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arXiv:0809.1932 · Physical Review A

Entanglement measures for intermediate separability of quantum states

Tsubasa Ichikawa, Toshihiko Sasaki, Izumi Tsutsui

Published May 11, 2009• 6 citations
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Abstract

We present a family of entanglement measures ${R}_{m}$ which act as indicators of separability of $n$-qubit quantum states into $m$ subsystems for arbitrary $2\ensuremath{\le}m\ensuremath{\le}n$. The measure ${R}_{m}$ vanishes if the state is separable into $m$ subsystems, and for $m=n$ it gives the Meyer-Wallach measure, while for $m=2$ it reduces, in effect, to the one introduced recently by Love et al. [Quantum Inf. Process. 6, 187 (2007)]. The measures ${R}_{m}$ are evaluated explicitly for the Greenberger-Horne-Zeilinger state and the $W$ state (and its modifications, the ${W}_{k}$ or Dicke states) to show that these globally entangled states exhibit rather distinct behaviors under the measures, indicating the utility of the measures ${R}_{m}$ for characterizing globally entangled states as well.

DOI
10.1103/physreva.79.052307

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We present a family of entanglement measures Rm{R}_{m}Rm​ which act as indicators of separability of nnn-qubit quantum states into mmm subsystems for arbitrary 2\ensuremath≤m\ensuremath≤n2\ensuremath{\le}m\ensuremath{\le}n2\ensuremath≤m\ensuremath≤n. The measure Rm{R}_{m}Rm​ vanishes if the state is separable into mmm subsystems, and for m=nm=nm=n it gives the Meyer-Wallach measure, while for m=2m=2m=2 it reduces, in effect, to the one introduced recently by Love et al. [Quantum Inf. Process. 6, 187 (2007)]. The measures Rm{R}_{m}Rm​ are evaluated explicitly for the Greenberger-Horne-Zeilinger state and the WWW state (and its modifications, the Wk{W}_{k}Wk​ or Dicke states) to show that these globally entangled states exhibit rather distinct behaviors under the measures, indicating the utility of the measures Rm{R}_{m}Rm​ for characterizing globally entangled states as well.
We investigate the dynamics of global entanglement, the Meyer-Wallach measure, under decoherence, analytically. We study two important classes of multipartite entangled states, the Greenberger-Horne-Zeilinger and the W\mathit{W}W state. We obtain exact results for various models of system-environment interactions (decoherence). Our results shows distinctly different scaling behavior for these initially entangled states indicating a relative robustness of the W\mathit{W}W state, consistent with previous studies.
Many-body quantum systems can be characterized using the notions of kkk-separability and entanglement depth. A quantum state is kkk-separable if it can be expressed as a mixture of kkk entangled subsystems, and its entanglement depth is given by the size of the largest entangled subsystem. In this paper we propose a multipartite entanglement measure that satisfies the following criteria: (i) It can be used with both pure and mixed states; (ii) it is encoded in a single element of the density matrix, so it does not require knowledge of the full spectrum of the density matrix; (iii) it can be applied to large systems; and (iv) it can be experimentally verified. The proposed method allows the certification of non-kkk-separability of a given quantum state. We show that the proposed method successfully classifies three-qubit systems into known stochastic local operations and classical communication (SLOCC) classes, namely, bipartite, WWW-, and Greenberger-Horne-Zeilinger-type entanglement. Furthermore, we characterize the non-kkk-separability in nine known SLOCC classes of four-qubit states and absolutely maximally entangled states for five and six qubits and for arbitrary-size-qubit Dicke states.
We explore the subtle relationships between partial separability and entanglement of subsystems in multiqubit quantum states and give experimentally accessible conditions that distinguish between various classes and levels of partial separability in a hierarchical order. These conditions take the form of bounds on the correlations of locally orthogonal observables. Violations of such inequalities give strong sufficient criteria for various forms of partial inseparability and multiqubit entanglement. The strength of these criteria is illustrated by showing that they are stronger than several other well-known entanglement criteria (the fidelity criterion, violation of Mermin-type separability inequalities, the Laskowski-\ifmmode \dot{Z}\else \.{Z}\fi{}ukowski criterion, and the D\"ur-Cirac criterion) and also by showing their great noise robustness for a variety of multiqubit states, including NNN-qubit Greenberger-Horne-Zeilinger states and Dicke states. Furthermore, for $N\ensuremath{\geqslant}3$ they can detect bound entangled states. For all these states, the required number of measurement settings for implementation of the entanglement criteria is shown to be only N+1N+1N+1. If one chooses the familiar Pauli matrices as single-qubit observables, the inequalities take the form of bounds on the antidiagonal matrix elements of a state in terms of its diagonal matrix elements.
The WWW subspace is defined as the subspace of the nnn-qubit Hilbert space spanned by states containing at most one excitation. In this work, a separability criterion is established for pure and mixed states supported in this subspace: if all reduced two-qubit subsystems are separable, then the entire nnn-qubit state is necessarily separable. This result motivates the identification of the sum of two-tangles as an entanglement measure for pure states supported in the WWW subspace. Furthermore, it is shown that the commonly used πππ-tangle becomes ineffective for the nnn-qubit WWW state as the system size increases, and vanishes in the large-nnn limit. To address this limitation, the sum of πππ-tangles is introduced, which, like the sum of two-tangles, successfully quantifies the entanglement of the nnn-qubit WWW state in the large-nnn limit. In addition, a new condition for entanglement measures is introduced, which facilitates the formulation of a well-behaved and physically meaningful entanglement measure.
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.
We propose a measure of entanglement between two subsystems of arbitrary (but finite) number of levels. This measure is invariant under unitary transformations of the subsystems and varies between 0 (product states) and 1 (maximum entangled states). The case of two two-level systems is worked out explicitly, where the known complementarity between one- and two-particle interference results in a very natural way from a sum rule. Generalizations to more than two subsystems are discussed. Here we show that for three particles there cannot exist a pure state that is completely characterized by three-particle entanglement alone. For the example of the Greenberger-Horne-Zeilinger (GHZ) state, an experimental setup is proposed, in which its corresponding two-particle entanglement would show up.