Asymptotic entropic uncertainty relations

Abstract

We analyze entropic uncertainty relations for two orthogonal measurements on a $N$-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix $U$ relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than these obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to establish the conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of an independent interest.

23 pages, 1 figure

References in corpus (6)

• Entropic Uncertainty Relations and their Applications
• Randomizing quantum states: Constructions and applications
• Strong Majorization Entropic Uncertainty Relations
• Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases
• General entropy-like uncertainty relations in finite dimensions
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Cited by in corpus (10)

• Advances in Quantum Cryptography
• Entropic Uncertainty Relations and their Applications
• Strong Majorization Entropic Uncertainty Relations
• State-independent Uncertainty Relations and Entanglement Detection in Noisy Systems
• Effects of Hawking radiation on the entropic uncertainty in a Schwarzschild space-time
• Certainty relations, mutual entanglement and non-displacable manifolds
• Conditional entropic uncertainty relations for Tsallis entropies
• Metric and classical fidelity uncertainty relations for random unitary matrices
• Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices
• Norms of structured random matrices