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arXiv:1905.04650 · Optics Express

Nearly maximal violation of the Mermin-Klyshko inequality with multimode entangled coherent states

Jian-Qi Sheng, Li-Hua Lin, Zhi-Rong Zhong, Shi-Biao Zheng

Published Oct 18, 2019• 3 citations

Abstract

Entangled coherent states for multiple bosonic modes, also referred to as multimode cat states, not only are of fundamental interest but also have practical applications. The nonclassical correlation among these modes is well characterized by the violation of the Mermin-Klyshko inequality. We here study Mermin-Klyshko inequality violations for such multi-mode entangled states with rotated quantum-number parity operators. It is shown that the Mermin-Klyshko signal obtained with these operators can approach the maximal value even when the average quantum number in each mode is only 1, and the inequality violation exponentially increases with the number of entangled modes. This is in distinct contrast with the framework based on displaced parity operators, with which a nearly maximal Mermin-Klyshko inequality violation requires the size of the cat state to be increased by about 15 times.

DOI: 10.1364/oe.27.031864

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Entangled coherent states for multiple bosonic modes, also referred to as multimode cat states, not only are of fundamental interest but also have practical applications. The nonclassical correlation among these modes is well characterized by the violation of the Mermin-Klyshko inequality. We here study Mermin-Klyshko inequality violations for such multi-mode entangled states with rotated quantum-number parity operators. It is shown that the Mermin-Klyshko signal obtained with these operators can approach the maximal value even when the average quantum number in each mode is only 1, and the inequality violation exponentially increases with the number of entangled modes. This is in distinct contrast with the framework based on displaced parity operators, with which a nearly maximal Mermin-Klyshko inequality violation requires the size of the cat state to be increased by about 15 times.

We investigate the violation of local realism in Bell tests involving homodyne measurements performed on multimode continuous-variable states. By binning the measurement outcomes in an appropriate way, we prove that the Mermin-Klyshko inequality can be violated by an amount that grows exponentially with the number of modes. Furthermore, the maximum violation allowed by quantum mechanics can be attained for any number of modes, albeit requiring a quantum state whose generation is hardly practicable. Interestingly, this exponential increase of the violation holds true even for simpler states, such as multipartite GHZ states. The resulting benefit of using more modes is shown to be significant in practical multipartite Bell tests by analyzing the increase of the robustness to noise with the number of modes. In view of the high efficiency achievable with homodyne detection, our results thus open a possible way to feasible loophole-free Bell tests that are robust to experimental imperfections. We provide an explicit example of a three-mode state (a superposition of coherent states) which results in a significantly high violation of the Mermin-Klyshko inequality (around 10%) with homodyne measurements.

We study the violation of Bell-Mermin-Klyshko (BMK) inequalities in initial quantum states of scalar fields in inflation. We show that the Bell inequality is maximally violated by the Bunch-Davies vacuum which is a two-mode squeezed state of a scalar field. However, we find that the violation of the BMK inequalities does not increase with the number of modes to measure. We then consider a non-Bunch-Davies vacuum expressed by a four-mode squeezed state of two scalar fields. Remarkably, we find that the violation of the BMK inequalities increases exponentially with the number of modes to measure. This shows that the BMK inequalities are useful to classify the initial quantum state of the Universe.

In this paper, we characterize the maximal violation of Ardehali's inequality of

n

qubits by showing that GHZ's states and the states obtained from them by local unitary transformations are the unique states that maximally violate the Ardehali's inequalities. This concludes that Ardehali's inequalities can be used to characterize maximally entangled states of

n

qubits, as the same as Mermin's and Bell-Klyshko's inequalities.

This paper proposes a generalized Bell-like inequality (GBI) for multiparticle entangled Schr\"odinger cat states of arbitrary spin

s

. Based on quantum probability statistics, the GBI and violation are formulated in a unified manner with the help of a state density operator, which can be separated into local and nonlocal parts. The local part gives rise to the inequality, while the nonlocal part is responsible for the violation. The GBI is not violated at all by the quantum average, except the spin-

1/2

entangled states. If the measuring outcomes are restricted in the subspace of the spin coherent state (SCS), namely, only the maximum spin values $\ifmmode\pm\else\textpm\fi{}s$, the GBI is still meaningful for the incomplete measurement. With the help of SCS quantum probability statistics, it is proved that the violation of GBI can occur only for half-integer spins and not integer spins. Moreover, the maximum violation bound depends on the number parity of the entangled particles, which is

1/2

for the odd particle numbers and 1 for the even numbers.