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Quantum Information Processing

Reversible projective measurement in quantum ensembles

Anatoly K. Khitrin, Michael Michalski, Jae‐Seung Lee

Published Dec 11, 2010• 5 citations
Publisher
DOI
10.1007/s11128-010-0219-5

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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) MMM, we form an operator TMT_MTM​ from the posterior label probabilities and show that its spectral projection-valued measure (PVM) ΠMΠ_MΠM​ satisfies IΠM(X:Y)≥IM(X:Y)I_{Π_M}(X{:}Y)\ge I_M(X{:}Y)IΠM​​(X:Y)≥IM​(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 fff-divergences; the proof below isolates the binary argument and makes the replacement M↦TM↦ΠMM\mapsto T_M\mapstoΠ_MM↦TM​↦ΠM​ explicit.