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arXiv:hep-ph/0010108

A generalized Bell-Inequality and Decoherence for the K^0\bar{K^0}-System

Beatrix C. Hiesmayr

Published Oct 11, 2000• 0 citations
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Abstract

First a generalized Bell-inequality for different times and for different quasi-spin states is developed. We focus on special quasi-spin eigenstates and times. The inequality based on a local realistic theory is violated by the CP-violating parameter, if the quantum theory is used to recalculate the probabilities. Next the quantum mechanical probabilities are modified by the decoherence approach which enables the initial state to factorize spontaneously. In this way we get a lower limit for the decoherence parameter ζ, which measures the degree of decoherence. This result is compared with the experimental value of the decoherence parameter ζdeduced from the data of the CPLEAR-experiment.

DOI
10.48550/arxiv.hep-ph/0010108

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Modern Physics Letters A•Sep 3, 2003•0 citations
First a generalized Bell-inequality for different times and for different quasi-spin states is developed. We focus on special quasi-spin eigenstates and times. The inequality based on a local realistic theory is violated by the CP-violating parameter, if the quantum theory is used to recalculate the probabilities. Next the quantum mechanical probabilities are modified by the decoherence approach which enables the initial state to factorize spontaneously. In this way we get a lower limit for the decoherence parameter ζ, which measures the degree of decoherence. This result is compared with the experimental value of the decoherence parameter ζdeduced from the data of the CPLEAR-experiment.
First a generalized Bell-inequality for dierent times and for dierent quasispin states is developed. We focus on special quasi-spin eigenstates and times. The inequality based on a local realistic theory is violated by the CP-violating parameter [1], if the quantum theory is used to recalculate the probabilities. Next the quantum mechanical probabilities are modied by the decoherence approach which enables the initial state to factorize spontaneously. In this way we get a lower limit for the decoherence parameter , which measures the degree of decoherence. This result is compared with the experimental value [2, 3] of the decoherence parameter deduced from the data of the CPLEAR-experiment [4].
In this talk we discuss the predictions of local realistic theories for the evolution of a K0-K0bar quantum entangled pair created in the decay of the phi-meson. It is shown, in agreement with Bell's theorem, that the most general local hidden-variable model fails in reproducing the whole set of quantum-mechanical observables. We achieve this conclusion by employing two different approaches. In the first approach, the local realistic observables are deduced from the most general premises concerning locality and realism, and Bell-like inequalities are not employed. The other approach makes use of Bell's inequalities. Under particular conditions for the detection times, within the first approach the discrepancy between quantum mechanics and local realism for the asymmetry parameter turns out to be not less than 20%. A similar incompatibility can be made evident by means of a Bell-type test, by employing a Clauser-Horne-Shimony-Holt's inequality written in terms of properly normalized observables. Because of its relatively low experimental accuracy, the data obtained by the CPLEAR collaboration do not yet allow a decisive test of local realism. Such a test, both with and without the use of Bell's inequalities, should be feasible in the future at the Frascati Phi-factory.
We investigate Bell inequalities for neutral kaon systems from $\ensuremath{\Phi}$ resonance decay to test local realism versus quantum mechanics. We emphasize the unitary time evolution of the states, which means we also include all decay product states, in contrast to other authors. Only this guarantees the use of the complete Hilbert space. We develop a general formalism for Bell inequalities including both arbitrary ``quasispin'' states and different times; finally, we analyze Wigner-type inequalities. They contain an additional term, a correction function h, as compared to the spin 1/21/21/2 or photon case, which changes considerably the possibility of quantum mechanics to violate the Bell inequality. Examples for special ``quasispin'' states are given, especially those that are sensitive to the CP\mathrm{CP}CP parameters $\ensuremath{\varepsilon}$ and ${\ensuremath{\varepsilon}}^{\ensuremath{'}}.$
We discuss the properties of two Bose-Einstein condensates in different spin states, represented quantum mechanically by a double Fock state. Individual measurements of the spins of the particles are performed in transverse directions (perpendicular to the spin quantization axis), giving access to the relative phase of the two macroscopically occupied states. Before the first spin measurement, the phase is completely undetermined; after a few measurements, a more and more precise knowledge of its value emerges under the effect of the quantum measurement process. This naturally leads to the usual notion of a quasiclassical phase (Anderson phase) and to an interesting transposition of the Einstein-Podolsky-Rosen argument to macroscopic physical quantities. The purpose of this paper is to discuss this transposition, as well as situations where the notion of a quasiclassical phase is no longer sufficient to account for the quantum results, and where significant violations of Bell-type inequalities are predicted. Quantum mechanically, the problem can be treated exactly: the probability for all sequences of results can be expressed in the form of a double integral, depending on all parameters that define the experiment (number of particles, number and angles of measurements). We discuss the differences between this case and the usual two-spin case. We discuss the effect of the many parameters that the experimenters can adjust for their measurements, starting with a discussion of the effect of the angles of measurement (the ``settings''), and then envisaging various choices of the functions that are used to obtain violation of Bell-Clauser-Horne-Shimony-Holt inequalities. We then discuss how the ``sample bias loophole'' (often also called ``efficiency loophole'') can be closed in this case, by introducing a preliminary sequence of measurements to localize the particles into ``measurement boxes.'' We finally show that the same nonlocal effects can be observed with distinguishable spins.
We consider strangeness correlations of the Einstein-Podolski-Rosen type in K0K0{K}^{0}{K}^{0}K0K0 pairs created in a ${J}^{\mathrm{PC}}{=1}^{\ensuremath{-}\ensuremath{-}}$ state as a function of time under the hypothesis that spontaneous decoherence takes place. We parametrize the degree of decoherence by a factor $(1\ensuremath{-}\ensuremath{\zeta})$ which multiplies the quantum-mechanical interference terms occurring in the amplitudes for like and unlike strangeness events and discuss the dependence of this procedure on the basis chosen in ${K}^{0}\ensuremath{-}{K}^{0}$ space to which the interference terms correspond. Consequently, all statements about the ``decoherence parameter'' $\ensuremath{\zeta}$ inferred from experimental data are basis dependent as well. We illustrate this point by estimating the value of $\ensuremath{\zeta}$ for the two bases KL,KS{{K}_{L}{,K}_{S}}KL​,KS​ and K0,K0{{K}^{0},{K}^{0}}K0,K0 with the help of recent data of the CPLEAR experiment.
John Ju Sakurai's classical book in quantum mechanics makes a very illuminative presentation that studies entangled states in a two spin s=1/2 particles system in a singlet state. A Bell's inequality emerges as a consequence. Bell's inequality is a relationship among observables that discriminates between Einstein's locality principle and the nonlocal point of view of orthodox quantum mechanics. Following Sakurai's style we propose, by making natural induction, a generalization for Bell's inequality for any two spin-s particles in a singlet state (s integer or half-integer). This inequality is expressed as a function of a θ parameter, which is a measure of the angle between two possible directions in which the spin is measured. Besides the expression for this general inequality we have found that: (a) for any finite half-integer spin Bell's inequality is violated for some interval of the θ-parameter. The right limit of this interval is fixed and equal to π/2, while the left one comes closer and closer to this value as spin number grows. A function fit shows clearly that the size of this θ-interval over which Bell's inequality is violated diminishes asymptotically to zero as 1/s1/2; (b) an analogous behavior for any finite integer spin. For large spins the disagreement between Einstein's locality principle and the nonlocal point of view in orthodox quantum mechanics disappears.