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arXiv:quant-ph/0312173 · Journal of Physics A Mathematical and General

Effect of the unpolarized spin state in spin-correlation measurement of two protons produced in the12C(d,2He) reaction

S. Hamieh

Published Feb 4, 2004• 0 citations
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Abstract

In this paper we discuss the effect of the unpolarized state in the spin-correlation measurement of the 1 S 0 two-proton state produced in the 12 C( d , 2 He) reaction at the Kernfysisch Versneller Instituut (KVI), Groningen. We show that in the presence of the unpolarized state the maximal violation of the Clauser, Horne, Shimony and Holth (CHSH)–Bell inequality is lower than the classical limit if the purity of the state is less than ∼70%. In particular, for the KVI experiment the violation of the CHSH–Bell inequality should be corrected by a factor of ∼10% from the pure 1 S 0 state.

DOI
10.1088/0305-4470/37/7/018

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In this paper we discuss the effect of the unpolarized state in the spin-correlation measurement of the 1 S 0 two-proton state produced in the 12 C( d , 2 He) reaction at the Kernfysisch Versneller Instituut (KVI), Groningen. We show that in the presence of the unpolarized state the maximal violation of the Clauser, Horne, Shimony and Holth (CHSH)–Bell inequality is lower than the classical limit if the purity of the state is less than ∼70%. In particular, for the KVI experiment the violation of the CHSH–Bell inequality should be corrected by a factor of ∼10% from the pure 1 S 0 state.
Arising out of the EPR debate, the Bell and Wigner inequalities of quantum theory are experimentally testable relations which address the question of quantum completeness. This thesis provides feasibility results for a fermionic test of the Bell inequality, and the first known attempt to investigate the completeness question through the Wigner relation. Such a test is made possible by the production of an entangled p-p singlet at the KVI research facility in Groningen, the Netherlands, through the reactions 12C(d,2He)12B and p(d,2He)n. The p-p spin-correlations are analyzable via the KVI's 2-pi polarimeter acceptance, which eliminates loopholes common to previous experiments. The results distinguish between a hidden variables and quantum mechanical description of the universe. Also presented is a critique of the GHZ argument against the existence of local hidden variables.
Abstract Violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality by a bipartite quantum state is now used in many quantum applications. However, the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-s measurements is still known only for s = 1 / 2 . In the present article, for an arbitrary state of two spin-s qudits, each of dimension d = 2 s + 1 ⩾ 2 , we introduce the notion of the spin-s correlation matrix, which has dimension 3 × 3 for all s ⩾ 1 2 ; establish its relation to the general correlation ( d 2 − 1 ) × ( d 2 − 1 ) matrix of this state within the generalized Pauli representation and derive in terms of the spin-s correlation matrix the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-s measurements in this state. Specifying this general expression for the two-qudit Greenberger–Horne–Zeilinger state, the nonlocal two-qudit Werner state, and some nonseparable pure two-qudit states, we find that, under local Alice and Bob high-spin ( s ⩾ 1 ) measurements in each of these nonseparable states, including the maximally entangled one, the CHSH inequality is not violated. Moreover, unlike the case of spin- 1 / 2 measurements, where each pure nonseparable two-qubit state violates the CHSH inequality and the maximal value of its CHSH expectation increases monotonically with a growth of its entanglement, the situation under high-spin measurements is quite different—for a pure two-qudit state with a higher degree of entanglement, the maximal value of the CHSH expectation turns out to be less than for a pure two-qudit state with lower entanglement and even for a separable one.
We study the violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for two-spin systems, of arbitrary spins j1{j}_{1}j1​ and j2{j}_{2}j2​, prepared in an entanglement of spin-coherent states of each of the spins. We show that the Bell-CHSH inequality is quite robustly violated for a wide range of values of the parameters that specify the spin-coherent states, and, for a particular choice of these parameters, maximal violations are obtained. That is, the violations can reach the Tsirelson bound, 222\sqrt{2}22​, for any choices of the spins.
We show that a pair of massive relativistic spin-1/2 particles prepared in a maximally entangled spin state in general is not capable of maximally violating the Clauser-Horne-Shimony-Holt (CHSH) version of Bell's inequalities without a postselection of the particles' momenta, representing a major difference in relation to nonrelativistic systems. This occurs because the quantization axis of the measurements performed on each particle depends on the particle velocity, such that it is not possible to define a reduced density matrix for the particles' spin. We also show that the amount of violation of the CHSH inequality depends on the reference frame and that in some frames the inequality may not be violated.
The Bell--Clauser-Horne-Shimony-Holt (BCHSH) inequality (in terms of correlation functions) of two qutrits is studied in detail by employing tritter measurements. A uniform formula for the maximum value of this inequality for tritter measurements is obtained. Based on this formula, we show that nonmaximally entangled states violate the BCHSH inequality more strongly than the maximally entangled one. This result is consistent with what was obtained by Ac\'{\i}n et al. [Phys. Rev. A 65,65,65, 052325 (2002)] using the Bell--Clauser-Horne inequality (in terms of probabilities).