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arXiv:2411.16875

Dynamic violation of Bell's inequalities in the angular momentum representation

Julio A. López-Saldívar, Octavio Castaños, S. Cordero, Ramón López–Peña, Eduardo Nahmad-Achar

Published Nov 25, 2024• 0 citations
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Abstract

A parametrization of density matrices of $d$ dimensions in terms of the raising $J_+$ and lowering $J_-$ angular momentum operators is established together with an implicit connection with the generalized Bloch-GellMann parameters. A general expression for the density matrix of the composite system of angular momenta $j_1$ and $j_2$ is obtained. In this matrix representation violations of the Bell-Clauser-Horne-Shimony-Holt inequalities are established for the $X$-states of a qubit-qubit, pure and mixed, composite system, as well as for a qubit-qutrit density matrix. In both cases maximal violation of the Bell inequalities can be reached, i.e., the Cirel'son limit. A correlation between the entanglement measure and a strong violation of the Bell factor is also given. For the qubit-qutrit composite system a time-dependent convex combination of the density matrix of the eigenstates of a two-particle Hamiltonian system is used to determine periodic maximal violations of the Bell's inequality.

DOI
10.48550/arxiv.2411.16875

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A parametrization of density matrices of ddd dimensions in terms of the raising J+J_+J+​ and lowering J−J_-J−​ angular momentum operators is established together with an implicit connection with the generalized Bloch-GellMann parameters. A general expression for the density matrix of the composite system of angular momenta j1j_1j1​ and j2j_2j2​ is obtained. In this matrix representation violations of the Bell-Clauser-Horne-Shimony-Holt inequalities are established for the XXX-states of a qubit-qubit, pure and mixed, composite system, as well as for a qubit-qutrit density matrix. In both cases maximal violation of the Bell inequalities can be reached, i.e., the Cirel'son limit. A correlation between the entanglement measure and a strong violation of the Bell factor is also given. For the qubit-qutrit composite system a time-dependent convex combination of the density matrix of the eigenstates of a two-particle Hamiltonian system is used to determine periodic maximal violations of the Bell's inequality.
A parametrization of density matrices of d dimensions in terms of the raising J + and lowering J − angular momentum operators is established together with an implicit connection with the generalized Bloch-GellMann parameters. A general expression for the density matrix of the composite system of angular momenta j 1 and j 2 is obtained. In this matrix representation violations of the Bell-Clauser-Horne-Shimony-Holt inequalities are established for the X -states of a qubit-qubit, pure and mixed, composite system, as well as for a qubit-qutrit density matrix. In both cases maximal violation of the Bell inequalities can be reached, i.e., the Cirel’son limit. A correlation between the entanglement measure and a strong violation of the Bell factor is also given. For the qubit-qutrit composite system a time-dependent convex combination of the density matrix of the eigenstates of a two-particle Hamiltonian system is used to determine periodic maximal violations of the Bell’s inequality.
Orbital angular momentum entangled photons emitted by a down-conversion source are shown to be in higher dimensional entangled states. This is experimentally confirmed by demonstrating a violation of Clauser-Horne-Shimony-Holt-type Bell inequalities. The experiment is performed in two steps. First, the two-photon two-dimensional entanglement is demonstrated for orbital angular momentum states by violating the usual CHSH Bell inequality. In a second step the setup is extended in order to show that the entanglement also holds in three dimensions. By using a generalized type of the CHSH Bell inequality, a violation of more than 18 standard deviations is achieved. Higher dimensional entangled states thus allow the realization of new types of quantum communication protocols. They also provide a more secure quantum cryptography scheme.
The paper deals with the structure of Bell’s inequalities (Clauser, Horne, Shimony, and Holt (CHSH) form). It is established that 2 is the universal bound for Bell’s correlations given by a general correlation duality on a complex linear space. Using geometric arguments the maximal violation of Bell’s inequalities is described on this abstract level. It is proved that Bell’s inequalities are maximally violated for general ∗-algebras and a faithful state exactly when the corresponding elements are the Pauli spin matrices. Interesting structural consequences of this result are derived. They demonstrate that the maximal violation requires a very “nonclassical” position of the local systems. Moreover, the maximal strength of Bell’s correlations for a general state with respect to nearly commuting subalgebras is characterized in terms of the spin systems and tracial properties of the local states.
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.
Cirel'son inequality states that the absolute value of the combination of quantum correlations appearing in the Clauser-Horne-Shimony-Holt (CHSH) inequality is bound by 2 square root of (2). It is shown that the correlations of two qubits belonging to a three-qubit system can violate the CHSH inequality beyond 2 square root of (2). Such a violation is not in conflict with Cirel'son's inequality because it is based on postselected systems. The maximum allowed violation of the CHSH inequality, 4, can be achieved using a Greenberger-Horne-Zeilinger state.