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AIP conference proceedings

Four qubit entangled pure States and K-way negativities

Naresh Kumar Sharma, S. Shelly Sharma, B. G. Sidharth, F. Honsell, O. Mansutti, Katepalli R. Sreenivasan, A. De Angelis

Published Jan 1, 2008• 0 citations
Publisher

Abstract

It is known that by Stochastic local operations and classical communication (SLOCC) a pure state of four qubits can be transformed to a state belonging to one of a set of nine families of states. Using the eigen vectors corresponding to negative eigenvalues of global partial transpose and the K‐way partial transposes, we construct measures of genuine four‐partite, tri‐partite, and bi‐partite entanglement for all the nine families of states.

DOI
10.1063/1.2947706

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It is known that by Stochastic local operations and classical communication (SLOCC) a pure state of four qubits can be transformed to a state belonging to one of a set of nine families of states. Using the eigen vectors corresponding to negative eigenvalues of global partial transpose and the K‐way partial transposes, we construct measures of genuine four‐partite, tri‐partite, and bi‐partite entanglement for all the nine families of states.
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 consider a single copy of a pure four-partite state of qubits and investigate its behavior under the action of stochastic local quantum operations assisted by classical communication (SLOCC). This leads to a complete classification of all different classes of pure states of four qubits. It is shown that there exist nine families of states corresponding to nine different ways of entangling four qubits. The states in the generic family give rise to Greenberger-Horne-Zeilinger-like entanglement. The other ones contain essentially two-or three-qubit entanglement distributed among the four parties. The concept of concurrence and 3-tangle is generalized to the case of mixed states of four qubits, giving rise to a seven-parameter family of entanglement monotones. Finally, the SLOCC operations maximizing all these entanglement monotones are derived, yielding the optimal single-copy distillation protocol.
We construct coefficient matrices of size ${2}^{\ensuremath{\ell}}$ by ${2}^{n\ensuremath{-}\ensuremath{\ell}}$ associated with pure nnn-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure nnn-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, and as a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up to permutations of the four qubits. We also discuss the completeness of the classification of four qubits into nine families.
It is a recent observation that entanglement classification for qubits is closely related to stochastic local operations and classical communication (SLOCC) invariants. Verstraete et al.[ Phys. Rev. A 65 (2002) 052112 ] showed that for pure states of four qubits there are nine different degenerate SLOCC entanglement classes. Li et al.[ Phys. Rev. A 76 (2007) 052311 ] showed that there are at feast 28 distinct true SLOCC entanglement classes for four qubits by means of the SLOCC invariant and semi-invariant. We give 16 different entanglement classes for four qubits by means of basic SLOCC invariants.
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
Abstract Quantifying genuine entanglement is a key task in quantum information theory. We study the quantification of genuine multipartite entanglement for four-qubit systems. Based on the concurrence of nine different classes of four-qubit states, with each class being closed under stochastic local operation and classical communication, we construct a concurrence tetrahedron. Proper genuine four-qubit entanglement measure is presented by using the volume of the concurrence tetrahedron. For non genuine entangled pure states, the four-qubit entanglement measure classifies the bi-separable entanglement. We show that the concurrence tetrahedron based measure of genuine four-qubit entanglement is not equivalent to the genuine four-partite entanglement concurrence. We illustrate the advantages of the concurrence tetrahedron by detailed examples.