Abstract
Entanglement in symmetric quantum states and the theory of copositive matrices are intimately related concepts. For the simplest symmetric states, i.e., the diagonal symmetric (DS) states, it has been shown that there exists a correspondence between exceptional (non-exceptional) copositive matrices and non-decomposable (decomposable) Entanglement Witnesses (EWs). Here we show that EWs of symmetric, but not DS, states can also be constructed from extended copositive matrices, providing new examples of bound entangled symmetric states, together with their corresponding EWs, in arbitrary odd dimensions.
Highlights
Entanglement and symmetry lie at the heart of quantum theory
In this work we have studied the connection between entanglement witnesses (EWs) and copositive matrices, showing how this class of matrices can be effectively used to detect entanglement in bipartite symmetric states
Our analysis has shown that, on the one hand, for dimension d ≥ 5, it is possible to certify bound entanglement in any family of symmetric states constructed by adding coherences to a PPT entangled diagonal symmetric states (PPTEDS) state
Summary
As shown in previous works [13, 19], deciding if a DS state ρDS is separable, is equivalent to check the membership of Md(ρDS) to the cone of completely positive matrices CPd, i.e., the cone formed by those d × d matrices Ad that admit a decomposition of the type Ad = BBT , where B is a d × k matrix, for some k > 1, with Bij ≥ 0, Bij ∈ R. A d × d copositive matrix H is said to be exceptional if, and only if, H cannot be decomposed as the sum of a positive semidefinite matrix (PSDd), and a symmetric entry-wise non-negative matrix (Nd), i.e., H ∈ COPd \ (PSDd + Nd).
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