Shor猜想成立：投影测量足以实现二元可访问信息
Shor's Conjecture Is True: Projective Measurements Suffice for Binary Accessible Information
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中文总结 AI 辅助
本文构造性地证明了Shor猜想：在任意有限维度下，投影测量（PVM）足以达到二元量子系综的可访问信息，并通过Jensen算子不等式给出了精确的凹变分公式。
中文摘要 AI 辅助
Shor猜想von Neumann测量能够达到每个二元量子系综的可访问信息。我们在任意有限维度下构造性地证明了该猜想。对于每个有限结果的positive operator-valued measure (POVM) $M$，我们从后验标签概率构造算子$T_M$，并证明其谱投影值测度(PVM) $\Pi_M$满足$I_{\Pi_M}(X{:}Y)\ge I_M(X{:}Y)$；每个秩一细化都保持该不等式。Jensen算子不等式的两个应用证明了该比较，并给出了可访问信息的精确凹变分公式。该结果是Fang、Fawzi和Fawzi关于measured $f$-divergences的一般定理的特例；下面的证明隔离了二元论证，并明确展示了替换$M\mapsto T_M\mapsto\Pi_M$。
英文摘要
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.
