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arXiv:1611.06723 · Journal of Physics A Mathematical and Theoretical

Full Bell locality of a noisy state for N ⩾ 3 nonlocally entangled qudits

Elena R. Loubenets

Published Sep 7, 2017• 1 citation
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Abstract

Abstract Bounds, expressed in terms of d and N , on full Bell locality of a quantum state for N ⩾ 3 nonlocally entangled qudits (of a dimension d ⩾ 2 ) mixed with white noise are known, to our knowledge, only within full separability of this noisy N -qudit state. For the maximal violation of general Bell inequalities by an N -partite quantum state, we specify the analytical upper bound expressed in terms of dilation characteristics of this state, and this allows us to find new general bounds in d , N , valid for all d ⩾ 2 and all N ⩾ 3 , on full Bell locality under generalized quantum measurements of (i) the N -qudit GHZ state mixed with white noise and (ii) an arbitrary N -qudit state mixed with white noise. The new full Bell locality bounds are beyond the known ranges for full separability of these noisy N -qudit states.

DOI
10.1088/1751-8121/aa84e8

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Bounds, expressed in terms of d and N, on full Bell locality of a quantum state for N≥3N\geq 3N≥3 nonlocally entangled qudits (of a dimension d≥2d\geq 2d≥2) mixed with white noise are known, to our knowledge, only within full separability of this noisy N-qudit state. For the maximal violation of general Bell inequalities by an N-partite quantum state, we specify the analytical upper bound expressed in terms of dilation characteristics of this state, and this allows us to find new general bounds in d,Nd, Nd,N, valid for all d≥2d\geq 2d≥2 and all N≥3N\geq 3N≥3, on full Bell locality under generalized quantum measurements of (i) the N-qudit GHZ state mixed with white noise and (ii) an arbitrary N-qudit state mixed with white noise. The new full Bell locality bounds are beyond the known ranges for full separability of these noisy N-qudit states.
Locality of a noisy state for a large number of nonlocally entangled qudits was analyzed in the literature only in the qubit case and only within separability of a noisy N-qubit state. In the present paper, we evaluate amounts of noise breaking the nonlocal character of statistical correlations of NNN entangled qudits by analyzing Bell inequalities violations via the general frame, the LqHV (local quasi hidden variable) modeling frame, introduced in [Loubenets: J. Math. Phys. 53, 022201 (2012)]. This allows us to derive the general exact bound on locality of a noisy N-qudit state which is new and important for all N≥3N\geq 3N≥3 and d≥2.d\geq 2.d≥2. We specify the locality noise thresholds for the maximally entangled state and arbitrary nonlocal states of N≥3N\geq 3N≥3 qudits and study their asymptotics for large NNN. In view of developing quantum information technologies with many particle quantum systems, this problem is of a particular interest.
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I derive separability inequalities for Bell correlations of observables in arbitrary pure or mixed NNN Qudit states in DND^NDN-dimensional state space. I find states (a continuum of states if D>3D>3D>3) including maximally entangled states which violate these inequalities by a factor 2N−12^{N-1}2N−1 ; local reality Bell inequalities are much weaker, their maximum violation being by a factor 2(N−1)/22^{(N-1)/2}2(N−1)/2. The separability inequalities allow tests of entanglement of unknown states using only the measured correlations .
We present a general approach for quantifying tolerance of a nonlocal N-partite state to any local noise under different classes of quantum correlation scenarios with arbitrary numbers of settings and outcomes at each site. This allows us to derive new precise bounds in d and N on noise tolerances for: (i) an arbitrary nonlocal N-qudit state; (ii) the N-qudit Greenberger–Horne–Zeilinger (GHZ) state; (iii) the N-qubit W state and the N-qubit Dicke states, and to analyse asymptotics of these precise bounds for large N and d .