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arXiv:0812.3736 · Physical Review A

Two destructive effects of decoherence on Bell inequality violation

Fedor Levkovich-Maslyuk

Published May 1, 2009• 4 citations
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Abstract

We consider a system of two spin-$\frac{1}{2}$ particles, initially in an entangled Bell state. If one of the particles is interacting with an environment (e.g., a collection of $N$ independent spins), the two-particle system undergoes decoherence. Using a simple model of decoherence, we show that this process has two consequences. First, the maximal amount by which the Clauser-Horne-Shimony-Holt inequality is violated decays to zero. Second, the set of directions of measurement for which the inequality is violated is reduced in the course of decoherence. The volume of that set is bounded above by $\text{const}\ifmmode\times\else\texttimes\fi{}{|r|}^{2}$, where $r$ is the decoherence factor. We obtain similar results for the case when each of the two particles is in interaction with a separate environment.

DOI
10.1103/physreva.79.054101

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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).
We show that the decoherence, which in the long run destroys quantum features of a system, can be used to reveal the entanglement in a two-qubit system. To this end, we consider a criterion that formally resembles the Clauser-Horne-Shimony-Holt (CHSH) inequality. In our case the local observables are set by the coupling of each qubit to the environmental noise, controlled with the dynamical decoupling method. We demonstrate that the constructed inequality is an entanglement criterion---it can only be violated by nonseparable initial two-qubit states, provided that the local noises are correlated. We also show that, for a given initial state, this entanglement criterion can be repurposed as a method of discriminating between Gaussian and non-Gaussian noise generated by the environment of the qubits. The latter application is important for ongoing research on using qubits to characterize the dynamics of environment that perturbs them and causes their decoherence.
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.
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