Proof of Shor's conjecture on the accessible information of quantum dichotomies
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.
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BibTeX
@misc{dallarno2026proof,
title = {{Proof of Shor's conjecture on the accessible information of quantum dichotomies}},
author = {Michele Dall'Arno},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20600},
eprint = {2609.20600},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20600},
}
RIS
TY - UNPB
TI - Proof of Shor's conjecture on the accessible information of quantum dichotomies
AU - Michele Dall'Arno
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.20600
UR - https://arxiv.org/abs/2609.20600
AB - 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.
ER -