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arXiv:0909.4443 · Physical Review A

Entanglement and discord of the superposition of Greenberger-Horne-Zeilinger states

Preeti Parashar, Swapan Rana

Published Mar 2, 2011• 39 citations
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Abstract

We calculate the analytic expression for geometric measure of entanglement for arbitrary superposition of two $N$-qubit canonical orthonormal Greenberger-Horne-Zeilinger (GHZ) states and the same for two $W$ states. In the course of characterizing all kinds of nonclassical correlations, an explicit formula for quantum discord (via relative entropy) for the former class of states has been presented. Contrary to the GHZ state, the closest separable state to the $W$ state is not classical. Therefore, in this case, the discord is different from the relative entropy of entanglement. We conjecture that the discord for the $N$-qubit $W$ state is $\mathrm{log}{}_{2}N$.

DOI
10.1103/physreva.83.032301

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We calculate the analytic expression for geometric measure of entanglement for arbitrary superposition of two NNN-qubit canonical orthonormal Greenberger-Horne-Zeilinger (GHZ) states and the same for two WWW states. In the course of characterizing all kinds of nonclassical correlations, an explicit formula for quantum discord (via relative entropy) for the former class of states has been presented. Contrary to the GHZ state, the closest separable state to the WWW state is not classical. Therefore, in this case, the discord is different from the relative entropy of entanglement. We conjecture that the discord for the NNN-qubit WWW state is log2N\mathrm{log}{}_{2}Nlog2​N.
Quantum correlations characterized by quantum entanglement and quantum discord play important roles in many quantum information processing. We study the relations among the entanglement of formation, concurrence, tangle, linear entropy-based classical correlation and von Neumann entropy-based classical correlation . We present analytical formulae of linear entropy-based classical correlation for arbitrary d⊗2d\otimes 2d⊗2 quantum states and von Neumann entropy-based classical correlation for arbitrary 2⊗22\otimes 22⊗2 rank-2 quantum states. From the von Neumann entropy-based classical correlation, we derive an explicit formula of quantum discord for arbitrary rank-2 two-qubit quantum states.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called XXX states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.
We investigate how much amount of Greenberger-Horne-Zeilinger (GHZ) entanglement is required in order to prepare a given multipartite state by local operations and classical communication (LOCC). We present a LOCC procedure that asymptotically converts GHZ states into an arbitrary multipartite pure state, whose conversion rate is given by the multipartite discord of the state. This reveals that the GHZ-entanglement cost of preparing a pure state is not higher than the multipartite discord of the state. It also provides an operational interpretation of multipartite discord for pure states, namely, the consumption rate of GHZ entanglement in the devised procedure.
We propose a global measure for quantum correlations in multipartite systems, which is obtained by suitably recasting the quantum discord in terms of relative entropy and local von Neumann measurements. The measure is symmetric with respect to subsystem exchange and is shown to be nonnegative for an arbitrary state. As an illustration, we consider tripartite correlations in the Werner-GHZ (Greenberger-Horne-Zeilinger) state and multipartite correlations at quantum criticality. In particular, in contrast with the pairwise quantum discord, we show that the global quantum discord is able to characterize the infinite-order quantum phase transition in the Ashkin-Teller spin chain.
In this paper, we study the entanglement classification of the two-dimensional subspace spanned by nnn-qubit Greenberger-Horne-Zeilinger (GHZ) and WWW states under stochastic local operations and classical communication (SLOCC). We show that for $n\ensuremath{\ge}4$ such a system can have infinite SLOCC inequivalent classes. Specifically, two states, ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+x{|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+y{|W\ensuremath{\rangle}}_{n}$, from the subspace spanned by nnn-qubit ${|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}$ are SLOCC equivalent if and only if (xy)n=1{(\frac{x}{y})}^{n}=1(yx​)n=1.
We introduce a class of generalized geometric measures of entanglement. For pure quantum states of N\mathit{N}N elementary subsystems, they are defined as the distances from the sets of K\mathit{K}K-separable states (K=2,…,N)(K=2,\dots{},N)(K=2,…,N). The entire set of generalized geometric measures provides a quantification and hierarchical ordering of the different bipartite and multipartite components of the global geometric entanglement, and allows discrimination among the different contributions. The extended measures are applied to the study of entanglement in different classes of N\mathit{N}N-qubit pure states. These classes include W\mathit{W}W and Greenberger-Horne-Zeilinger (GHZ) states, and their symmetric superpositions; symmetric multimagnon states; cluster states; and, finally, asymmetric generalized W\mathit{W}W-like superposition states. We discuss in detail a general method for the explicit evaluation of the multipartite components of geometric entanglement, and we show that the entire set of geometric measures establishes an ordering among the different types of bipartite and multipartite entanglement. In particular, it determines a consistent hierarchy between GHZ and W\mathit{W}W states, clarifying the original result of Wei and Goldbart that W\mathit{W}W states possess a larger global entanglement than GHZ states. Furthermore, we show that all multipartite components of geometric entanglement in symmetric states obey a property of self-similarity and scale invariance with the total number of qubits and the number of qubits per party.