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arXiv:2205.02674 · Physical Review A

Geometric interpretation of the Clauser-Horne-Shimony-Holt inequality of nonmaximally entangled states

Johannes Seiler, Thomas Strohm, Wolfgang P. Schleich

Published Sep 24, 2021• 16 citations
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Abstract

We show that for pure and mixed states the problem of maximizing the correlation measure in the Clauser-Horne-Shimony-Holt inequality reduces to maximizing the perimeter of a parallelogram enclosed by an ellipse characterized by the entanglement contained in the bipartite system. Our geometrical description is valid for a nonmaximally entangled state. We also determine the corresponding optimal measurements.

DOI
10.1103/physreva.104.032218

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We show that for pure and mixed states the problem of maximizing the correlation measure in the Clauser-Horne-Shimony-Holt inequality reduces to maximizing the perimeter of a parallelogram enclosed by an ellipse characterized by the entanglement contained in the bipartite system. Our geometrical description is valid for a nonmaximally entangled state. We also determine the corresponding optimal measurements.
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Nonmaximally entangled states are pivotal for quantum technologies, offering enhanced noise resilience and resource efficiency for quantum communication tasks. However, the maximum violation of the multisetting Clauser–Horne–Shimony–Holt (CHSH) inequality in nonmaximally entangled systems has yet to be explored, primarily due to the algebraic complexity associated with such systems. Herein, it is demonstrated that the maximization of the correlation measure in the three‐setting CHSH inequality, for both pure and mixed states, can be elegantly reformulated as a geometric optimization problem: maximizing the perimeter of a triangle inscribed within an ellipse. Leveraging this analytical insight, the maximum violation of the three‐setting CHSH inequality using photonic orbital angular momentum entanglement and spatial light modulators for precise intensity measurements is experimentally realized, observing that the Bell parameter diminishes to the classical bound as entanglement decreases. The findings shed new light on optimal measurement strategies for nonmaximally entangled states, demonstrating conclusively that phase‐only measurements are insufficient for achieving maximum violations in such states. Thus, the work may offer valuable guidance for harnessing nonmaximally entangled states in practical quantum technologies.
The Bell--Clauser-Horne-Shimony-Holt (BCHSH) inequality (in terms of correlation functions) of two qutrits is studied in detail by employing tritter measurements. A uniform formula for the maximum value of this inequality for tritter measurements is obtained. Based on this formula, we show that nonmaximally entangled states violate the BCHSH inequality more strongly than the maximally entangled one. This result is consistent with what was obtained by Ac\'{\i}n et al. [Phys. Rev. A 65,65,65, 052325 (2002)] using the Bell--Clauser-Horne inequality (in terms of probabilities).
Maximally entangled states should maximally violate the Bell inequality. It is proved that all two-qubit states that maximally violate the Bell-Clauser-Horne-Shimony-Holt inequality are exactly Bell states and the states obtained from them by local transformations. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy. The argument is extended to the three-qubit system. Since all states obtained by local transformations of a maximally entangled state are equally valid entangled states, we thus give the characterizations of maximally entangled states in both the two-qubit and three-qubit systems in terms of the Bell inequality.
Imagine two parties, Alice and Bob, who share an entangled quantum state. A well-established result that if Alice performs a two-outcome measurement on the portion of the state in her possession and Bob does likewise, they are able to produce correlations that cannot be reproduced by any classical theory. The allowed classical correlations can be expressed quantitatively by the Bell inequalities. Here we propose new families of Bell inequalities, as a generalization of the Clauser-Horne-Shimony-Holt inequality and show that the maximum violation of these Bell inequalities allowed by quantum theory cannot be attained by a bipartite quantum system having support on a qubit at each site.
We construct a Bell inequality from the Clauser-Horne-Shimony-Holt inequality for two qubits that provides a stronger bound on the correlations of entangled states than allowed by the CHSH inequality. The argument involved here can be generalized to nnn qubits. The inequalities obtained are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits.