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arXiv:0806.3984 · Physical Review Letters

Conjectured Strong Complementary Information Tradeoff

Joseph M. Renes, Jean-Christian Boileau

Published Jul 9, 2009• 293 citations
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Abstract

We conjecture a new entropic uncertainty principle governing the entropy of complementary observations made on a system given side information in the form of quantum states, generalizing the entropic uncertainty relation of Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)]. We prove a special case for certain conjugate observables by adapting a similar result found by Christandl and Winter pertaining to quantum channels [IEEE Trans. Inf. Theory 51, 3159 (2005)], and discuss possible applications of this result to the decoupling of quantum systems and for security analysis in quantum cryptography.

DOI
10.1103/physrevlett.103.020402

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We conjecture a new entropic uncertainty principle governing the entropy of complementary observations made on a system given side information in the form of quantum states, generalizing the entropic uncertainty relation of Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)]. We prove a special case for certain conjugate observables by adapting a similar result found by Christandl and Winter pertaining to quantum channels [IEEE Trans. Inf. Theory 51, 3159 (2005)], and discuss possible applications of this result to the decoupling of quantum systems and for security analysis in quantum cryptography.
We conjecture uncertainty relations that restrict correlations between the results of measurements performed by two separate parties on a shared quantum state. The first uncertainty relation bounds the sum of two mutual informations when one party measures a single observable and the other party measures one of two observables. The uncertainty relation does not follow from the Maassen-Uffink uncertainty relation and is much stronger than the Hall uncertainty relation derived from the latter. The second uncertainty relation bounds the sum of two mutual informations when each party measures one of two observables. We provide numerical evidence for the validity of the conjectured uncertainty relations, and we prove them for large classes of states and observables.
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The entropic uncertainty principle in the form proven by Maassen and Uffink yields a fundamental inequality that is prominently used in many places all over the field of quantum information theory. In this paper, we provide a family of versatile generalizations of this relation. Our proof methods build on a deep connection between entropic uncertainties and interpolation inequalities for the doubly stochastic map that links probability distributions in two measurement bases. In contrast to the original relation, our generalization also incorporates the von Neumann entropy of the underlying quantum state. These results can be directly used to bound the extractable randomness of a source-independent quantum random number generator in the presence of fully quantum attacks, to certify entanglement between trusted parties, or to bound the entanglement of a system with an untrusted environment.
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