Shor's Conjecture Is True: Projective Measurements Suffice for Binary Accessible Information

Abstract: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) $\Pi_M$ satisfies $I_{\Pi_M}(X{:}Y)\ge I_M(X{:}Y)$; 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 $M\mapsto T_M\mapsto\Pi_M$ explicit.

arXiv:2609.19003 [quant-ph]

(or arXiv:2609.19003v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.19003 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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