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.
Einstein, A., Podolsky, B. and Rosen, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review , 47, 777-780. https://doi.org/10.1103/physrev.47.777
Bell, J.S. (1964) On the Einstein Podolsky Rosen Paradox. Physics Physique Fizika , 1, 195-200. https://doi.org/10.1103/physicsphysiquefizika.1.195
Bell, J.S. (1966) On the Problem of Hidden Variables in Quantum Mechanics. Reviews of Modern Physics , 38, 447-452. https://doi.org/10.1103/revmodphys.38.447
Bohm, D. and Aharonov, Y. (1957) Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky. Physical Review , 108, 1070-1076. https://doi.org/10.1103/physrev.108.1070
Clauser, J.F. and Shimony, A. (1978) Bell’s Theorem. Experimental Tests and Implications. Reports on Progress in Physics , 41, 1881-1927. https://doi.org/10.1088/0034-4885/41/12/002
Freedman, S.J. and Clauser, J.F. (1972) Experimental Test of Local Hidden-Variable Theories. Physical Review Letters , 28, 938-941. https://doi.org/10.1103/physrevlett.28.938
Aspect, A., Grangier, P. and Roger, G. (1982) Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment : A New Violation of Bell’s Inequalities. Physical Review Letters , 49, 91-94. https://doi.org/10.1103/physrevlett.49.91
Lo, T.K. and Shimony, A. (1981) Proposed Molecular Test of Local Hidden-Variables Theories. Physical Review A , 23, 3003-3012. https://doi.org/10.1103/physreva.23.3003
Aspect, A. (1976) Proposed Experiment to Test the Nonseparability of Quantum Mechanics. Physical Review D , 14, 1944-1951. https://doi.org/10.1103/physrevd.14.1944
Khalfin, L.A. and Tsirelson, B.S. (1992) Quantum/Classical Correspondence in the Light of Bell’s Inequalities. Foundations of Physics , 22, 879-948. https://doi.org/10.1007/bf01889686
Cirel’son, B.S. (1980) Quantum Generalizations of Bell’s Inequality. Letters in Mathematical Physics , 4, 93-100. https://doi.org/10.1007/bf00417500
Buhrman, H. and Massar, S. (2005) Causality and Tsirelson’s Bounds. Physical Review A , 72, Article ID: 052103. https://doi.org/10.1103/physreva.72.052103
Gisin, N. (1991) Bell’s Inequality Holds for All Non-Product States. Physics Letters A , 154, 201-202. https://doi.org/10.1016/0375-9601(91)90805-i
Leslie, N., Devin, J. and Lynn, T.W. (2019) Maximal LELM Distinguishability of Qubit and Qutrit Bell States Using Projective and Non-Projective Measurements. https://arxiv.org/abs/1903.02655
Méndez Martínez, J.M. (2023) Nonlocal Correlations of a Fully Entangled Qubit-Qutrit Bell Scenario. Physics Letters A , 492, Article ID: 129216. https://doi.org/10.1016/j.physleta.2023.129216
Bernal, A., Casas, J.A. and Moreno, J.M. (2024) Optimal Bell Inequalities for Qubitqudit Systems. https://arxiv.org/abs/2404.02092
Sorella, S.P. (2024) Bell’s and Mermin’s Inequalities, Entangled Coherent States and Unitary Operators. International Journal of Theoretical Physics , 63, Article No. 227. https://doi.org/10.1007/s10773-024-05764-y
Gisin, N. and Peres, A. (1992) Maximal Violation of Bell’s Inequality for Arbitrarily Large Spin. Physics Letters A , 162, 15-17. https://doi.org/10.1016/0375-9601(92)90949-m
Peruzzo, G. and Sorella, S.P. (2023) Entanglement and Maximal Violation of the CHSH Inequality in a System of Two Spins J: A Novel Construction and Further Observations. Physics Letters A , 474, Article ID: 128847. https://doi.org/10.1016/j.physleta.2023.128847
Peres, A. (1992) Finite Violation of a Bell Inequality for Arbitrarily Large Spin. Physical Review A , 46, 4413-4414. https://doi.org/10.1103/physreva.46.4413
Gerry, C.C. and Albert, J. (2005) Finite Violations of a Bell Inequality for High Spin: An Optical Realization. Physical Review A , 72, Article ID: 043822. https://doi.org/10.1103/physreva.72.043822
Werner, R.F. (1989) Quantum States with Einstein-Podolsky-Rosen Correlations Admitting a Hidden-Variable Model. Physical Review A , 40, 4277-4281. https://doi.org/10.1103/physreva.40.4277
Rau, A.R.P. (2009) Algebraic Characterization of x -States in Quantum Information. Journal of Physics A : Mathematical and Theoretical , 42, Article ID: 412002. https://doi.org/10.1088/1751-8113/42/41/412002
Kelleher, C., Holweck, F., Lévay, P. and Saniga, M. (2021) X-States from a Finite Geometric Perspective. Results in Physics , 22, Article ID: 103859. https://doi.org/10.1016/j.rinp.2021.103859
Rau, A.R.P. (2000) Manipulating Two-Spin Coherences and Qubit Pairs. Physical Review A , 61, Article ID: 032301. https://doi.org/10.1103/physreva.61.032301
Brüning, E., Mäkelä, H., Messina, A. and Petruccione, F. (2012) Parametrizations of Density Matrices. Journal of Modern Optics , 59, 1-20. https://doi.org/10.1080/09500340.2011.632097
Horodecki, R. (2021) Quantum Information. Acta Physica Polonica A , 139, 197-2018. https://doi.org/10.12693/aphyspola.139.197
Clauser, J.F., Horne, M.A., Shimony, A. and Holt, R.A. (1969) Proposed Experiment to Test Local Hidden-Variable Theories. Physical Review Letters , 23, 880-884. https://doi.org/10.1103/physrevlett.23.880
Plenio, M.B. (2005) Logarithmic Negativity: A Full Entanglement Monotone That Is Not Convex. Physical Review Letters , 95, Article ID: 090503. https://doi.org/10.1103/physrevlett.95.090503
Verstraete, F. and Wolf, M.M. (2002) Entanglement versus Bell Violations and Their Behavior under Local Filtering Operations. Physical Review Letters , 89, Article ID: 170401. https://doi.org/10.1103/physrevlett.89.170401
Wootters, W.K. (1998) Entanglement of Formation of an Arbitrary State of Two Qubits. Physical Review Letters , 80, 2245-2248. https://doi.org/10.1103/physrevlett.80.2245
Schlienz, J. and Mahler, G. (1995) Description of Entanglement. Physical Review A , 52, 4396-4404. https://doi.org/10.1103/physreva.52.4396
Bennett, C.H., DiVincenzo, D.P., Smolin, J.A. and Wootters, W.K. (1996) Mixed-state Entanglement and Quantum Error Correction. Physical Review A , 54, 3824-3851. https://doi.org/10.1103/physreva.54.3824
Rose, M.E. (2011) Elementary Theory of Angular Momentum. Dover Publications.
Messiah, A. (1981) Angular Momentum in Quantum Mechanics. Amsterdam.
Cirel’son, B.S. (1980) Quantum Generalizations of Bell’s Inequality. Letters in Mathematical Physics , 4, 93-100. https://doi.org/10.1007/bf00417500