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arXiv:1602.04645 · International Journal of Quantum Information

On the existence of a local quasi hidden variable (LqHV) model for eachN-qudit state and the maximal quantum violation of Bell inequalities

Elena R. Loubenets

Published May 19, 2016• 7 citations
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Abstract

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.

DOI
10.1142/s0219749916400104

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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.
We introduce for a general correlation scenario a new simulation model, a local quasi hidden variable (LqHV) model, where locality and the measure-theoretic construction inherent to a local hidden variable (LHV) model are preserved but positivity of a simulation measure is dropped. We specify a necessary and sufficient condition for LqHV modelling and, based on this, prove that every quantum correlation scenario admits a LqHV simulation. Via the LqHV approach, we construct analogs of Bell-type inequalities for an N-partite quantum state and find a new analytical upper bound on the maximal violation by an N-partite quantum state of S1 × ⋯ × SN-setting Bell-type inequalities – either on correlation functions or on joint probabilities and for outcomes of an arbitrary spectral type, discrete or continuous. This general analytical upper bound is expressed in terms of the new state dilation characteristics introduced in the present paper and not only traces quantum states admitting an S1 × ⋯ × SN-setting LHV description but also leads to the new exact numerical upper estimates on the maximal Bell violations for concrete N-partite quantum states used in quantum information processing and for an arbitrary N-partite quantum state. We, in particular, prove that violation by an N-partite quantum state of an arbitrary Bell-type inequality (either on correlation functions or on joint probabilities) for S settings per site cannot exceed (2S − 1)N − 1 even in case of an infinite dimensional quantum state and infinitely many outcomes.
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.
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.
We prove the existence for each Hilbert space of the two new quasi hidden variable (qHV) models, statistically noncontextual and context-invariant, reproducing all the von Neumann joint probabilities via non-negative values of real-valued measures and all the quantum product expectations—via the qHV (classical-like) average of the product of the corresponding random variables. In a context-invariant model, a quantum observable X can be represented by a variety of random variables satisfying the functional condition required in quantum foundations but each of these random variables equivalently models X under all joint von Neumann measurements, regardless of their contexts. The proved existence of this model negates the general opinion that, in terms of random variables, the Hilbert space description of all the joint von Neumann measurements for dimH≥3 can be reproduced only contextually. The existence of a statistically noncontextual qHV model, in particular, implies that every N-partite quantum state admits a local quasi hidden variable model introduced in Loubenets [J. Math. Phys. 53, 022201 (2012)]. The new results of the present paper point also to the generality of the quasi-classical probability model proposed in Loubenets [J. Phys. A: Math. Theor. 45, 185306 (2012)].
Noncommuting observables cannot be simultaneously measured; however, under local hidden variable models, they must simultaneously hold premeasurement values, implying the existence of a joint probability distribution. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Fréchet bounds limiting the joint probabilities, concluding that the latter may be negative. We use symmetrization, justified heuristically and then more carefully via the Moyal characteristic function, to find the quantum operator corresponding to the product of noncommuting observables. This is then used to construct Quasi-Bell inequalities, Bell inequalities containing products of noncommuting observables, on two qubits. These inequalities place limits on the local hidden variable models that define joint probabilities for noncommuting observables. We find that the Quasi-Bell inequalities have a quantum to classical violation as high as 32 on two qubit, higher than conventional Bell inequalities. The result demonstrates the theoretical importance of noncommutativity in the nonlocality of quantum mechanics and provides an insightful generalization of Bell inequalities.