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Modern Physics Letters B

SPIN-PARITY EFFECT IN VIOLATION OF BELL'S INEQUALITIES

Zhigang Song, J.-Q. Liang, L. F. Wei

Published Dec 18, 2013• 5 citations
Publisher

Abstract

Analytic formulas of Bell correlations are derived in terms of quantum probability statistics under the assumption of measuring outcome-independence and the Bell's inequalities (BIs) are extended to general bipartite-entanglement macroscopic quantum-states (MQS) of arbitrary spins. For a spin-½ entangled state we find analytically that the violations of BIs really resulted from the quantum nonlocal correlations. However, the BIs are always satisfied for the spin-1 entangled MQS. More generally the quantum nonlocality does not lead to the violation for the integer spins since the nonlocal interference effects cancel each other by the quantum statistical-average. Such a cancellation no longer exists for the half-integer spins due to the nontrivial Berry phase, and thus the violation of BIs is understood remarkably as an effect of geometric phase. Specifically, our generic observation of the spin-parity effect can be experimentally tested with the entangled photon-pairs.

DOI
10.1142/s0217984914500043

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Analytic formulas of Bell correlations are derived in terms of quantum probability statistics under the assumption of measuring outcome-independence and the Bell's inequalities (BIs) are extended to general bipartite-entanglement macroscopic quantum-states (MQS) of arbitrary spins. For a spin-½ entangled state we find analytically that the violations of BIs really resulted from the quantum nonlocal correlations. However, the BIs are always satisfied for the spin-1 entangled MQS. More generally the quantum nonlocality does not lead to the violation for the integer spins since the nonlocal interference effects cancel each other by the quantum statistical-average. Such a cancellation no longer exists for the half-integer spins due to the nontrivial Berry phase, and thus the violation of BIs is understood remarkably as an effect of geometric phase. Specifically, our generic observation of the spin-parity effect can be experimentally tested with the entangled photon-pairs.
We in this Letter derive analytic formulas of Bell correlations in terms of quantum probability statistics under the assumption of measuring outcome-independence. For a spin-1/2 singlet state we find analytically that the violations of Bell-type inequalities are really related to the quantum non-local correlations. However, the Bell and Clauser-Horne-Shimony-Holt (CHSH) inequalities are always satisfied for the spin-1 singlet states. More generally the quantum non-locality does not lead to the violation of Bell and CHSH inequalities for the integer-spin singlet since the non-local interference effects cancel each other by the quantum statistical-average. Such a cancellation no longer exists for the half-integer spin singlets due to the nontrivial Berry phase, and thus the relevant Bell-type inequalities can be violated. Specifically, our generic observations can be experimentally tested with the entangled photon-pairs. Our arguments are based on the spin-singlet states, but could be generalized to other bipartite quantum states.
Bell’s inequalities (BIs) derived in terms of quantum probability statistics are extended to general bipartite-entangled states of arbitrary spins with parallel polarization. The original formula of Bell for the two-spin singlet is slightly modified in the parallel configuration, while the inequality formulated by Clauser–Horne–Shimony–Holt (CHSH) remains unchanged. The violation of BIs indeed resulted from the quantum nonlocal correlation for spin-[Formula: see text] case. However, the inequalities are always satisfied for the spin-1 entangled states regardless of parallel or antiparallel polarizations of two spins. The spin parity effect originally demonstrated with the antiparallel spin-polarizations (Z. Song, J.-Q. Liang and L.-F. Wei, Mod. Phys. Lett. B 28 (2013) 145004) still exists for the parallel case. The quantum nonlocality does not lead to the violation for integer spins due to the cancellation of nonlocal interference effects by the quantum statistical average. Again, the violation of BIs seems to be a result of the measurement-induced nontrivial Berry phase (BP) for half-integer spins.
In terms of quantum probability statistics, the Bell inequality (BI) and its violation are extended to spin-[Formula: see text] entangled Schrödinger cat state (called the Bell cat state) with both parallel and antiparallel spin-polarizations. Except the spin-1/2, the BI is never ever violated by the Bell cat states with the measuring outcomes including entire Hilbert space. If, on the other hand, measuring outcomes are restricted in the subspace of spin coherent state (SCS), a universal Bell-type inequality (UBI), [Formula: see text], is formulated in terms of the local realistic model. We observe a spin parity effect that the UBI can be violated only by the Bell cat states of half-integer but not the integer spins. The violation of UBI is seen to be a direct result of nontrivial Berry phase between the SCSs of south- and north-pole gauges for half-integer spin, while the geometric phase is trivial for the integer spins. A maximum violation bound of UBI is found as [Formula: see text], which is valid for arbitrary half-integer spin-[Formula: see text] states.
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.
This paper proposes a generalized Bell-like inequality (GBI) for multiparticle entangled Schrödinger-cat--states of arbitrary spin-sss. Based on quantum probability statistics the GBI and violation are formulated in an unified manner with the help of state density operator, which can be separated to local and non-local parts. The local part gives rise to the inequality, while the non-local part is responsible for the violation. The GBI is not violated at all by quantum average except the spin-1/21/21/2 entangled states. If the measuring outcomes are restricted in the subspace of spin coherent state (SCS), namely, only the maximum spin values ±s\pm s±s, the GBI is still meaningful for the incomplete measurement. With the help of SCS quantum probability statistics, it is proved that the violation of GBI can occur only for half-integer spins but not integer spins. Moreover, the maximum violation bound depends on the number parity of entangled particles, that it is 1/21/21/2 for the odd particle-numbers while 111 for even numbers.
Quantum nonlocality is presented often as the most remarkable and inexplicable phenomenon known to modern science. It has been known already for a long time that the probabilistic models used to prove Bell and Clauser-Horn-Shimony-Holt inequalities (BI-CHSH) for spin polarization correlation experiments (SPCE) are incompatible with the experimental protocols of SPCE. In particular these models use the same common probability space, joint probability distributions and/or conditional independence to describe coincidence experiments in incompatible experimental settings. Strangely enough these results are not known or simply neglected. This is why we will once again reanalyze Bell locality assumptions and show that they have nothing to do with the notion of Einsteinian locality therefore their violation should not be called quantum nonlocality but rather quantum non-Kolmogorovness or quantum contextuality. Moreover if local variables describing the measuring instruments are correctly taken into account then BI-CHSH can no longer be proven and one can easily construct non-signaling probabilistic models able to reproduce the predictions of QT. The violation of BI-CHSH is considered usually as a proof that a quantum state is entangled. Since BI-CHSH are violated also in some experiments from outside the domain of quantum physics therefore the entanglement is not exclusively a quantum phenomenon. In order to further demystify these notions we show that one can prepare two macroscopic systems in such a way that simple realizable local experiments on these systems violate BI. In view of these arguments the further testing of BI-CHSH inequalities in search for the loopholes does not seem to be necessary.