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Locality of a noisy N-qudit state for large N

Elena R. Loubenets

Published Nov 21, 2016• 0 citations
OpenAlex

Abstract

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 $N$ 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 $N\geq 3$ and $d\geq 2.$ We specify the locality noise thresholds for the maximally entangled state and arbitrary nonlocal states of $N\geq 3$ qudits and study their asymptotics for large $N$. In view of developing quantum information technologies with many particle quantum systems, this problem is of a particular interest.

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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.
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.
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Last years, bounds on the maximal quantum violation of general Bell inequalities were intensively discussed in the literature via different mathematical tools. In the present paper, we analyze quantum violation of general Bell inequalities via the LqHV (local quasi hidden variable) modelling framework, correctly reproducing the probabilistic description of every quantum correlation scenario. The LqHV mathematical framework allows us to derive for all d and N a new upper bound (2d − 1)N−1 on the maximal violation by an N-qudit state of all general Bell inequalities, also, new upper bounds on the maximal violation by an N-qudit state of general Bell inequalities for S settings per site. These new upper bounds essentially improve all the known precise upper bounds on quantum violation of general multipartite Bell inequalities. For some S, d, and N, the new upper bounds are attainable.
Abstract For the maximal violation of all Bell inequalities by an arbitrary pure two-qudit state of any dimension, we derive a new lower bound expressed via the concurrence of this pure state. This new lower bound and the upper bound on the maximal Bell violation, found in Loubenets and Namkung (2022 J. Phys. A: Math. Theor. 55 285301) and also expressed via the concurrence, analytically quantify Bell nonlocality of a pure two-qudit state via its entanglement, in particular, prove explicitly that entanglement of a pure two-qudit state is necessary and sufficient for its Bell nonlocality. By re-visiting the pure two-qubit case, we also find and rigorously prove the new results on correlation properties of an arbitrary pure two-qubit state.
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 .
We specify the local quasi hidden variable (LqHV) model reproducing the probabilistic description of all N-partite joint von Neumann measurements on an N-qudit state. Via this local probability model, we derive a new upper bound on the maximal violation by an N-qudit state of Bell inequalities of any type (either on correlation functions or on joint probabilities) for [Formula: see text] observables per site. This new upper bound not only improves for all [Formula: see text] [Formula: see text] and d the corresponding results available for general Bell inequalities in the literature but also, for the N-qubit case with two observables per site, reduces exactly to the attainable upper bound known for quantum violations of correlation [Formula: see text] setting Bell inequalities in a dichotomic case.