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arXiv:1209.1755 · Physical Review A

Small violations of full-correlation Bell inequalities for multipartite pure random states

Raphael C. Drumond, Roberto I. Oliveira

Published Jul 24, 2012• 7 citations
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Abstract

We estimate the probability of random $N$-qudit pure states violating full-correlation Bell inequalities with two dichotomic observables per site. These inequalities can show violations that grow exponentially with $N$, but we prove this is not the typical case. For many-qubit states the probability to violate any of these inequalities by an amount that grows linearly with $N$ is vanishingly small. If each system's Hilbert space dimension is larger than two, on the other hand, the probability of seeing any violation is already small. For the qubits case we discuss, furthermore, the consequences of this result for the probability of seeing arbitrary violations (i.e., of any order of magnitude) when experimental imperfections are considered.

DOI
10.1103/physreva.86.012117

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We estimate the probability of random NNN-qudit pure states violating full-correlation Bell inequalities with two dichotomic observables per site. These inequalities can show violations that grow exponentially with NNN, but we prove this is not the typical case. For many-qubit states the probability to violate any of these inequalities by an amount that grows linearly with NNN is vanishingly small. If each system's Hilbert space dimension is larger than two, on the other hand, the probability of seeing any violation is already small. For the qubits case we discuss, furthermore, the consequences of this result for the probability of seeing arbitrary violations (i.e., of any order of magnitude) when experimental imperfections are considered.
For any finite number of parts, measurements, and outcomes in a Bell scenario, we estimate the probability of random N-qudit pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension, the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the local Hilbert space dimension goes to infinity.
For any finite number of parts, measurements and outcomes in a Bell scenario we estimate the probability of random NNN-qudddit pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the the local Hilbert space dimension goes to infinity.
A multipartite quantum state violates a Bell inequality asymptotically if, after jointly processing by general local operations an arbitrarily large number of copies of it, the result violates the inequality. In the bipartite case we show that asymptotic violation of the Clauser-Horne-Shimony-Holt inequality is equivalent to distillability. Hence, bound entangled states do not violate it. In the multipartite case we consider the complete set of full-correlation Bell inequalities with two dichotomic observables per site. We show that asymptotic violation of any of these inequalities by a multipartite state implies that pure-state entanglement can be distilled from it, although the corresponding distillation protocol may require that some of the parties join into several groups. We also obtain the extreme points of the set of distributions generated by measuring N quantum systems with two dichotomic observables per site.
We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck’s constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized Greenberger-Horne-Zeilinger (GHZ) states is always bounded so that, in contrast to many other contexts, GHZ states do not lead to extremal quantum correlations in this case. In order to derive all these physical consequences, we will have to obtain new mathematical results in the theories of operator spaces and tensor norms. In particular, we will prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras. Finally, we will relate the existence of diagonal states leading to unbounded violations with a long-standing open problem in the context of Banach algebras.
Consider a scenario where NNN separated quantum systems are measured, each with one among two possible dichotomic observables. Assume that the NNN events corresponding to the choice and performance of the measurement in each site are space-like separated. In the present paper, the correlations among the measurement outcomes that arise in this scenario are analyzed. It is shown that all extreme points of this convex set are attainable by measuring NNN-qubit pure-states with projective observables. This result allows the possibility of using known algorithms in order decide whether some correlations are achievable within quantum mechanics or not. It is also proven that if an NNN-partite state ρρρ violates a given Bell inequality, then, ρρρ can be transformed by stochastic local operations into an NNN-qubit state that violates the same Bell inequality by an equal or larger amount.