Simple permutation-based measure of quantum correlations and maximally-3-tangled states

Abstract

Quantities invariant under local unitary transformations are of natural interest in the study of entanglement. This paper deduces and studies a particularly simple quantity that is constructed from a combination of two standard permutations of the density matrix, namely, realignment and partial transpose. This bipartite quantity, denoted here as R12, vanishes on large classes of separable states including classical-quantum correlated states, while being maximum for only maximally entangled states. It is shown to be naturally related to the 3-tangle in three-qubit states via their two-qubit reduced density matrices. Upper and lower bounds on concurrence and negativity of two-qubit density matrices for all ranks are given in terms of R12. Ansatz states satisfying these bounds are given and verified using various numerical methods. In the rank-2 case, it is shown that the states satisfying the lower bound on R12 versus concurrence define a class of three-qubit states that maximize the tripartite entanglement (the 3-tangle) given an amount of entanglement between a pair of them. The measure R12 is conjectured, via numerical sampling, to be always larger than the concurrence and negativity. In particular, this is shown to be true for the physically interesting case of X states.