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arXiv:quant-ph/0310102 · Physical Review A

Multicomponent Bell inequality and its violation for continuous-variable systems

Jing‐Ling Chen, Chunfeng Wu, L. C. Kwek, Dagomir Kaszlikowski, Marek Żukowski, C. H. Oh

Published Mar 22, 2005• 12 citations
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Abstract

Multicomponent correlation functions are developed by utilizing $d$-outcome measurements. Based on multicomponent correlation functions, we propose a Bell inequality for bipartite $d$-dimensional systems. Violation of the Bell inequality for continuous-variable (CV) systems is investigated. The violation of maximally entangled states can exceed the Cirel'son bound; the maximal violation is 2.969 81. For finite values of the squeezing parameter, the violation strength of CV states increases with dimension $d$. Numerical results show that the violation strength of CV states with finite squeezing parameters is stronger than that of maximally entangled states.

DOI
10.1103/physreva.71.032107

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Multicomponent correlation functions are developed by utilizing ddd-outcome measurements. Based on multicomponent correlation functions, we propose a Bell inequality for bipartite ddd-dimensional systems. Violation of the Bell inequality for continuous-variable (CV) systems is investigated. The violation of maximally entangled states can exceed the Cirel'son bound; the maximal violation is 2.969 81. For finite values of the squeezing parameter, the violation strength of CV states increases with dimension ddd. Numerical results show that the violation strength of CV states with finite squeezing parameters is stronger than that of maximally entangled states.
We propose a Bell inequality for high-dimensional bipartite systems obtained by binning local measurement outcomes and show that it is tight. We find a binning method for even ddd-dimensional measurement outcomes for which this Bell inequality is maximally violated by maximally entangled states. Furthermore, we demonstrate that the Bell inequality is applicable to continuous variable systems and yields strong violations for two-mode squeezed states.
We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four-dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two-dimensional systems, i.e., qubits, are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate ``measurements'' in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.
We generalize Bell's inequalities to biparticle systems with continuous quantum variables. This is achieved by introducing the Bell operator in perfect analogy to the usual spin-1/2 systems. It is then demonstrated that two-mode squeezed vacuum states display quantum nonlocality by using the generalized Bell operator. In particular, the original Einstein-Podolsky-Rosen entangled states, which are the limiting case of the two-mode squeezed vacuum states, can maximally violate Bell's inequality due to Clauser, Horne, Shimony and Holt. The experimental aspect of our scheme and nonlocality of arbitrary biparticle entangled pure states of continuous variables are briefly considered.
A Bell inequality is a fundamental test to rule out local hidden variable model descriptions of correlations between two physically separated systems. There have been a number of experiments in which a Bell inequality has been violated using discrete-variable systems. We demonstrate a violation of Bell's inequality using continuous variable quadrature measurements. By creating a four-mode entangled state with homodyne detection, we recorded a clear violation with a Bell value of B=2.31±0.02. This opens new possibilities for using continuous variable states for device independent quantum protocols.
In the present article, based on the formalism introduced in [Loubenets, J. Math. Phys. 53, 022201 (2012)], we derive for a pure bipartite quantum state a new upper bound on its maximal violation of general Bell inequalities. This new bound indicates that, for an infinite dimensional pure bipartite state with a finite sum of its Schmidt coefficients, violation of any general Bell inequality is bounded from above by the value independent on a number of settings and a type of outcomes, continuous or discrete, specific to this Bell inequality. As an example, we apply our new general results to specifying upper bounds on the maximal violation of general Bell inequalities by infinite dimensional bipartite states having the Bell states like forms comprised of two binary coherent states ∣α⟩,∣−α⟩|α\rangle ,|-α\rangle∣α⟩,∣−α⟩, with α>0α>0α>0. We show that, for each of these bipartite coherent states, the maximal violation of general Bell inequalities cannot exceed the value 333 and analyse numerically the dependence of the derived analytical upper bounds on a parameter α>0α>0α>0.