A geometric characterization of invertible quantum measurement maps

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Abstract

A geometric characterization is given for invertible quantum measurement maps. Denote by S(H) the convex set of all states (i.e., trace 1 positive operators) on Hilbert space H with dimH, and [ρ1,ρ2] the line segment joining two elements ρ1,ρ2 in S(H). It is shown that a bijective map ϕ:S(H)S(H) satisfies ϕ([ρ1,ρ2])[ϕ(ρ1),ϕ(ρ2)] for any ρ1,ρ2S if and only if ϕ has one of the following formsρMρMtr(MρM)orρMρTMtr(MρTM), where M is an invertible bounded linear operator and ρT is the transpose of ρ with respect to an arbitrarily fixed orthonormal basis.

Keywords

Quantum states
Quantum measurement
Segment preserving maps

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This work is partially supported by National Natural Science Foundation of China (11171249, 11271217, 11201329), a grant to International Cooperating Research from Shanxi (2011081039). Li was also supported by a USA NSF grant and a HK RCG grant.