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arXiv:2512.11233 · IEEE Transactions on Information Theory

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

Khac Duc An Thai, Michele Dall'Arno

quant-phSubmitted Dec 12, 2025• Announced Dec 15, 2025• 0 citations

Abstract

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.

Journal reference: IEEE Transactions on Information Theory (Volume: 72, Issue: 10, October 2026)
DOI: 10.1109/TIT.2026.3715995
Comments: 10 pages, 1 figure

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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.

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.

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)

M

, we form an operator

T_M

from the posterior label probabilities and show that its spectral projection-valued measure (PVM)

Π_M

satisfies

I_{Π_M}(X{:}Y)\ge I_M(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

f

-divergences; the proof below isolates the binary argument and makes the replacement

M\mapsto T_M\mapstoΠ_M

explicit.

Shor algorithm dynamics of quantum computation states are analysed from the classical and the quantum information theory points of view. The Shannon entropy is interpreted as the degree of information accessibility through measurement, while the von Neumann entropy is employed to measure the quantum information of entanglement. The intelligence of a state with respect to a subset of qubits is defined. The intelligence of a state is maximal if the gap between the Shannon and the von Neumann entropy for the chosen result qubits is minimal. We prove that the quantum Fourier transform creates maximally intelligent states with respect to the first n qubits for Shor's problem, since it annihilates the gap between the classical and quantum entropies for the first n qubits of every output state.

The accessible information acc(E) of an ensemble E is the maximum mutual information between a random variable encoded into quantum states, and the probabilistic outcome of a quantum measurement of the encoding. Accessible information is extremely difficult to characterize analytically; even bounds on it are hard to place. The celebrated Holevo bound states that accessible information cannot exceed chi(E), the quantum mutual information between the random variable and its encoding. However, for general ensembles, the gap between the acc(E) and chi(E) may be arbitrarily large. We consider the special case of a binary random variable, which often serves as a stepping stone towards other results in information theory and communication complexity. We give explicit lower bounds on the the accessible information acc(E) of an ensemble E = {(p, rho_0), (1-p, rho_1)}, with 0 <= p <= 1, as functions of p and chi(E). The bounds are incomparable in the sense that they surpass each other in different parameter regimes. Our bounds arise by measuring the ensemble according to a complete orthogonal measurement that preserves the fidelity of the states rho_0,rho_1. As an intermediate step, therefore, we give new relations between the two quantities acc(E), chi(E) and the fidelity B(rho_0,rho_1).