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arXiv:1811.10231 · Physical Review A

Classical limit of entangled states of two angular momenta

Marek Kuś, Jan Mostowski, Joanna Pietraszewicz

Published May 14, 2019• 2 citations
arXivHTMLPDF

Abstract

We consider a system of two particles, each with large angular momentum $j$, in the singlet state. The probabilities of finding projections of the angular momenta on selected axes are determined. The generalized Bell inequalities involve these probabilities and we study them using statistical methods. We show that most of Bell's inequalities cannot be violated, or are violated only marginally, in the limit $j\ensuremath{\rightarrow}\ensuremath{\infty}$. The precision required to confirm a violation appears to be difficult to achieve. In practice, the quantum system, in spite of being entangled, becomes indistinguishable from its classical counterpart.

DOI
10.1103/physreva.99.052112

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We consider a system of two particles, each with large angular momentum jjj, in the singlet state. The probabilities of finding projections of the angular momenta on selected axes are determined. The generalized Bell inequalities involve these probabilities and we study them using statistical methods. We show that most of Bell's inequalities cannot be violated, or are violated only marginally, in the limit j\ensuremath→\ensuremath∞j\ensuremath{\rightarrow}\ensuremath{\infty}j\ensuremath→\ensuremath∞. The precision required to confirm a violation appears to be difficult to achieve. In practice, the quantum system, in spite of being entangled, becomes indistinguishable from its classical counterpart.
We investigate two-particle phase-space distributions in classical mechanics constructed to be the analogs of quantum-mechanical angular-momentum eigenstates. We obtain the phase-space averages of specific observables related to the projection of the particles' angular momentum along axes with different orientations, and show that the ensuing correlation function violates Bell's inequality. The key to the violation resides in choosing observables impeding the realization of the joint measurements whose existence is required in the derivation of the inequalities. This situation can have statistical (detection related) or dynamical (interaction related) underpinnings, but non-locality does not play any role.
For a system composed of two particles, Bell's theorem asserts that averages of physical quantities determined from local variables must conform to a family of inequalities. In this work we show that a classical model containing a local probabilistic interaction in the measurement process can lead to a violation of the Bell inequalities. We first introduce two-particle phase-space distributions in classical mechanics constructed to be the analogs of quantum-mechanical angular momentum eigenstates. These distributions are then employed in four schemes characterized by different types of detectors measuring the angular momenta. When the model includes an interaction between the detector and the measured particle leading to ensemble dependencies, the relevant Bell inequalities are violated if the total angular momentum is required to be conserved. The violation is explained by identifying assumptions made in the derivation of Bell's theorem that are not fulfilled by the model, in particular noncommutativity of single-particle measurements. We discuss to what extent a violation of these assumptions is a faithful marker of nonlocality.
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.
Some new, powerful, and very simple Bell inequalities are derived. They establish that for two spin-sss particles in the singlet state, the quantum-theoretic Einstein-Podolsky-Rosen correlations for spin measurements along all pairs of directions from a set of NNN axes are incompatible with local realism for any set of distinct coplanar axes when N=3N=3N=3, and for any set of distinct axes whatever when N=4N=4N=4, with the possible exception of sets restricted by one of the constraints $\stackrel{^}{a}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{b}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{c}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{d}=0$. These results hold for any value of the spin.
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.
The original formula of Bell inequality (BI) in terms of two-spin singlet has to be modified for the entangled-state with parallel spin polarization. Based on classical statistics of the particle-number correlation, we prove in this paper an extended BI, which is valid for two-spin entangled states with both parallel and antiparallel polarizations. The BI and its violation can be formulated in a unified formalism based on the spin coherent-state quantum probability statistics with the state-density operator, which is separated to the local and non-local parts. The local part gives rise to the BI, while the violation is a direct result of the non-local quantum interference between two components of entangled state. The Bell measuring outcome correlation denoted by PBP_{B}PB​ is always less than or at most equal to one for the local realistic model (PBlc≤1P_{B}^{lc}\leq1PBlc​≤1) regardless of the specific superposition coefficients of entangled state. Including the non-local quantum interference the maximum violation of BI is found as PBmax⁡P_{B}^{\max}PBmax​ =2=2=2, which, however depends on state parameters and three measuring directions as well. Our result is suitable for entangled photon pairs.