Reduction of the Density Matrix and Generalized Bell Inequalities

Reinhard F. Werner

Luigi Schiavulli

V. A. Andreev

The right hand side of Bell inequalities don’t depend on a form of any two‐particle spin state. In the case of the generalized Bell‐Clauser‐Horne‐Schimony‐Holt inequalities the right hand side depend on a form of concrete two‐particle state. The left hand sides of these inequalities depend on four arbitrary vectors defined in three‐dimensional space. They define directions on which spins of particles forming a correlated pair are projected. Our aim is to find such vectors that left‐hand side of the inequality should takes its maximum value. In other words on these vectors the inequality transforms into equality. It is shown that it can be done with the help of special reduction of density matrix of two‐state spin state.

Chang‐shui Yu, He‐Shan Song

In this paper, we present a generalized Bell inequality for mixed states. The distinct characteristic is that the inequality has variable bound depending on the decomposition of the density matrix. The inequality has been shown to be more refined than the previous Bell inequality. It is possible that a separable mixed state can violate the Bell inequality.

Julio A. López-Saldívar, Octavio Castaños, S. Cordero +2 more

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.