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arXiv:0912.2232 · Physical Review A

Bound on Hardy’s nonlocality from the principle of information causality

Ali Ahanj, Samir Kunkri, Ashutosh Rai, Ramij Rahaman, Pramod S. Joag

Published Mar 4, 2010• 24 citations
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Abstract

Recently, the principle of nonviolation of information causality [Nature 461, 1101 (2009)] has been proposed as one of the foundational properties of nature. We explore the Hardy's nonlocality theorem for two-qubit systems, in the context of generalized probability theory, restricted by the principle of nonviolation of information causality. Applying a sufficient condition for information causality violation, we derive an upper bound on the maximum success probability of Hardy's nonlocality argument. We find that the bound achieved here is higher than that allowed by quantum mechanics but still much less than what the no-signaling condition permits. We also study the Cabello type nonlocality argument (a generalization of Hardy's argument) in this context.

DOI
10.1103/physreva.81.032103

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Recently, the principle of nonviolation of information causality [Nature 461, 1101 (2009)] has been proposed as one of the foundational properties of nature. We explore the Hardy's nonlocality theorem for two-qubit systems, in the context of generalized probability theory, restricted by the principle of nonviolation of information causality. Applying a sufficient condition for information causality violation, we derive an upper bound on the maximum success probability of Hardy's nonlocality argument. We find that the bound achieved here is higher than that allowed by quantum mechanics but still much less than what the no-signaling condition permits. We also study the Cabello type nonlocality argument (a generalization of Hardy's argument) in this context.
The amount of nonlocality in quantum theory is limited compared to that allowed in generalized no-signaling theory [S. Popescu and D. Rohrlich, Found. Phys. 24, 379 (1994)]. This feature, for example, gets manifested in the amount of Bell inequality violation as well as in the degree of success probability of Hardy's (Cabello's) nonlocality argument. Physical principles like information causality and macroscopic locality have been proposed for analyzing restricted nonlocality in quantum mechanics, viz. explaining the Cirel'son bound. However, these principles are not very successful in explaining the maximum success probability of Hardy's as well as Cabello's argument in quantum theory. Here we show that a recently proposed physical principle, namely local orthogonality, does better by providing a tighter upper bound on the success probability for Hardy's nonlocality. This bound is relatively closer to the corresponding quantum value compared to the bounds achieved from other principles.
Study of non-local correlations in terms of Hardy's argument has been quite popular in quantum mechanics. Hardy's non-locality argument depends on some kind of asymmetry, but a two-qubit maximally entangled state, being symmetric, does not exhibit this kind of non-locality. Here we ask the following question: can this feature be explained by some principle outside quantum mechanics? The no-signaling condition does not provide a solution. But, interestingly, the information causality principle (Pawlowski et al 2009 Nature 461 1101) offers an explanation. It shows that any generalized probability theory which gives completely random results for local dichotomic observable, cannot provide Hardy's non-local correlation if it is restricted by a necessary condition for respecting the information causality principle. In fact, the applied necessary condition imposes even more restrictions on the local randomness of measured observable. Still, there are some restrictions imposed by quantum mechanics that are not reproduced from the considered information causality condition.
We give an analytic quantitative relation between Hardy’s non-locality and Bell operator. We find that Hardy’s non-locality is a sufficient condition for the violation of Bell inequality, the upper bound of Hardy’s non-locality allowed by information causality just corresponds to Tsirelson bound of Bell inequality and the upper bound of Hardy’s non-locality allowed by the principle of no-signaling just corresponds to the algebraic maximum of Bell operator. Then we study the Cabello’s argument of Hardy’s non-locality (a generalization of Hardy’s argument) and find a similar relation between it and violation of Bell inequality. Finally, we give a simple derivation of the bound of Hardy’s non-locality under the constraint of information causality with the aid of the above derived relation between Hardy’s non-locality and Bell operator.
We give an analytic quantitative relation between Hardy's non-locality and Bell operator. We find that Hardy's non-locality is a sufficient condition for the violation of Bell inequality, the upper bound of Hardy's non-locality allowed by information causality just corresponds to Tsirelson bound of Bell inequality and the upper bound of Hardy's non-locality allowed by the principle of no-signaling just corresponds to the algebraic maximum of Bell operator. Then we study the Cabello's argument of Hardy's non-locality (a generalization of Hardy's argument) and find a similar relation between it and violation of Bell inequality. Finally, we give a simple derivation of the bound of Hardy's non-locality under the constraint of information causality with the aid of the above derived relation between Hardy's non-locality and Bell operator.
We give an analytic quantitative relation between Hardy's non-locality and Bell operator. We find that Hardy's non-locality is a sufficient condition for the violation of Bell inequality, the upper bound of Hardy's non-locality allowed by information causality just corresponds to Tsirelson bound of Bell inequality and the upper bound of Hardy's non-locality allowed by the principle of no-signaling just corresponds to the algebraic maximum of Bell operator. Then we study the Cabello's argument of Hardy's non-locality (a generalization of Hardy's argument) and find a similar relation between it and violation of Bell inequality. Finally, we give a simple derivation of the bound of Hardy's non-locality under the constraint of information causality with the aid of the above derived relation between Hardy's non-locality and Bell operator.