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arXiv:2306.07739 · Physics Open

Probing Mermin’s inequalities violations through pseudospin operators

Philipe De Fabritiis, I. Roditi, Silvio Paolo sorella

Published Aug 16, 2023• 3 citations
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Abstract

The violation of Mermin’s inequalities is analyzed by making use of two different Bell setups built with pseudospin operators. Employing entangled states defined by means of squeezed and coherent states, the expectation value of Mermin’s polynomials Mn is evaluated for n=3 and n=4. In each case, we analyze the correlator 〈Mn〉 and identify the set of parameters leading to the violation of Mermin’s inequalities and to the saturation of the bound predicted by Quantum Mechanics.

DOI
10.1016/j.physo.2023.100177

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The violation of Mermin’s inequalities is analyzed by making use of two different Bell setups built with pseudospin operators. Employing entangled states defined by means of squeezed and coherent states, the expectation value of Mermin’s polynomials Mn is evaluated for n=3 and n=4. In each case, we analyze the correlator 〈Mn〉 and identify the set of parameters leading to the violation of Mermin’s inequalities and to the saturation of the bound predicted by Quantum Mechanics.
In this paper, we introduce the motivation for Bell inequalities and give some background on two different types: CHSH and Mermin's inequalities. We present a proof for each and then show that certain quantum states can violate both of these inequalities. We introduce a new result stating that for four given measurement directions of spin, two respectively from Alice and two from Bob, which are able to produce a violation of the Bell inequality for an arbitrary shared quantum state will also violate the Bell inequality for a maximally entangled state. Then we provide another new result that characterizes all of the two-qubit states that violate Mermin's inequality.
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Chen et al (2002 Phys. Rev. Lett. 88 040406) proposed a method to measure the entanglement of a bipartite quantum system involving continuous variables. The approach is based upon the construction of a pseudospin operator. An important difference with the spin case is an infinite degeneracy of the eigenkets associated with the pseudospin. In this paper, we analyze the role played by such a degeneracy in the study of the locality of the two-mode-squeezed vacuum state. We construct other representations for the pseudospin and study the corresponding Bell inequalities.
Mermin’s inequalities are investigated in a quantum field theory framework by using von Neumann algebras built with Weyl operators. We devise a general construction based on the Tomita-Takesaki modular theory and use it to compute the vacuum expectation value of the Mermin operator, analyzing the parameter space and explicitly exhibiting a violation of Mermin’s inequalities. Therefore, relying on the power of modular operators, we are able to demonstrate that Mermin’s inequalities are violated when examined within the vacuum state of a scalar field theory. Published by the American Physical Society 2024
A possibility to reveal the entanglement in generalized n-qubit two-parameter GHZ states, as well as in any n-qubit states, with the help of the Mermin and Ardehali inequalities from the collection generally called the Mermin–Ardehali–Belinskii–Klyshko inequalities has been studied. Formulas for the calculation of the Mermin and Ardehali correlation functions in any quantum n-qubit states are derived, and criteria of the violation of corresponding inequalities by specific states are obtained. A set of states that are absolutely insensitive to the Mermin and Ardehali operators is revealed. Modified Mermin and Ardehali operators are proposed, the set of which makes it possible to extend the class of n-qubit states, in which quantum correlations can be revealed.