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Photonics Research

Entanglement and nonlocality in a coupled-cavity system

Zhe-Yong Zhang, Jianping Ding, Hui‐Tian Wang

Published Apr 24, 2017• 5 citations
Publisher

Abstract

No instrument is able to measure directly the quantum entanglement of a system. However, both theory and experiment, following the well-known Bell inequality, reveal the existence of the entanglement phenomenon in quantum mechanics. To examine the characterization of quantum entanglement, here we present a two-site cavity system, in which each cavity contains a Λ-type three-level atom and the two sites are identical and coupled with each other. We investigate and calculate the bipartite entanglement entropy of the system for the ground states. For photons of different types, corresponding to the two ground states of the atom, we investigate the correlations and violation of the Bell inequality.

DOI
10.1364/prj.5.000224

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No instrument is able to measure directly the quantum entanglement of a system. However, both theory and experiment, following the well-known Bell inequality, reveal the existence of the entanglement phenomenon in quantum mechanics. To examine the characterization of quantum entanglement, here we present a two-site cavity system, in which each cavity contains a Λ-type three-level atom and the two sites are identical and coupled with each other. We investigate and calculate the bipartite entanglement entropy of the system for the ground states. For photons of different types, corresponding to the two ground states of the atom, we investigate the correlations and violation of the Bell inequality.
Bipartite entanglement entropies are calculated for the ground state of the two-excitation subspace in a two-site coupled cavity model. Each region in the phase diagram (atomic insulator, polaritonic insulator, photonic superfluid, and polaritonic superfluid) is found to be characterized by unique entanglement properties. In particular, the polaritonic superfluid region exhibits genuine 4-partite entanglement among the two atoms and two-cavity fields. The difference in entanglement properties between the small- and large-hopping limits provides further evidence that the transitions in the two limits are qualitatively different and that the large-hopping limit cannot be described by a simple Bose-Hubbard-type model.
Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, Gisin's Theorem ), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio R = 1/Tr(ρ 2 ) or their maximum eigenvalue λ max . This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so-called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four-qubit systems, where nonlocality for generalized Greenberger–Horne–Zeilinger states of arbitrary number of parties is computed.
There have been studies on the formation of quantum-nonlocal states in two spatially separate cavities. We suggest a nonlocal test for a field prepared in two cavities. We couple classical driving fields with cavities where a nonlocal state is prepared. Two independent two-level atoms are then sent through respective cavities to interact off-resonantly with the cavity fields. The atomic states are measured after the interaction. Bell's inequality can be tested by the joint probabilities of the two-level atoms being in their excited or ground states. We find that a violation of Bell's inequality can also be tested using a single atom sequentially interacting with the two cavities. Potential experimental errors are also considered. We show that with the present experimental condition of 5% error in the atomic velocity distribution, the violation of Bell's inequality can be measured.
A well-known manifestation of quantum entanglement is that it may lead to correlations that are inexplicable within the framework of a locally causal theory---a fact that is demonstrated by the quantum violation of Bell inequalities. The precise relationship between quantum entanglement and the violation of Bell inequalities is, however, not well understood. While it is known that entanglement is necessary for such a violation, it is not clear whether all entangled states violate a Bell inequality, even in the scenario where one allows joint operations on multiple copies of the state and local filtering operations before the Bell experiment. In this paper we show that all entangled states, or more precisely, all not-fully-separable states of arbitrary Hilbert space dimension and arbitrary number of parties, violate a Bell inequality when combined with another state which on its own cannot violate the same Bell inequality. This result shows that quantum entanglement and quantum nonlocality are in some sense equivalent, thus giving an affirmative answer to the aforementioned open question. It follows from our result that two entangled states that are apparently useless in demonstrating quantum nonlocality via a specific Bell inequality can be combined to give a Bell violation of the same inequality. Explicit examples of such activation phenomenon are provided.
Entanglement, the Einstein-Podolsky-Rosen (EPR) paradox and Bell’s failure of local-hiddenvariable (LHV) theories are three historically famous forms of “quantum nonlocality”. We give experimental criteria for these three forms of nonlocality in multi-particle systems, with the aim of better understanding the transition from microscopic to macroscopic nonlocality. We examine the nonlocality of N separated spin J systems. First, we obtain multipartite Bell inequalities that address the correlation between spin values measured at each site, and then we review spin squeezing inequalities that address the degree of reduction in the variance of collective spins. The latter have been particularly useful as a tool for investigating entanglement in Bose-Einstein condensates (BEC). We present solutions for two topical quantum states: multi-qubit Greenberger-Horne-Zeilinger (GHZ) states, and the ground state of a two-well BEC.
Bell inequality is a mathematical inequality derived using the assumptions of locality and realism. Its violation guarantees the existence of quantum correlations in a quantum state. Bell inequality acts as an entanglement witness in the sense that a pure bipartite quantum state, having nonvanishing entanglement, always violates a Bell inequality. We construct Bell-type inequalities for product states in quantum coherence theory for different measures of coherence, and find that the maximally entangled states violate these inequalities. We further show that Bell-type inequalities for relative entropy of coherence is violated by all two-qubit pure entangled states, serving as an entanglement witness.