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

TL;DR

This paper investigates Shor's conjecture on whether the maximum classical information from binary quantum encodings is achieved by von Neumann measurements, providing new insights and disproving some assumptions in the qubit case.

Contribution

It disproves the monotonicity of accessible information with guessing probability in binary quantum systems and introduces a state-dependent extremality characterization for qubit dichotomies.

Findings

Disproved the claimed monotonicity of accessible information in the binary case.

Provided a state-dependent extremality characterization for qubit measurements.

Tightened bounds on accessible information for qubit dichotomies.

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.