Preservers of maximally entangled states

Abstract

The linear structure of the real space spanned by maximally entangled states is investigated, and used to completely characterize those linear maps preserving the set of maximally entangled states on M m ⊗ M m , where M m denotes the space of m × m complex matrices. Aside from a degenerate rank one map, such preservers are generated by a change of orthonormal basis in each tensor factor, interchanging the two tensor factors, and the transpose operator.

On the preservers of maximally entangled states

“…This generalizes (the invertible case of) Poon's result from [5], which can be stated within our framework as follows:…”

“…The following proposition generalizes Remark 2.3 in [5]. Proof: Let M be a rank-1, trace-1, positive operator in span(MES).…”

“…Recently, there has been work done on linear preservers within the context of quantum information theory. Two relevant papers to this endeavor are [2] and the more recent paper [5], whose results we generalize here.…”

“…Theorem 1.2. [5] A linear map Φ : span(MES X ,X ) → span(MES X ,X ) preserves MES X ,X if and only if it has one of the following forms:…”

On a Matrix Inequality Related to the Distillability Problem

“…The distillability problem [ 1 , 2 ] asks whether the above-mentioned conversion succeeds for any mixed states. It has been a main open problem in quantum information [ 1 ] for a long time, since the distillability problem lies at the heart of entanglement theory [ 3 , 4 , 5 ] and is related to the separability problem extensively studied by the quantum information community recently [ 6 , 7 , 8 ]. There have been some attempts at the problem in the past years [ 1 , 2 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 ].…”