The automorphism group of separable states in quantum information theory

Abstract

We show that the linear group of automorphism of Hermitian matrices which preserves the set of separable states is generated by natural automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. We apply our results to preservers of the product numerical range.

Preservers of maximally entangled states

“…If case (b) occurs for every pairing of Q i and Q j , i = j, we get four orthogonal rank r projections Q 1 , Q 2 , Q 3 , Q 4 in M 2r , a contradiction. Hence case (a) must occur at least once.…”

“…Recently there has been work done on finding and classifying linear preservers of various properties or sets related to quantum information theory. One paper of particular relevance is [4], in which the authors classify linear preservers of separable states (a related paper is [7]). However, they work under the more restrictive assumption that the linear map is surjective, an assumption we shall not require.…”

Quantum measurements and maps preserving strict convex combinations and pure states

“…Let H m be the real linear space of all m × m Hermitian matrices and let P m be the set of all rank-1 m × m projection matrices. The next lemma comes from [3] which can be viewed as a characterization of linear preservers of pure states. Also, ref.…”

“…Many researchers pay their attention to the problem of characterizing the maps on the states; ref. [1,2,3,6].…”

Additive preservers of tensor product of rank one hermitian matrices

“…. , m k ) is the set of all bijective linear maps on k i=1 H mi that leave invariant the set k i=1 P mi (see [1], [5]). Thus, our work concerning linear and additive maps on…”