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 $R_{12}$, 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 $R_{12}$. Ansatz states satisfying these bounds are given and verified using various numerical methods. In rank-2 case it is shown that the states satisfying the lower bound on $R_{12}$ {\it vs} concurrence define a class of three qubit states that maximizes the tripartite entanglement (the 3-tangle) given an amount of entanglement between a pair of them. The measure $R_{12}$ 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.