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arXiv:2609.20600

Proof of Shor's conjecture on the accessible information of quantum dichotomies

Michele Dall'Arno

quant-phSubmitted Sep 17, 2026• Announced Sep 18, 2026• 0 citations• Revised Oct 23, 2026 (v2)
v1 · Sep 17, 2026v2 · Oct 23, 2026
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Abstract

The accessible information of any given quantum ensemble quantifies the maximum amount of Shannon information that can be extracted from the ensemble by any quantum measurement. Almost three decades ago, Shor conjectured that the accessible information of any quantum dichotomy, that is, an ensemble of two states, is attained by a projective measurement. Recently, a proof of this conjecture restricted to the qubit case was published by Keil. Here, we conclusively settle this longstanding open problem. First, we show that Shor's conjecture follows, in arbitrary dimension, from a recent result by Fang, Fawzi and Fawzi, and we provide a self-contained, elementary proof. Second, for any given dichotomy and measurement, we provide the explicit construction of a projective measurement that outperforms such a measurement in extracting information from the given dichotomy. Third, while the computation of the accessible information is known to be non-convex in general, we frame the computation of the accessible information of dihotomies as a convex problem.

Comments
This submission is being withdrawn because it is superseeded by arXiv:2609.27992

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The accessible information of any given quantum ensemble quantifies the maximum amount of Shannon information that can be extracted from the ensemble by any quantum measurement. Almost three decades ago, Shor conjectured that the accessible information of any quantum dichotomy, that is, an ensemble of two states, is attained by a projective measurement. Recently, a proof of this conjecture restricted to the qubit case was published by Keil. Here, we conclusively settle this longstanding open problem. First, we show that Shor's conjecture follows, in arbitrary dimension, from a recent result by Fang, Fawzi and Fawzi, and we provide a self-contained, elementary proof. Second, for any given dichotomy and measurement, we provide the explicit construction of a projective measurement that outperforms such a measurement in extracting information from the given dichotomy. Third, while the computation of the accessible information is known to be non-convex in general, we frame the computation of the accessible information of dihotomies as a convex problem.
Around the turn of the century, Shor formulated his well-known and still-open conjecture stating that the accessible information of any quantum dichotomy, that is the maximum amount of classical information that can be decoded from a binary quantum encoding, is attained by a von Neumann measurement. A quarter of a century later, new developments on the Lorenz curves of quantum dichotomies in the field of quantum majorization and statistical comparison may provide the key to unlock such a longstanding open problem. Here, we first investigate the tradeoff relations between accessible information and guessing probability in the binary case, thus disproving the claimed monotonicity of the former quantity in the latter that, if true, would have settled Shor's problem in the qubit case. Our second result is to provide a state-dependent generalization of extremality for quantum measurements, to characterize state-dependent extremality for qubit dichotomies, and to apply such results to tighten previous results on the accessible information of qubit dichotomies.
Shor conjectured that a von Neumann measurement attains the accessible information of every binary quantum ensemble. We prove the conjecture constructively in arbitrary finite dimension. For every finite-outcome positive operator-valued measure (POVM) MMM, we form an operator TMT_MTM​ from the posterior label probabilities and show that its spectral projection-valued measure (PVM) ΠMΠ_MΠM​ satisfies IΠM(X:Y)≥IM(X:Y)I_{Π_M}(X{:}Y)\ge I_M(X{:}Y)IΠM​​(X:Y)≥IM​(X:Y); every rank-one refinement retains the inequality. Two applications of Jensen's operator inequality prove the comparison and yield an exact concave variational formula for the accessible information. The result is a special case of the general theorem of Fang, Fawzi, and Fawzi on measured fff-divergences; the proof below isolates the binary argument and makes the replacement M↦TM↦ΠMM\mapsto T_M\mapstoΠ_MM↦TM​↦ΠM​ explicit.
The accessible information and the informational power quantify the maximum amount of information that can be extracted from a quantum ensemble and by a quantum measurement, respectively. Here, we investigate the tradeoff between the accessible information (informational power, respectively) and the purity of the states of the ensemble (the elements of the measurement, respectively). Under any given lower bound on the purity, 1) we compute the minimum informational power and show that it is attained by the depolarized uniformly-distributed measurement; and 2) we give a lower bound on the accessible information. Under any given upper bound on the purity, 1) we compute the maximum accessible information and show that it is attained by an ensemble of pairwise commuting states with at most two distinct non-null eigenvalues; and 2) we give a lower bound on the maximum informational power. The present results provide, as a corollary, novel sufficient conditions for the tightness of the Jozsa-Robb-Wootters lower bound to the accessible information.
The accessible information quantifies the amount of classical information that can be extracted from an ensemble of quantum states. Analogously, the informational power quantifies the amount of classical information that can be extracted by a quantum measurement. For both quantities, we provide upper and lower bounds that depend only on the dimension of the system, and we prove their tightness. In the case of symmetric informationally complete (SIC) ensembles and measurements, stronger bounds are provided and their tightness proved for qubits and qutrits. From our upper bounds, we notice, perhaps surprisingly, that the statistics generated by SIC ensembles or measurements in arbitrary dimension, though optimal for tomographic purposes, in fact never contain more than just one bit of information, the rest being constituted by completely random bits. On the other hand, from our lower bounds, we obtain an explicit strategy beating the so-called pretty-good one for the extraction of mutual information in the case of SIC ensembles and measurements.
The problem of estimating quantity of information, which can be obtained in the process of binary consecutive measuring a qubit state, is considered in the paper. The studied quantity is expressed as Shannon mutual information between the initial qubit state and the outcomes of a consecutive measurement. It is demonstrated that the maximum of information is reached by projective measurements. The maximum quantity of accessible information is calculated. It is proved that in the case of arbitrary binary test the maximum is reached asymptotically for consecutive measurements.
Abstract Computing accessible information for an ensemble of quantum states is a basic problem in quantum information theory. We show that the recently obtained optimality criterion (A.S. Holevo, Lobachevskii J. Math., 43, 1646–1650 (2022)), when applied to specific ensembles of states leads to nontrivial tight entropy inequalities that are discrete relatives of the famous log-Sobolev inequality. In this light, the hypothesis of globally information-optimal measurement for an ensemble of equiangular equiprobable states (quantum pyramids) (B.-G. Englert and J. Řeháček, J. Mod. Optics 57 (3), 218–226 (2010)) is reconsidered and the corresponding entropy inequalities are proposed. Via the optimality criterion, this suggests also an approach to the proof of the conjectures concerning globally information-optimal observables for quantum pyramids.