Two destructive effects of decoherence on Bell inequality violation

Abstract

We consider a system of two spin-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 CHSH 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 const ·|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.In a model of local hidden variables (LHVs) the statistical correlations of measurements on a composite physical system must obey certain bounds, called Bell inequalities [1,2]. It is well known that for some quantum systems the observables can be chosen in such a way that at least one of the inequalities is violated [1,2,3]. Therefore, certain quantum systems cannot be described by an LHV model. Clauser, Horne, Shimony and Holt (CHSH) [2] obtained a Bell type inequality which provided a way to experimentally test the existence of nonlocal correlations for such systems. For a pair of spin-1 2 particles (such a particle can represent a qubit) the CHSH inequality can be written in the following form:Here, E(a, b) denotes the expectation value of the product A 1 (a) · A 2 (b), where A k (a) is the result of a measurement of the kth particle's spin projection in the direction a (A k takes the values of ±1). Inequality (1) must hold if the results of spin measurements are described by an LHV model. Moreover, if Eq. (1) is not violated for arbitrary vectors a, a ′ , b, b ′ , then there exists [4] an LHV model describing the results which are obtained when a single ideal measurement is performed on each of the particles. Consider now the singlet state of the two-particle system:where |↑ and |↓ denote, respectively, the states with spin up or down along the Z axis. The operator which corresponds to the left-hand side of the CHSH inequality is given bywhere a, a ′ , b, b ′ are unit vectors in R 3 , σ i are the Pauli matrices, a · σ = 3 i=1 a i σ i . Hence, the CHSH inequality can be written as | ψ|B CHSH |ψ | ≤ 2. State (2) does not admit an LHV model, as the vectors a, a ′ , b, b ′ can be chosen in such a way that | ψ|B CHSH |ψ | = 2 √ 2.(4) * flev@ms2.inr.ac.ru 1

The dynamics of Bell nonlocality for three-qubit system in independent reservoirs

“…The research of quantum correlation dynamics in noise environments is a basic and significant problem in the quantum decoherence theory. In the last twenty years the decoherence dynamics of quantum correlations including quantum entanglement [1][2][3][4][5][6][7], Bell nonlocality measured by the violation of the famous Clauser-Horne-Shimony-Holt (CHSH) inequality [8] and its relation to quantum entanglement [9][10][11][12][13][14][15][16], quantum discord [17,18] for the bipartite systems have been studied intensely, which is also generalized to the multipartite system for quantum entanglement [19][20][21][22][23][24][25][26] and the Bell nonlocality [27][28][29][30][31][32][33][34]. It has been shown that the Bell nonlocality may be stricter than quantum entanglement in the bipartite system [9][10][11][12][13][14][15][16] and tripartite system [32].…”