum one. Zero eigenvalues stay as zero weights; no division by them is needed.

For each pure vector v, rectangular Gram invariance gives C_Delta(|v><v|)=Q_cl(r), where r_y=sum_b |v(y,b)|^2. The stage-1 kernel bound integrates to Q_cl(r)<=1-r_x for every label x. Convexity of C_Delta therefore gives

$$ Q(X|B)\le\sum_{x,j}w_{x,j}(1-r_x^{(x,j)}) =1-\sum_xp_x\operatorname{Re}\operatorname{Tr}(M_x\rho_x). $$

The last equality follows from Tr(P_x G_x)=Tr(M_x Y_x), with the same original outcome label x. The least-upper-bound property now gives Q(X|B)<=1-p_guess; no maximizing measurement is assumed to exist. Finally log p_guess<=p_guess-1 proves the logarithmic bound.

A valid finite ensemble violating the target would refute the conjecture. A mismatch in the POVM set, omitted measurement labels or an unproved pinching-gap inequality would invalidate this proof route. An estimate for only pure or orthogonal conditional states would not settle the assertion.

The settlement theorem is POVMGram.cq_conditional_subentropy_guessing. Its conditionalSubentropy is exactly the subentropy of the repository's recordState minus that of mixtureState. GuessingProbability defines the genuine finite-POVM success set and its supremum, and proves nonemptiness, boundedness, positivity and the supremum conversion. ClassicalQuantumGram provides the canonical Gram equality and the pure label-specific bound. PinchingGap proves the actual convex pinching-gap inequality.

The frozen StrongSubadditivityRefutation.subentropyLike_eq identification states that every SubentropyLike family equals the canonical subentropy on every density state. Substitution at recordState and mixtureState transfers this conditional inequality to every such family, including repeated and zero spectra.

PROVED — settlement. For every finite ensemble, $Q(X|B)\le1-p_{\rm guess}\le-\ln p_{\rm guess}$. Zero probabilities, mixed or singular conditional states and degenerate spectra are included.

PROVED — decisive mechanism. The Gaussian representation turns the pinching gap into an integral of finite entropy Jensen gaps. Their joint convexity gives convexity of $C_\Delta(G)=Q(\Delta G)-Q(G)$. Rectangular Gram duality identifies the conditional subentropy of a c-q record with this same gap on $V^V$, while $VV^$ is the mixture. The argument retains one common state and its actual block weights.

PROVED — general pinching bound. For every finite block pinching in the product-coordinate setting, every positive semidefinite trace-one $G$ obeys

$$ 0\le C_\Delta(G)\le Q_{\rm cl}(p), \qquad p_x=\operatorname{Tr}(P_xG). $$

The upper bound is proved by Subadditivity.pinching_upper: decompose $G$ into normalized pure states, apply convexity of the pinching gap, identify each pure gap with classical subentropy of its block masses, and apply concavity to those masses. No bipartite marginal assumption is needed for this estimate. The corresponding unnormalized positive-block inequality also includes zero blocks without dividing by their traces.

PROVED — stronger conditional bound. Label-preserving POVM Gram pieces give $Q(X|B)\le1-p_{\rm guess}\le-\ln p_{\rm guess}$ for every finite ensemble. The linear upper bound is stronger than the logarithmic bound in Conjecture 2. It follows by bounding against every POVM and passing to the supremum; neither a maximizing POVM nor pure conditional states are needed. Thus Conjecture 2 follows alongside bipartite subadditivity from the same convex-gap mechanism, rather than from an assumption of strong subadditivity.

PROVED — boundary by refutation. Strong subadditivity fails, as established by the repository's StrongSubadditivityRefutation. The bipartite inequality therefore does not supply the strong-subadditivity step needed for a general multipartite conditional chain rule.

OPEN — sharpness. Characterize equality in the pinching bound and in $Q(X|B)\le1-p_{\rm guess}$, and determine the best bound at a prescribed nontrivial guessing probability. The estimates above alone do not settle these optimality questions.

OPEN — multipartite extensions. Determine which multipartite conditional inequalities survive under additional structural hypotheses and whether label-preserving Gram decompositions give sharper bounds with several classical registers. Any proposed extension must respect the existing strong-subadditivity counterexample. No new multipartite conditional claim is settled here.

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