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Journal of Taiyuan University of Technology

Nonlinear Maps Preserving Convex Combination of Tensor Products of Operators

Liu Lian

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

In this paper,we give a structure feature of convex combination preserving bijective maps which send separable states to separable states of bipartite system.It is shown that such a map Φ has the form of Φ(ρσ)=(ST)Ψ(ρσ)(ST)*Tr((ST)Ψ(ρσ)(ST)*),where Ψ is one of the identity,transpose,partial transpose,or a combination of above mentioned map with the swap.

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In this paper,we give a structure feature of convex combination preserving bijective maps which send separable states to separable states of bipartite system.It is shown that such a map Φ has the form of Φ(ρσ)=(ST)Ψ(ρσ)(ST)Tr((ST)Ψ(ρσ)(ST)),where Ψ is one of the identity,transpose,partial transpose,or a combination of above mentioned map with the swap.
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.
We study separability properties in a 5-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states, which can be written as linear combinations of permutation operators, or, equivalently, commute with unitaries of the form UxUxU. We compute explicitly the following subsets: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition.
In the convex set of all 3 ⊗ 3 states with positive partial transposes, we show that one can take two extreme points whose convex combinations belong to the interior of the convex set. Their convex combinations may be even in the interior of the convex set of all separable states. In general, we need at least mn extreme points to get an interior point by their convex combination, for the case of the convex set of all m ⊗ n separable states. This shows a sharp distinction between PPT states and separable states. We also consider the same questions for positive maps and decomposable maps.
We study separability properties in a five-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states that can be written as linear combinations of permutation operators, or equivalently, commute with unitaries of the form $U\ensuremath{\bigotimes}U\ensuremath{\bigotimes}U.$ We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition. Tripartite entanglement is investigated in terms of the relative entropy of tripartite entanglement and of the trace norm.
We use classical results from the theory of linear preserver problems to characterize operators that send the set of pure states with Schmidt rank no greater than k back into itself, extending known results characterizing operators that send separable pure states to separable pure states. We also provide a new proof of an analogous statement in the multipartite setting. We use these results to develop a bipartite version of a classical result about the structure of maps that preserve rank-1 operators and then characterize the isometries for two families of norms that have recently been studied in quantum information theory. We see, in particular, that for k ≥ 2 the operator norms induced by states with Schmidt rank k are invariant only under local unitaries, the swap operator and the transpose map. However, in the k = 1 case there is an additional isometry: the partial transpose map.