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Physical Review A

Maximally entangled coherent states and strong violations of Bell-type inequalities

Christopher C. Gerry, Jihane Mimih, Adil Benmoussa

Published Aug 17, 2009• 38 citations
Publisher

Abstract

Recently, Wildfeuer et al. [Phys. Rev. A 76, 052101 (2007)] studied possible experiments demonstrating nonlocal correlation effects through the violation of various Bell-type inequalities by maximally path-entangled number states of the form $(|N⟩|0⟩+{e}^{i\ensuremath{\varphi}}|0⟩|N⟩)/\sqrt{2}$, the so-called $N00N$ states, and some strong violations were found. In this paper, we re-examine the same Bell-type inequalities with respect to maximally path-entangled coherent states of the form $\mathcal{N}({|\ensuremath{\alpha}⟩}_{a}{|0⟩}_{b}+{e}^{i\ensuremath{\Phi}}{|0⟩}_{a}{|\ensuremath{\alpha}\text{ }{e}^{i\ensuremath{\theta}}⟩}_{b})$. We find in many cases even stronger violations of the Bell-type inequalities than appear to be possible with the $N00N$ states.

DOI
10.1103/physreva.80.022111

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Recently, Wildfeuer et al. [Phys. Rev. A 76, 052101 (2007)] studied possible experiments demonstrating nonlocal correlation effects through the violation of various Bell-type inequalities by maximally path-entangled number states of the form (∣N⟩∣0⟩+ei\ensuremathφ∣0⟩∣N⟩)/2(|N⟩|0⟩+{e}^{i\ensuremath{\varphi}}|0⟩|N⟩)/\sqrt{2}(∣N⟩∣0⟩+ei\ensuremathφ∣0⟩∣N⟩)/2​, the so-called N00NN00NN00N states, and some strong violations were found. In this paper, we re-examine the same Bell-type inequalities with respect to maximally path-entangled coherent states of the form N(∣\ensuremathα⟩a∣0⟩b+ei\ensuremathΦ∣0⟩a∣\ensuremathα ei\ensuremathθ⟩b)\mathcal{N}({|\ensuremath{\alpha}⟩}_{a}{|0⟩}_{b}+{e}^{i\ensuremath{\Phi}}{|0⟩}_{a}{|\ensuremath{\alpha}\text{ }{e}^{i\ensuremath{\theta}}⟩}_{b})N(∣\ensuremathα⟩a​∣0⟩b​+ei\ensuremathΦ∣0⟩a​∣\ensuremathα ei\ensuremathθ⟩b​). We find in many cases even stronger violations of the Bell-type inequalities than appear to be possible with the N00NN00NN00N states.
We show that nonlocal correlation experiments on the two spatially separated modes of a maximally path-entangled number state may be performed. They lead to a violation of a Clauser-Horne Bell inequality for any finite photon number NNN. We also present an analytical expression for the two-mode Wigner function of a maximally path-entangled number state and investigate a Clauser-Horne-Shimony-Holt Bell inequality for such a state. We test other Bell-type inequalities. Some are violated by a constant amount for any NNN.
Recent numerical investigations [K. P\'al and T. V\'ertesi, Phys. Rev. A 82, 022116 (2010)] suggest that the I3322 inequality, arguably the simplest extremal Bell inequality after the CHSH inequality, has a very rich structure in terms of the entangled states and measurements that maximally violate it. Here we show that for this inequality the maximally entangled state of any dimension achieves the same violation than just a single EPR pair. In contrast, stronger violations can be achieved using higher dimensional states which are less entangled. This shows that the maximally entangled state is not the most nonlocal resource, even when one restricts attention to the most simple extremal Bell inequalities.
Entanglement of quasiclassical (coherent) states of two harmonic oscillators leads to striking quantum effects and is useful for quantum technologies. These effects and applications are closely related to nonlocal correlations inherent in these states, manifested by the violation of Bell inequalities. With previous frameworks, this violation is limited by the size of the system, which does not approach the maximum, even when the amount of entanglement approaches its maximum. Here, we propose a new version of Bell correlation operators, with which a nearly maximal violation can be obtained, as long as the associated entanglement approximates to the maximum. Consequently, the revealed nonlocality is significantly stronger than those with previous frameworks for a wide range of the system size. We present a new scheme for realizing the gate necessary for measurement of the nonlocal correlations. In addition to the use in test of quantum nonlocality, this gate is useful for quantum information processing with coherent states.
The possibility of producing entangled superpositions of strong coherent states is discussed. A recent proposal by Howell and Yeazell (2000 Phys. Rev. A 62 012102) of a device that entangles two strong coherent states is modified. It is shown that the modified scheme can generate non-classical states that violate Bell inequality. Moreover, a detailed analysis of the effect of losses and decoherence on the degree of entanglement is accomplished. It reveals the high sensitivity of the device to any disturbances and the fragility of generated states.
In a recent article [Phys. Rev. A 54, 1793 (1996)] Krenn and Zeilinger investigated the conditioned two-particle correlations for the subensemble of data obtained by selecting the results of the spin measurements by two observers 1 and 2 with respect to the result found in the corresponding measurement by a third observer. In this paper we write out explicitly the condition required in order for the selected results of observers 1 and 2 to violate Bell's inequality for general measurement directions $e{\ensuremath{\rightarrow}}_{1},$ $e{\ensuremath{\rightarrow}}_{1}^{\ensuremath{'}},$ $e{\ensuremath{\rightarrow}}_{2},$ $e{\ensuremath{\rightarrow}}_{2}^{\ensuremath{'}},$ and $e{\ensuremath{\rightarrow}}_{3}.$ It is shown that there are infinitely many sets of directions giving the maximum level of violation. Further, we extend the analysis by the authors to the class of triorthogonal states $|\ensuremath{\Psi}〉{=c}_{1}|{z}_{1}〉|{z}_{2}〉|{z}_{3}〉{+c}_{2}|\ensuremath{-}{z}_{1}〉|\ensuremath{-}{z}_{2}〉|\ensuremath{-}{z}_{3}〉.$ It is found that a maximal violation of Bell's inequality occurs provided the corresponding three-particle state yields a direct (``all or nothing'') nonlocality contradiction.