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arXiv:quant-ph/9909016 · Physical Review A

Nonlocal correlations are generic in infinite-dimensional bipartite systems

Rob Clifton, Hans Halvorson, Adrian Kent

Published Mar 7, 2000• 17 citations
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Abstract

It was recently shown that nonseparable density operators on the Hilbert space ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2}$ are trace norm dense if either factor space has infinite dimension. We show here that nonlocal states, i.e., states whose correlations cannot be reproduced by any local hidden variable model, are also dense. Our constructions distinguish between the case $\mathrm{dim}{\mathcal{H}}_{1}=\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ where we show that states violating the Clauser-Horne-Shimony-Holt (CHSH) inequality are dense, and the case $\mathrm{dim}{\mathcal{H}}_{1}<\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ where we identify open neighborhoods of nonseparable states that do not violate the CHSH inequality but show that states with a subtler form of nonlocality (often called ``hidden'' nonlocality) remain dense.

DOI
10.1103/physreva.61.042101

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It was recently shown that nonseparable density operators on the Hilbert space H1\ensuremath⨂H2{\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2}H1​\ensuremath⨂H2​ are trace norm dense if either factor space has infinite dimension. We show here that nonlocal states, i.e., states whose correlations cannot be reproduced by any local hidden variable model, are also dense. Our constructions distinguish between the case dimH1=dimH2=\ensuremath∞,\mathrm{dim}{\mathcal{H}}_{1}=\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},dimH1​=dimH2​=\ensuremath∞, where we show that states violating the Clauser-Horne-Shimony-Holt (CHSH) inequality are dense, and the case dimH1<dimH2=\ensuremath∞,\mathrm{dim}{\mathcal{H}}_{1}<\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},dimH1​<dimH2​=\ensuremath∞, where we identify open neighborhoods of nonseparable states that do not violate the CHSH inequality but show that states with a subtler form of nonlocality (often called ``hidden'' nonlocality) remain dense.
Given a bipartite quantum system represented by a Hilbert space ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2},$ we give an elementary argument to show that if either $\mathrm{dim}{\mathcal{H}}_{1}=\ensuremath{\infty}$ or $\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ then the set of nonseparable density operators on ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2}$ is trace-norm dense in the set of all density operators (and the separable density operators nowhere dense). This result complements recent detailed investigations of separability, which show that when $\mathrm{dim}{\mathcal{H}}_{i}<\ensuremath{\infty}$ for i=1,2,i=1,2,i=1,2, there is a separable neighborhood (perhaps very small for large dimensions) of the maximally mixed state.
For a nonseparable bipartite quantum state violating the Clauser–Horne–Shimony–Holt (CHSH) inequality, we evaluate amounts of noise breaking the quantum character of its statistical correlations under any generalized quantum measurements of Alice and Bob. Expressed in terms of the reduced states, these new threshold bounds can be easily calculated for any concrete bipartite state. A noisy bipartite state, satisfying the extended CHSH inequality and the perfect correlation form of the original Bell inequality for any quantum observables, neither necessarily admits a local hidden variable model nor exhibits the perfect correlation of outcomes whenever the same quantum observable is measured on both 'sides'.
For a nonseparable bipartite quantum state violating the Clauser-Horne-Shimony-Holt (CHSH) inequality, we evaluate amounts of noise breaking the quantum character of its statistical correlations under any generalized quantum measurements of Alice and Bob. Expressed in terms of the reduced states, these new threshold bounds can be easily calculated for any concrete bipartite state. A noisy bipartite state, satisfying the extended CHSH inequality and the perfect correlation form of the original Bell inequality for any quantum observables, neither necessarily admits a local hidden variable model nor exhibits the perfect correlation of outcomes whenever the same quantum observable is measured on both ”sides”. 1
We find two two-qubit bipartite states ${\ensuremath{\rho}}_{1},{\ensuremath{\rho}}_{2}$ such that arbitrarily many copies of one or the other cannot exhibit nonlocal correlations in a two-setting--two-outcome Bell scenario. However, the bipartite state ${\ensuremath{\rho}}_{1}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{2}$ violates the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).] by an amount of 2.023. We also identify a CHSH-local state $\ensuremath{\rho}$ such that ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}2}$ is CHSH inequality-violating. The tools employed can be easily adapted to find instances of nonlocality activation in arbitrary Bell scenarios.
The assumption of local realism, in a Bell locality scenario, imposes nontrivial conditions on the Shannon entropies of the associated probability distributions, expressed by linear entropic Bell inequalities. In principle, these entropic inequalities provide necessary but not sufficient criteria for the existence of a local hidden variable model reproducing the correlations, as, for example, the paradigmatic nonlocal Popescu-Rohrlich (PR) box is entropically not different from a classically correlated box. In this paper we show that for the nnn-cycle scenario, entropic inequalities completely characterize the set of local correlations. In particular, every nonsignaling box which violates the Clauser-Horne-Shimony-Holt (CHSH) inequality---including the PR box---can be locally modified so that it also violates the entropic version of CHSH inequality. As we show, any nonlocal probabilistic model when appropriately mixed with a local model, violates an entropic inequality, thus evidencing a very peculiar kind of nonlocality. As the nnn-cycle captures equally well both the notion of local realism introduced by Bell and that of noncontextuality presented by the Kochen-Specker theorem, the results are also valid for noncontextuality scenarios.
We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to arbitrarily high-dimensional systems. Based on this generalization, we construct the general CHSH inequality for bipartite quantum systems of arbitrarily high dimensionality, which takes the same simple form as CHSH inequality for two dimensions. This inequality is optimal in the same sense as the CHSH inequality for two-dimensional systems, namely, the maximal amount by which the inequality is violated consists of the maximal resistance to noise. We also discuss the physical meaning and general definition of the correlation functions. Furthermore, by giving another specific set of the correlation functions with the same physical meaning, we realize the inequality presented by Collins et al. [Phys. Rev. Lett. 88, 040404 (2002)]].