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arXiv:0707.1164 · Physical Review A

Quantum coherences, K -way negativities and multipartite entanglement

S. Shelly Sharma, Naresh Kumar Sharma

Published Apr 25, 2008• 16 citations
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Abstract

A characterization of multipartite quantum states having $N$ subsystems, based on negativities of matrices obtained by selective partial transposition of the state operator, is proposed. The $K$-way partial transpose with respect to a subsystem is constructed by imposing constraints involving the states of $K$ subsystems of the multipartite composite system. The $K$-way negativity, defined as the negativity of $K$-way partial transpose, quantifies the $K$-way coherences of the composite system. For an $N$-partite system the fraction of $K$-way negativity $(2\ensuremath{\le}K\ensuremath{\le}N)$, contributing to global negativity, is obtained. The entanglement measures for a given state $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}$ are identified as the partial $K$-way negativities of the corresponding canonical state.

DOI
10.1103/physreva.77.042117

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A characterization of multipartite quantum states having NNN subsystems, based on negativities of matrices obtained by selective partial transposition of the state operator, is proposed. The KKK-way partial transpose with respect to a subsystem is constructed by imposing constraints involving the states of KKK subsystems of the multipartite composite system. The KKK-way negativity, defined as the negativity of KKK-way partial transpose, quantifies the KKK-way coherences of the composite system. For an NNN-partite system the fraction of KKK-way negativity (2\ensuremath≤K\ensuremath≤N)(2\ensuremath{\le}K\ensuremath{\le}N)(2\ensuremath≤K\ensuremath≤N), contributing to global negativity, is obtained. The entanglement measures for a given state \ensuremathρ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}\ensuremathρ\ifmmode^\elseˆ\fi​ are identified as the partial KKK-way negativities of the corresponding canonical state.
Partial transposition of state operator is a well-known tool to detect quantum correlations between two parts of a composite system. In this paper, the global partial transpose (GPT) is linked to conceptually multipartite underlying structures in a state---the negativity fonts. If KKK-way negativity fonts with nonzero determinants exist, then selective partial transposition of a pure state, involving KKK of the NNN qubits ($K\ensuremath{\leqslant}N$), yields an operator with negative eigenvalues, identifying KKK-body correlations in the state. Expansion of GPT in terms of KKK-way partially transposed (KPT) operators reveals the nature of intricate intrinsic correlations in the state. Classification criteria for multipartite entangled states based on the underlying structure of global partial transpose of canonical state are proposed. The number of NNN-partite entanglement types for an NNN-qubit system is found to be ${2}^{N\ensuremath{-}1}\ensuremath{-}N+2$, while the number of major entanglement classes is ${2}^{N\ensuremath{-}1}\ensuremath{-}1$. Major classes for three- and four-qubit states are listed. Subclasses are determined by the number and type of negativity fonts in canonical states.
We propose to quantify three-qubit entanglement using global negativity along with KKK-way negativities, where K=2K=2K=2 and 3. The principle underlying the definition of KKK-way negativity for pure and mixed states of NNN subsystems is a positive partial transpose sufficient condition. However, KKK-way partial transpose with respect to a subsystem is defined so as to shift the focus to KKK-way coherences instead of KKK subsystems of the composite system. A quantum state of a three-qubit system is characterized by the coherences measured by global, two-way, and three-way negativities. For a canonical state of three-qubit system, entanglement measures for genuine tripartite entanglement, W-like entanglement, and bipartite entanglement can be related to two-way and three-way negativities.
A classification of N-partite states, based on K-way negativities (K=2 to N), is proposed. The K-way partial transpose with respect to a subsystem is defined so as to shift the focus to K-way coherences instead of K subsystems of the composite system. For an N-partite system, the fraction of K-way negativity contributing to global negativity, is obtained. After minimizing K-way negativities through local unitary qubit rotations, a combined analysis of 2-way, 3-way and global negativities is shown to provide distinct measures of genuine tripartite, W-state like and bipartite entanglement, for three qubit composite system. To illustrate the point, entanglement of three qubit GHZ class states, W-class states, three boson state and noisy states is analysed. While genuine N-partite entanglement of a composite system is generated by N-way coherences, N-partite entanglement in general can be present due to (K
It was shown by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] that by stochastic local operations and classical communication, a pure state of four qubits can be transformed to a state belonging to one of a set of nine families of states. By using selective partial transposition, we construct partial KKK-way negativities to measure the genuine four-partite, tripartite, and bipartite entanglements of single-copy states belonging to the nine families of four-qubit states. Partial KKK-way negativities are polynomial functions of local invariants characterizing each family of states, as such, entanglement monotones.
We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\ensuremath{\rho}$ is entirely determined by the diagonal elements of $\ensuremath{\rho}$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\ensuremath{\rho}$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.