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arXiv:1402.1143 · Physical Review A

Quantum and classical entropic uncertainty relations

Kamil Korzekwa, Matteo Lostaglio, David H. Jennings, Terry Rudolph

Published Apr 29, 2014• 71 citations
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Abstract

How much of the uncertainty in predicting measurement outcomes for noncommuting quantum observables is genuinely quantum mechanical? We provide a natural decomposition of the total entropic uncertainty of two noncommuting observables into a classical component and an intrinsically quantum mechanical component. We show that the total quantum component in a state is never lower or upper bounded by any state-independent quantities, but instead admits ``purity-based'' lower bounds that generalize entropic formulations such as the Maassen-Uffink relation. These relations reveal a nontrivial interplay between quantum and classical randomness in any finite-dimensional state.

DOI
10.1103/physreva.89.042122

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Measurement outcomes of a quantum state can be genuinely random (unpredictable) according to the basic laws of quantum mechanics. The Heisenberg-Robertson uncertainty relation puts constraints on the accuracy of two noncommuting observables. The existing uncertainty relations adopt variance or entropic measures, which are functions of observed outcome distributions, to quantify the uncertainty. According to recent studies of quantum coherence, such uncertainty measures contain both classical (predictable) and quantum (unpredictable) components. In order to extract out the quantum effects, we define quantum uncertainty to be the coherence of the state on the measurement basis. We discover a quantum uncertainty relation using coherence between two measurement noncommuting bases. Furthermore, we analytically derive the quantum uncertainty relation for the qubit case with three widely adopted coherence measures, the relative entropy of coherence, the coherence of formation, and the l1{l}_{1}l1​ norm of coherence.
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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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