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arXiv:quant-ph/0609012 · Physical Review A

Two-way and three-way negativities of three-qubit entangled states

S. Shelly Sharma, Naresh Kumar Sharma

Published Jul 24, 2007• 21 citations
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Abstract

We propose to quantify three-qubit entanglement using global negativity along with $K$-way negativities, where $K=2$ and 3. The principle underlying the definition of $K$-way negativity for pure and mixed states of $N$ subsystems is a positive partial transpose sufficient condition. However, $K$-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. 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.

DOI
10.1103/physreva.76.012326

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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
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 $(2\ensuremath{\le}K\ensuremath{\le}N)$, 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 KKK-way negativities of the corresponding canonical state.
We obtain, analytically, the global negativity, partial KKK-way negativities (K=2,3)(K=2,3)(K=2,3), Wootter's tangle, and three-tangle for the generic three-qubit canonical state. It is found that the product of global negativity and partial three-way negativity is equal to the three-tangle, while the partial two-way negativity is related to the tangle of qubit pairs. We also calculate similar quantities for the state canonical to a single-parameter (0<q<1)(0<q<1)(0<q<1) pure state which is a linear combination of a Greenberger-Horne-Zeilinger state and a WWW state. In this case, for q=0.62685q=0.626\phantom{\rule{0.2em}{0ex}}85q=0.62685, the state has zero three-tangle and zero three-way negativity, having only WWW-like entanglement. The difference between the product of global and partial three-way negativities and the three-tangle for a given state is a quantitative measure of two-qubit coherences transformed by unitary transformations on the canonical state into three-qubit coherences. The global negativity and partial KKK-way negativities, obtained by selective partial transpositions on multiqubit state operators, satisfy inequalities which for three qubits are equivalent to the Coffman-Kundu-Wootter inequality.
The detection and classification of entanglement properties in a two-qubit and a multi-qubit system is a topic of great interest. This topic has been extensively studied, and as a result, we discovered various approaches for detecting and classifying multi-qubit, in particular three-qubit entangled states. The emphasis of this work is on a formalism of methods for the detection and classification of bipartite as well as multipartite quantum systems. We have used the method of structural physical approximation of partially transposed matrix (SPA-PT) for the detection of entangled states in arbitrary dimensional bipartite quantum systems. Also, we have proposed criteria for the classification of all possible stochastic local operations and classical communication (SLOCC) inequivalent classes of a pure and mixed three-qubit state using the SPA-PT map. To quantify entanglement, we have defined a new measure of entanglement based on the method of SPA-PT, which we named as "structured negativity". We have shown that this measure can be used to quantify entanglement for negative partial transposed entangled states (NPTES). Since the methods for detection, classification and quantification of entanglement, defined in this thesis are based on SPA-PT, they may be realized in an experiment.
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.