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arXiv:0912.1941 · Physical Review Letters

Operator Space Theory: A Natural Framework for Bell Inequalities

Marius Junge, Carlos Palazuelos, David Pérez-Garcı́a, Ignacio Villanueva, Michael M. Wolf

Published Apr 29, 2010• 101 citations
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Abstract

In this Letter we show that the field of operator space theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular, regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local, Hilbert space dimension n can violate a Bell inequality by a factor of order sqrt[n]/(log{2}n) when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates, and communication complexity are given.

DOI
10.1103/physrevlett.104.170405

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In this Letter we show that the field of operator space theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular, regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local, Hilbert space dimension n can violate a Bell inequality by a factor of order sqrt[n]/(log{2}n) when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates, and communication complexity are given.
Abstract: In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order Ω n log2 n when observables with n possible outcomes are used. A central tool in the anal-ysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative Lp embedding theory. As a consequence of this result, we obtain better Hilbert space dimension witnesses and quantum violations of Bell inequalities with better resistance to noise. 1.
The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say that a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given for the quantum violation of these Bell inequalities in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. This makes the Bell inequality resistant to the detection loophole, while a normalized Bell inequality is resistant to general local noise. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bound techniques. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities.
We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four-dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two-dimensional systems, i.e., qubits, are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate ``measurements'' in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.
Quantum theory violates Bell's inequality, but not to the maximum extent that is logically possible. We derive inequalities (generalizations of Cirel'son's inequality) that quantify the upper bound of the violation, both for the standard formalism and the formalism of generalized observables (POVMs). These inequalities are quantum analogues of Bell inequalities, and they can be used to test the quantum version of locality. We discuss the nature of this kind of locality. We also go into the relation of our results to an argument by Popescu and Rohrlich [Found Phys. 24, 379 (1994)] that there is no general connection between the existence of Cirel'son's bound and locality.
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