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International Journal of Quantum Information

QUANTUM CORRELATIONS IN SUCCESSIVE SPIN MEASUREMENTS

Ali Ahanj, Pramod S. Joag, Sibasish Ghosh A

Published Dec 1, 2007• 2 citations
Publisher

Abstract

In this paper, we consider a hidden variable theoretical description of successive measurements of non-commuting spin observables on an input spin-s state. Although these spin observables are non-commuting, they act on different states, and so the joint probabilities for the outputs of successive measurements are well-defined. We show that, in this scenario, hidden variable theory (HVT) leads to Bell-type inequalities for the correlation between the outputs of successive measurements. We account for the maximum violation of these inequalities by quantum correlations (i.e. the correlations of successive measurements on a quantum state) by varying the spin value and the number of successive measurements. Our approach can be used to obtain a measure of the deviation of Quantum Mechanics from the theory obeying realism and time-locality, in terms of the amount of classical information needed to be transferred between successive measurements in order to simulate the above-mentioned correlations in successive measurements.

DOI
10.1142/s0219749907003298

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In this paper, we consider a hidden variable theoretical description of successive measurements of non-commuting spin observables on an input spin-s state. Although these spin observables are non-commuting, they act on different states, and so the joint probabilities for the outputs of successive measurements are well-defined. We show that, in this scenario, hidden variable theory (HVT) leads to Bell-type inequalities for the correlation between the outputs of successive measurements. We account for the maximum violation of these inequalities by quantum correlations (i.e. the correlations of successive measurements on a quantum state) by varying the spin value and the number of successive measurements. Our approach can be used to obtain a measure of the deviation of Quantum Mechanics from the theory obeying realism and time-locality, in terms of the amount of classical information needed to be transferred between successive measurements in order to simulate the above-mentioned correlations in successive measurements.
In this paper we present a new approach for testing QM against the realism aspect of hidden variable theory (HVT). We consider successive measurements of non-commuting operators on a input spin sss state. The key point is that, although these operators are non-commuting, they act on different states so that the joint probabilities for the outputs of successive measurements are well defined. We show that, in this scenario HVT leads to Bell type inequalities for the correlation between the outputs of successive measurements. We account for the maximum violation of these inequalities by quantum correlations by varying spin value and the number of successive measurements. Our approach can be used to obtain a measure of the deviation of QM from realism say in terms of the amount of information needed to be transferred between successive measurements in order to classically simulate the quantum correlations.
Abstract. In conformity with our previous work on the measurement process in quantum mechanics in terms of first passage random walks in Hilbert space through interactions with the measuring devices, we here consider the problem of measurement of entangled states at spatially separated measuring devices. We find that Bell’s inequality need not be an obstacle if the set of hidden variables include local ones, and we present a simple model that includes specific probability distributions of the source and local hidden variables, giving the quantum mechanical result for the expectation value of the spin-spin correlation for arbitrary orientations of the frames of measurement, which is known to violate Bell’s relation for chosen angles of orientation. Local measurements and entangled states 2 1.
In the derivation of Bell's inequalities, probability distribution is supposed to be a function of only hidden variable. We point out that the true implication of the probability distribution of Bell's correlation function is the distribution of the joint measurement outcomes on the two sides. So it is a function of both hidden variable and settings. In this case, Bell's inequalities fail. Our further analysis shows that Bell's locality holds neither for dependent events nor for independent events. We think that the measurements of EPR pairs are dependent events, thus violation of Bell's inequalities cannot rule out the existence of local hidden variable. In order to explain the results of EPR-type experiments, we suppose that polarization entangled photon pair can be composed of two circularly or linearly polarized photons with correlated hidden variables, and a couple of experiments of quantum measurement are proposed. The first uses delayed measurement on one photon of the EPR pair to demonstrate directly whether measurement on the other could have any non-local influence on it. Then several experiments are suggested to reveal the components of polarization entangled photon pair. The last one uses successive polarization measurements on a pair of EPR photons to show that two photons with a same quantum state will behave in the same way under the same measuring condition.
We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no-hidden-variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem.
We formulate stochastic noncontextual hidden-variable theories. In such theories the hidden state \ensuremath{\lambda} specifies not definite values, but expectation values A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}), . . . of observables A, . . . and noncontextuality means that A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}) is independent of which other commuting observables commuting with A are measured together with A. We show via Bell inequalities that such theories conflict with quantum theory and propose a two-photon experimental test. We also show that for a single particle of spin (2n{2}^{\mathit{n}}2n-1)/2, quantum violation of classical noncontextuality grows exponentially with n.