Bipartite quantum systems: on the realignment criterion and beyond

Abstract

Inspired by the 'computable cross norm' or 'realignment' criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states is derived. We conjecture that these conditions, which are weaker than the parent criterion, can be strengthened in such a way to obtain a new family of criteria that are independent of the original one. This conjecture is supported by numerical examples for the low dimensional cases. These ideas can be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement.

Unitary quantum gates, perfect entanglers, and unistochastic maps

“…The symbol U R has a unique meaning if a concrete decomposition of the total dimension, L = M N , is specified. Similar reorderings of matrices were considered by Hill et al [51,52] while investigating CP maps and also in [53][54][55][56][57] to analyze separability of mixed quantum states.…”

On the Relation Between Schmidt Coefficients and Entanglement

“…We have already mentioned the fact that the set of the Schmidt coefficients of a bipartite state is uniquely determined by this state; on the other hand, one can show that a set of Schmidt coefficients does not identify a unique quantum state (or, more generally, a unique linear operator). As it turns out that there is a relation between the Schmidt coefficients and entanglement [9,10], it is worth wondering to what extent these coefficients alone are able to witness the presence of entanglement and, in this spirit, introducing a suitable 'Schmidt equivalence relation' (see [10]). 2 Observe that for any ρ ∈ D(H) -since d a=1 µ 2 a = ρ, ρ HS = tr ρ 2 -we have:…”

“…Recently, in ref. [10], it has been proposed to consider the (elementary) symmetric polynomials in the Schmidt coefficients of a state ρ ∈ D(H), namely:…”

Separability criteria based on the realignment of density matrices and reduced density matrices

“…After that, the generalizations of CCNR criterion were investigated in [19]. In [20], the authors made use of the symmetric function of Schmidt coefficients to improve the CCNR criterion further. Recently, the CCNR criterion was used to study the entanglement conditions for any two-mode continuous-variable state with permutational symmetry [21].…”