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AIP conference proceedings

Quantum Tomography and Verification of Generalized Bell-CHSH Inequalities

V. A. Andreev

Published Jan 1, 2005• 1 citation
Publisher

Abstract

It is shown that verification procedure of Bell inequalities consists of three parts. The first stage of verification procedure is consist of measurements of a set of spin correlations. Using them one can reconstruct a density matrix of two‐particle state. It can be done with the help of quantum tomography technique. At the second stage one must construct a generalized Bell inequality corresponding to this state. This inequality must be constructed in two forms: within the framework of the traditional formulation of quantum mechanics and in the frames of a local hidden variables theory. Only after that the results of experimental measurements can be compared with theoretical predictions. It is a third stage of verification.

DOI
10.1063/1.1874556

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It is shown that verification procedure of Bell inequalities consists of three parts. The first stage of verification procedure is consist of measurements of a set of spin correlations. Using them one can reconstruct a density matrix of two‐particle state. It can be done with the help of quantum tomography technique. At the second stage one must construct a generalized Bell inequality corresponding to this state. This inequality must be constructed in two forms: within the framework of the traditional formulation of quantum mechanics and in the frames of a local hidden variables theory. Only after that the results of experimental measurements can be compared with theoretical predictions. It is a third stage of verification.
The right hand side of Bell inequalities don’t depend on a form of any two‐particle spin state. In the case of the generalized Bell‐Clauser‐Horne‐Schimony‐Holt inequalities the right hand side depend on a form of concrete two‐particle state. The left hand sides of these inequalities depend on four arbitrary vectors defined in three‐dimensional space. They define directions on which spins of particles forming a correlated pair are projected. Our aim is to find such vectors that left‐hand side of the inequality should takes its maximum value. In other words on these vectors the inequality transforms into equality. It is shown that it can be done with the help of special reduction of density matrix of two‐state spin state.
Bell’s inequality can be traced back to a classical probabilistic formula that is invalid clearly, while spin correlation formula in quantum mechanics to a formula that has confirmed both by facts and quantum mechanics. Bell’s two misunderstandings about Bell’s theorem are pointed out: Firstly, the hidden variable theory Bell’s used, which is a special form of such theories, is regarded as a general one. Secondly, there are two theses obtainable from Bell’s hypotheses; one is compatible with quantum mechanics, the other leads to Bell's inequality. Unfortunately, the former is regarded as a property of local hidden variable theory, while the latter as self-evident. G.Lochak has revealed the first thesis, and thereby he pointed out that Bell’s inequality has nothing to do with locality, but he did not find the second one, so that he still analyzed this problem starting from hidden variable theory, which is actually without regard to Bell’s inequality.
The Bell inequality two-particle spin states are considered. It is shown that violation of this inequality at experimental verifications is connected with the fact that it is proved for some arbitrary random variables, but in experimental verification random variables of special type are used. A new inequality is constructed. It contains a correlation coefficient of random variables, measured at the experiment, and does not have to be violated at experimental verification. For factorizable and separable states it coincides with the usual Bell inequality.
Bell inequalities were meant to test quantum mechanics vs local hidden variable models, but can also be used to verify entanglement. For entanglement verification purposes one assumes the validity of quantum mechanics as well as quantum descriptions of one's measurements. With the help of these assumptions it is possible to derive a strengthened Bell inequality whose violation implies entanglement. We generalize known examples of such inequalities by relating the expectation value of the Bell operator to a particular quantitative measure of entanglement, namely, the negativity. Moreover, we obtain statistics illustrating the fact that violating a given (strengthened or not) Bell inequality is a much more rare feat for a quantum state of two qubits than it is to be entangled.
The Bell theorem and inequality were derived as consequences of seemingly reasonable physical and statistical hypotheses. Bell’s assumptions were used to deduce cross-correlations of three spin measurements on two entangled particles neglecting non-commutation. The assumed correlation functions, later confirmed for certain quantum measurements, violate the Bell inequality. The present paper reviews a more general derivation of the Bell inequality showing that it is identically satisfied by finite data sets whether deterministic or random, after assuming merely that they exist. It is thereafter concerned with the consequences of this result for interpretations of the inequality that result in its violation. A primary finding is that correlation functions have differing forms due to quantum commutation, non-commutation, and conditions of measurement, and result in satisfaction of the Bell inequality used consistently with its derivation. A stochastic process having the same correlation function for all variable pairs is shown to be inconsistent with experimentally reported data. The logic of the three and four variable inequalities is shown to be similar. Finally the inequalities in probabilities are shown to follow from those in correlations with quantum mechanical results satisfying either when properly implemented.
Standard Bell inequalities apply to correlations arising when two or more macroscopically separated systems are each subjected to a single ideal measurement. While these inequalities demonstrate that quantum mechanics and local hidden variable theories yield incompatible predictions for pure states, they fail for mixed states: For mixed states, correlations arising from a single ideal measurement on each system may obey standard Bell inequalities, yet when each system is subjected to a sequence of ideal measurements the correlations are nonlocal. For some situations even this last procedure fails, and we must consider more complex (``nonideal'') measurements.