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Journal of Physics A Mathematical and Theoretical

Quantum measurement and maps preserving convex combinations of separable states

Jinchuan Hou, Liang Liu

Published May 4, 2012• 5 citations
Publisher

Abstract

A map ϕ between convex sets is said to be convex combination preserving if, for any elements x , y in its domain and any number 0 < t < 1, there is some s with 0 < s < 1, such that ϕ( tx + (1 − t ) y ) = s ϕ( x ) + (1 − s )ϕ( y ). Every quantum measurement described by a measurement operator M introduces a map (i.e. if ) that is convex combination preserving. In this paper, we give a characterization of convex combination preserving bijective maps on the set of all separable states in a bipartite quantum system H 1 ⊗ H 2 and show that, under a mild additional condition, such a map is a composition of an invertible local quantum measurement (i.e. the map of the form with S , T invertible) and some of the following maps: the transpose, the partial transpose and the swap.

DOI
10.1088/1751-8113/45/20/205305

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A map ϕ between convex sets is said to be convex combination preserving if, for any elements x , y in its domain and any number 0 < t < 1, there is some s with 0 < s < 1, such that ϕ( tx + (1 − t ) y ) = s ϕ( x ) + (1 − s )ϕ( y ). Every quantum measurement described by a measurement operator M introduces a map (i.e. if ) that is convex combination preserving. In this paper, we give a characterization of convex combination preserving bijective maps on the set of all separable states in a bipartite quantum system H 1 ⊗ H 2 and show that, under a mild additional condition, such a map is a composition of an invertible local quantum measurement (i.e. the map of the form with S , T invertible) and some of the following maps: the transpose, the partial transpose and the swap.
Let H and K be the complex Hilbert spaces and mathcal SH) theconvex set of all quantum states (i.e., the positive operators withtrace one) on H. In this paper, the convex combination preservingmaps from mathcal SH) into mathcal SK)∪0and theirconnection to quantum measurements are discussed. It is shown that amap Φ:mathcal SH )→ mathcal SK) with 2≤ dimH≤∞ preserves pure states and strict convex combinations ifand only if one of the following holds: (1) there exists a purestate σ0 on K such that Φ(ρ)= σ0 for allρ∈ mathcal SH); (2) Φ is an injective selective quantummeasurement map up to the transpose, i.e., there exists an injective bounded linear operator M: H→ K such thatΦ(ρ)=fracMρ M^*Tr(Mρ M^*)
Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten ppp -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a C∗C^{\ast }C∗ -algebra.