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

##### Locator

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.022344

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##### Citation

Bhosale, U. T., & Lakshminarayan, A. (2016). Simple permutation-based measure of
quantum correlations and maximally-3-tangled states.* Physical Review A,* *94*(2):
022344. doi:10.1103/PhysRevA.94.022344.

Cite as: http://hdl.handle.net/11858/00-001M-0000-002B-A25E-4

##### 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 the rank-2 case, it is shown that the states satisfying the lower bound on R-12 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 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.