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arXiv:1804.01199 · Journal of Mathematical Physics

Entropic uncertainty relations under localizations on discrete quantum groups

Sang-Gyun Youn

Published Jul 1, 2018• 1 citation
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Abstract

The uncertainty principle has been established within the framework of locally compact quantum groups in recent years. This paper demonstrates that entropic uncertainty relations can be strengthened under localizations on discrete quantum groups, which is the case if the dual compact quantum group G is the free orthogonal quantum group ON+ with N ≥ 3 or if G admits an infinite Λ(p) set with p > 2. On the other hand, this paper explains the reason why such phenomena do not appear when G is one of the connected semisimple compact Lie groups, O2+, and the quantum SU(2) groups. Also, we discuss the divergence of entropic uncertainty relations together with some explicit explanations.

DOI
10.1063/1.5037820

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The uncertainty principle has been established within the framework of locally compact quantum groups in recent years. This paper demonstrates that entropic uncertainty relations can be strengthened under localizations on discrete quantum groups, which is the case if the dual compact quantum group G is the free orthogonal quantum group ON+ with N ≥ 3 or if G admits an infinite Λ(p) set with p > 2. On the other hand, this paper explains the reason why such phenomena do not appear when G is one of the connected semisimple compact Lie groups, O2+, and the quantum SU(2) groups. Also, we discuss the divergence of entropic uncertainty relations together with some explicit explanations.
We present a generalization of Hirschman's entropic uncertainty principle for locally compact Abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finite-dimensional quantum group algebras. In the non-unimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that when restricted to q-traces of discrete quantum groups, the relative entropy with respect to the Haar weight reduces to the canonical entropy of the random walk generated by the state.
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We present the entropic uncertainty relations for multiple measurement settings which demonstrate the uncertainty principle of quantum mechanics. Those uncertainty relations are obtained for both cases with and without the presence of quantum memory, and can be proven by a unified method. Our results recover some well known entropic uncertainty relations for two observables, which show the uncertainties about the outcomes of two incompatible measurements. The bounds of those relations which quantify the extent of the uncertainty take concise forms and are easy to calculate. Those uncertainty relations might play important roles in the foundations of quantum theory. Potential experimental demonstration of those entropic uncertainty relations is discussed.
A prominent formulation of the uncertainty principle identifies the fundamental quantum feature that no particle may be prepared with certain outcomes for both position and momentum measurements. Often the statistical uncertainties are thereby measured in terms of entropies providing a clear operational interpretation in information theory and cryptography. Recently, entropic uncertainty relations have been used to show that the uncertainty can be reduced in the presence of entanglement and to prove security of quantum cryptographic tasks. However, much of this recent progress has been focused on observables with only a finite number of outcomes not including Heisenberg’s original setting of position and momentum observables. Here, we show entropic uncertainty relations for general observables with discrete but infinite or continuous spectrum that take into account the power of an entangled observer. As an illustration, we evaluate the uncertainty relations for position and momentum measurements, which is operationally significant in that it implies security of a quantum key distribution scheme based on homodyne detection of squeezed Gaussian states.