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arXiv:hep-ph/0101356 · Physical Review A

Bell inequalities for entangled kaons and their unitary time evolution

Reinhold A. Bertlmann, Beatrix C. Hiesmayr

Published May 17, 2001• 119 citations
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Abstract

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 $1/2$ 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 $\mathrm{CP}$ parameters $\ensuremath{\varepsilon}$ and ${\ensuremath{\varepsilon}}^{\ensuremath{'}}.$

DOI
10.1103/physreva.63.062112

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We investigate Bell inequalities for neutral kaon systems from \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ 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 $1/2$ 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 \ensuremathε\ensuremath{\varepsilon}\ensuremathε and \ensuremathε\ensuremath′.{\ensuremath{\varepsilon}}^{\ensuremath{'}}.\ensuremathε\ensuremath′.
We analyze the premises of recent propositions to test local realism via the Bell inequalities using neutral kaons from $\ensuremath{\Phi}$ resonance decays as entangled Einstein-Podolsky-Rosen pairs. We pay special attention to the derivation of the Bell inequalities, or related expressions, for unstable and oscillating kaon ``quasispin'' states and to the possibility of the actual identification of these states through their associated decay modes. We discuss an indirect method to extract probabilities to find these states by combining experimental information with theoretical input. However, we still find inconsistencies in previous derivations of the Bell inequalities. We show that the identification of the quasispin states via their associated decay mode does not allow the free choice to perform different tests on them, a property which is crucial to establish the validity of any Bell inequality in the context of local realism. In view of this we propose a different kind of Bell inequality in which the free choice or adjustability of the experimental setup is guaranteed. We also show that the proposed inequalities are violated by quantum mechanics.
We extend the use of Bell inequalities to $\ensuremath{\Phi}\ensuremath{\rightarrow}{K}^{0}{\overline{K}}^{0}$ decays by exploiting analogies and differences to the well-known and experimentally verified singlet-spin case. Contrasting with other analyses, our Bell inequalities are violated by quantum mechanics and can strictly be derived from local realistic theories. In principle, quantum mechanics could then be tested using unstable, oscillating states governed by a $\mathrm{CP}\ensuremath{-}\mathrm{violating}$ Hamiltonian.
We shortly review a couple of proposals of Bell inequality tests with entangled pairs of neutral kaons from φ resonance decays.The suggested experiments are discussed in the light of the essential requirements for a genuine discrimination between local hidden-variable theories (i.e., local realism) and quantum mechanics.We clarify the logical limitations of these tests in refuting the whole class of local realistic models.Detailed analyses imply that a Bell-type test with neutral kaons free from the detection loophole requires a few % strangeness detection efficiencies and very high efficiencies for the detection of the kaon decay products.
The nonlocal property of quantum mechanics can be nicely tested in high energy physics; in particular, the neutral kaon pairs as produced at DAΦΦΦNE, Frascati, are very well suited. The analogies of kaons as compared to polarized photons or spin--1/2 particles --the kaonic qubit feature-- are reviewed. However, there are also fundamental differences which occur due to the kaon time evolution and due to internal symmetries; in particular, the violation of CP symmetry is related to the violation of Bell inequalities. Two type of Bell inequalities for kaons are presented, one for the variation of the ``quasi--spin'' and the other for different detection times of the kaon.
We derive a Bell inequality based on a generalized quasiprobability function, which is parametrized by one nonpositive real value. Two types of known Bell inequalities formulated in terms of the Wigner and the QQQ functions are included as limiting cases. We investigate violations of our Bell inequalities for single-photon entangled states and two-mode squeezed vacuum states when varying the detector efficiency. We show that the Bell inequality for the QQQ function allows the lowest detection efficiency for violations of local realism.
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.