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arXiv:1005.4453 · Physical Review A

Conditions for entanglement in multipartite systems

Mark S. Hillery, Ho Trung Dung, Hongjun Zheng

Published Jun 17, 2010• 47 citations
arXivHTMLPDF

Abstract

We introduce two entanglement conditions that take the form of inequalities involving expectation values of operators. These conditions are sufficient conditions for entanglement; that is, if they are satisfied the state is entangled, but if they are not, one can say nothing about the entanglement of the state. These conditions are quite flexible, because the operators in them are not specified, and they are particularly useful in detecting multipartite entanglement. We explore the range of utility of these conditions by considering a number of examples of entangled states, and seeing under what conditions entanglement in them can be detected by the inequalities presented here.

DOI
10.1103/physreva.81.062322

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We introduce two entanglement conditions that take the form of inequalities involving expectation values of operators. These conditions are sufficient conditions for entanglement; that is, if they are satisfied the state is entangled, but if they are not, one can say nothing about the entanglement of the state. These conditions are quite flexible, because the operators in them are not specified, and they are particularly useful in detecting multipartite entanglement. We explore the range of utility of these conditions by considering a number of examples of entangled states, and seeing under what conditions entanglement in them can be detected by the inequalities presented here.
There is no single test that will determine whether a general quantum state is separable or entangled. There are, however, a number of sufficient conditions, including the partial-transpose condition and entanglement witnesses. Here we present a class of sufficient conditions based on expectation values of non-hermitian operators. Initially we examine bipartite systems. The application of these conditions to multimode states of the electromagnetic field and to spin systems is discussed. We show that our original conditions can be strengthened if we make them invariant under certain classes of local unitary transformations. We then go on to study multipartite systems and present conditions that can be used to show that a state is not totally separable. Some simple applications of these conditions are also presented.
We present a way of identifying all kinds of entanglement for three-qubit pure states in terms of the expectation values of Pauli operators. The necessary and sufficient conditions to classify the fully separable, biseparable, and genuine entangled states are explicitly given. The approach can be generalized to multipartite high-dimensional cases. For three-qubit mixed states, we propose two kinds of inequalities in terms of the expectation values of complementary observables. One inequality has advantages in entanglement detection of the quantum state with positive partial transpositions, and the other is able to detect genuine entanglement. The results give an effective method for experimental entanglement identification.
We propose a sufficient entanglement criterion for the multipartite systems based on quantum Fisher information. It complements the well-established positive partial transpose criterion in the sense that it is also able to detect bound entangled states. Moreover, it improves the well-known upper bound on the quantum Fisher information of separable multiqudit states introduced by Toth et al. [G. Toth and T. Vertesi, Phys. Rev. Lett. 120, 020506 (2018)] and hence detects a greater number of metrologically useful entangled states. This criterion is formulated as a simple inequality which must be fulfilled by the quantum Fisher information of separable states of multipartite systems. Violation of this inequality by a state reveals its entanglement and accordingly confirms its usefulness for quantum metrology. The efficiency of this criterion for detection of entanglement is illustrated through some examples.
We present a set of inequalities for detecting quantum entanglement of $2\ensuremath{\bigotimes}d$ quantum states. For $2\ensuremath{\bigotimes}2$ and $2\ensuremath{\bigotimes}3$ systems, the inequalities give rise to sufficient and necessary separability conditions for both pure and mixed states. For the case of d>3d>3d>3, these inequalities are necessary conditions for separability, which detect all entangled states that are not positive under partial transposition and even some entangled states with positive partial transposition. These inequalities are given by mean values of local observables and present an experimental way of detecting the quantum entanglement of $2\ensuremath{\bigotimes}d$ quantum states and even multiqubit pure states.
I present methods for detecting entanglement around symmetric Dicke states. In particular, I consider N-qubit symmetric Dicke states with N/2 excitations. I show that for large N these states have the smallest overlap possible with states without genuine multipartite entanglement. Thus these states are particularly well suited for the experimental examination of multipartite entanglement. I present fidelity-based entanglement witness operators for detecting multipartite entanglement around these states. I then consider entanglement criteria, somewhat similar to the spin squeezing criterion, based on the moments or variances of the collective spin operators. Surprisingly, these criteria are based on an upper bound for variances for separable states. I present criteria detecting entanglement in general and criteria detecting only genuine multipartite entanglement. The collective operator measured for this criteria is an important physical quantity: Its expectation value essentially gives the intensity of the radiation when a coherent atomic cloud emits light.
Understanding the relation between nonlocality and entanglement is one of the fundamental problems in quantum physics. In the bipartite case, it is known that these two phenomena are inequivalent, as there exist entangled states of two parties that do not violate any Bell inequality. However, except for a single example of an entangled three-qubit state that has a local model, almost nothing is known about such a relation in multipartite systems. We provide a general construction of genuinely multipartite entangled states that do not display genuinely multipartite nonlocality, thus proving that entanglement and nonlocality are inequivalent for any number of parties.