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Cambridge University Press eBooks

Maximal Violation of the Original Bell Inequality: Two-Qubit and -Qutrit States

Andrei Yu. Khrennikov

Published Nov 28, 2024• 0 citations
Publisher

Abstract

In this chapter we introduce the general class of symmetric two-qubit statesguaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomeswhenever some spin observable is measured at both sites. It is proventhat, for all states from this class, the maximal violation of the original Bellinequality (OB) is upper bounded by 3/2 and specify the two-qubit stateswhere this quantum upper bound is attained. The case of two-qutrit statesis more complicated. Here, for all two-qutrit states, we obtain the same upperbound 3/2 for violation of the original Bell inequality under Alice and Bobspin measurements. But it has not yet been shown that this quantum upperbound is the least one. The experimental consequences of this mathematicalstudy are discussed.

DOI
10.1017/9781009313469.010

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In this chapter we introduce the general class of symmetric two-qubit statesguaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomeswhenever some spin observable is measured at both sites. It is proventhat, for all states from this class, the maximal violation of the original Bellinequality (OB) is upper bounded by 3/2 and specify the two-qubit stateswhere this quantum upper bound is attained. The case of two-qutrit statesis more complicated. Here, for all two-qutrit states, we obtain the same upperbound 3/2 for violation of the original Bell inequality under Alice and Bobspin measurements. But it has not yet been shown that this quantum upperbound is the least one. The experimental consequences of this mathematicalstudy are discussed.
We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the maximal violation of the original Bell inequality is upper bounded by 3 2 and specify the two-qubit states where this quantum upper bound is attained. The case of two-qutrit states is more complicated. Here, for all two-qutrit states, we obtain the same upper bound 3 2 for violation of the original Bell inequality under Alice and Bob spin measurements, but we have not yet been able to show that this quantum upper bound is the least one. We discuss experimental consequences of our mathematical study.
Abstract For an even qudit dimension we introduce a class of two-qudit states exhibiting perfect correlations/anticorrelations and prove via the generalized Gell–Mann representation that, for each two-qudit state from this class, the maximal violation of the original Bell inequality is bounded from above by the value 3/2—the upper bound attained on some two-qubit states. We show that the two-qudit Greenberger–Horne–Zeilinger (GHZ) state with an arbitrary even exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results are important steps towards proving in general the 3/2 upper bound on quantum violation of the original Bell inequality. The latter would imply that similarly as the Tsirelson upper bound specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound specifies the quantum analog of the original Bell inequality for all bipartite quantum states with perfect correlations/ anticorrelations. Possible consequences for the experimental tests on violation of the original Bell inequality are briefly discussed.
We introduce a method of investigating qutrit nonclassicality by translating qutrit operators to symmetric two-qubit operators. We show that this procedure sheds light on the discrepancy between maximal qutrit entanglement and maximal nonclassicality of qutrit correlations. Namely, we express Bell operators corresponding to qutrit Bell inequalities in terms of symmetric two-qubit operators and analyze the maximal quantum violation of a given Bell inequality from the qubit perspective. As an example we show that the two-qutrit Collins-Gisin-Linden-Massar-Popescu (CGLMP) Bell inequality can be seen as a combination of Mermin's and Clauser-Horne-Shimony-Holt CHSH) qubit Bell inequalities, and therefore the optimal state violating this combination differs from the one which corresponds to the maximally entangled state of two qutrits. In addition, we discuss the same problem for a three-qutrit inequality. We also demonstrate that the maximal quantum violation of the CGLMP inequality follows from complementarity of correlations.
Abstract In the present paper, based on the general analytical expression [arXiv:2412.03470] for the maximum of the CHSH expectation under local Alice and Bob spin-s measurements in a two-qudit state of dimension d = 2s + 1, s≥1/2, we analyze whether or not, under spin-1 measurements in an arbitrary two-qutrit state, the CHSH inequality is violated. We find analytically for a variety of pure nonseparable two-qutrit states and also, numerically for 1, 000, 000 randomly generated pure nonseparable two-qutrit states, that, under local Alice and Bob spin-1 measurements in each of these nonseparable states, including maximally entangled, the CHSH inequality is not violated. These results together with the spectral decomposition of a mixed state lead us to the Conjecture that, under local Alice and Bob spin-1 measurements, every nonseparable two-qutrit state, pure or mixed, does not violate the CHSH inequality. For a variety of pure two-qutrit states, we further find the values of their concurrence and compare them with the values of their spin-1 CHSH parameter, which determines violation or nonviolation by a two-qutrit state of the CHSH inequality under spin-1 measurements. This comparison indicates that, in contrast to spin- 1 2 measurements, where the spin- 1 2 CHSH parameter of a pure two-qubit state is increasing monotonically with a growth of its entanglement, for a pure two-qutrit state, this is not the case. In particular, for the two-qutrit GHZ state, which is maximally entangled, the spin-1 CHSH parameter is equal to 8 9 , while, for some separable pure two-qutrit states, this parameter can be equal to unity. Moreover, for the two-qutrit Horodecki state, the spin-1 CHSH parameter is equal to 4 2 / 21 < 1 regardless of the entanglement type of this mixed state.
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.
Last years, bounds on the maximal quantum violation of general Bell inequalities were intensively discussed in the literature via different mathematical tools. In the present paper, we analyze quantum violation of general Bell inequalities via the LqHV (local quasi hidden variable) modelling framework, correctly reproducing the probabilistic description of every quantum correlation scenario. The LqHV mathematical framework allows us to derive for all d and N a new upper bound (2d − 1)N−1 on the maximal violation by an N-qudit state of all general Bell inequalities, also, new upper bounds on the maximal violation by an N-qudit state of general Bell inequalities for S settings per site. These new upper bounds essentially improve all the known precise upper bounds on quantum violation of general multipartite Bell inequalities. For some S, d, and N, the new upper bounds are attainable.