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Nonlocal correlations are generic 1 n infinitedimensional bipartite systems

Rob Clifton

Published Sep 23, 2009• 0 citations
Publisher

Abstract

Abstract The observables of a bipartite quantum system are represented by the set of all self adjoint operators on the tensor product of two Hilbert spaces H1 ® ‘H2, whose dimensions we shall denote by d1 and d2, taking d1 :S: d2 without loss of generality. It is well known that when d1 2: 2 the states of the system can be nonseparable, and it is this possibility that much of the new technology associated with quantum information and computation theory relies upon. Prompted by concerns about whether the very noisy mixed states exploited by certain models of NMR quantum computing are truly nonseparable (Braunstein et al. 1999, Linden & Popescu 2001), de­ tailed investigations have shown that, whenever d2 < oo, there is always an open neighborhood of separable states surrounding the maximally mixed state (d1d2)­1 I® I (Zyczkowski et al. 1998, Caves

DOI
10.1093/oso/9780199270156.003.0010

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Abstract The observables of a bipartite quantum system are represented by the set of all self adjoint operators on the tensor product of two Hilbert spaces H1 ® ‘H2, whose dimensions we shall denote by d1 and d2, taking d1 :S: d2 without loss of generality. It is well known that when d1 2: 2 the states of the system can be nonseparable, and it is this possibility that much of the new technology associated with quantum information and computation theory relies upon. Prompted by concerns about whether the very noisy mixed states exploited by certain models of NMR quantum computing are truly nonseparable (Braunstein et al. 1999, Linden & Popescu 2001), de­ tailed investigations have shown that, whenever d2 < oo, there is always an open neighborhood of separable states surrounding the maximally mixed state (d1d2)­1 I® I (Zyczkowski et al. 1998, Caves
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.
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.
While the interest in multipartite nonlocality has grown in recent years, its existence in large quantum systems is difficult to confirm experimentally. This is mostly due to the inadequacy of standard multipartite Bell inequalities to many-body systems: such inequalities usually rely on expectation values involving many parties and require an individual addressing of each party. A recent work [J. Tura et al. Science 344, 6189 (2014)] proposed simpler Bell inequalities overcoming such difficulties, opening the way for the detection of Bell correlations with trusted collecti
I introduce a framework in which a variety of probabilistic theories can be defined, including classical and quantum theories, and many others. From two simple assumptions, a tensor product rule for combining separate systems can be derived. Certain features, usually thought of as specifically quantum, turn out to be generic in this framework, meaning that they are present in all except classical theories. These include the nonunique decomposition of a mixed state into pure states, a theorem involving disturbance of a system on measurement (suggesting that the possibility of secure key distribution is generic), and a no-cloning theorem. Two particular theories are then investigated in detail, for the sake of comparison with the classical and quantum cases. One of these includes states that can give rise to arbitrary nonsignaling correlations, including the superquantum correlations that have become known in the literature as nonlocal machines or Popescu-Rohrlich boxes. By investigating these correlations in the context of a theory with well-defined dynamics, I hope to make further progress with a question raised by Popescu and Rohrlich, which is why does quantum theory not allow these strongly nonlocal correlations? The existence of such correlations forces much of the dynamics in this theory to be, in a certain sense, classical, with consequences for teleportation, cryptography, and computation. I also investigate another theory in which all states are local. Finally, I raise the question of what further axiom(s) could be added to the framework in order to identify quantum theory uniquely, and hypothesize that quantum theory is optimal for computation.