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Title: Entropic uncertainty relations and their applications

Authors:
; ; ;
Publication Date:
Sponsoring Org.:
USDOE
OSTI Identifier:
1342817
Resource Type:
Journal Article: Publisher's Accepted Manuscript
Journal Name:
Reviews of Modern Physics
Additional Journal Information:
Journal Volume: 89; Journal Issue: 1; Related Information: CHORUS Timestamp: 2017-02-07 09:37:29; Journal ID: ISSN 0034-6861
Publisher:
American Physical Society
Country of Publication:
United States
Language:
English

Citation Formats

Coles, Patrick J., Berta, Mario, Tomamichel, Marco, and Wehner, Stephanie. Entropic uncertainty relations and their applications. United States: N. p., 2017. Web. doi:10.1103/RevModPhys.89.015002.
Coles, Patrick J., Berta, Mario, Tomamichel, Marco, & Wehner, Stephanie. Entropic uncertainty relations and their applications. United States. doi:10.1103/RevModPhys.89.015002.
Coles, Patrick J., Berta, Mario, Tomamichel, Marco, and Wehner, Stephanie. Mon . "Entropic uncertainty relations and their applications". United States. doi:10.1103/RevModPhys.89.015002.
@article{osti_1342817,
title = {Entropic uncertainty relations and their applications},
author = {Coles, Patrick J. and Berta, Mario and Tomamichel, Marco and Wehner, Stephanie},
abstractNote = {},
doi = {10.1103/RevModPhys.89.015002},
journal = {Reviews of Modern Physics},
number = 1,
volume = 89,
place = {United States},
year = {Mon Feb 06 00:00:00 EST 2017},
month = {Mon Feb 06 00:00:00 EST 2017}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1103/RevModPhys.89.015002

Citation Metrics:
Cited by: 25works
Citation information provided by
Web of Science

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  • Even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols, they remain ill understood. Here, we construct special sets of up to 2n+1 mutually unbiased bases (MUBs) in dimension d=2{sup n}, which have particularly beautiful symmetry properties derived from the Clifford algebra. More precisely, we show that there exists a unitary transformation that cyclically permutes such bases. This unitary can be understood as a generalization of the Fourier transform, which exchanges two MUBs, to multiple complementary aspects. We proceed to prove a lower bound for min-entropic entropic uncertainty relations for any set ofmore » MUBs and show that symmetry plays a central role in obtaining tight bounds. For example, we obtain for the first time a tight bound for four MUBs in dimension d=4, which is attained by an eigenstate of our complementarity transform. Finally, we discuss the relation to other symmetries obtained by transformations in discrete phase space and note that the extrema of discrete Wigner functions are directly related to min-entropic uncertainty relations for MUBs.« less
  • There is a renewed interest in the uncertainty principle, reformulated from the information theoretic point of view, called the entropic uncertainty relations. They have been studied for various integrable systems as a function of their quantum numbers. In this work, focussing on the ground state of a nonlinear, coupled Hamiltonian system, we show that approximate eigenstates can be constructed within the framework of adiabatic theory. Using the adiabatic eigenstates, we estimate the information entropies and their sum as a function of the nonlinearity parameter. We also briefly look at the information entropies for the highly excited states in the system.
  • We derive a collection of separability conditions for bipartite systems of dimension dxd which is based on the entropic version of the uncertainty relations. A detailed analysis of the two-qubit case is given by comparing the new separability conditions with existing criteria.
  • We discuss the relationship between entropic uncertainty relations and entanglement. We present two methods for deriving separability criteria in terms of entropic uncertainty relations. In particular, we show how any entropic uncertainty relation on one part of the system results in a separability condition on the composite system. We investigate the resulting criteria using the Tsallis entropy for two and three qubits.
  • We prove tight entropic uncertainty relations for a large number of mutually unbiased measurements. In particular, we show that a bound derived from the result by Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)] for two such measurements can in fact be tight for up to {radical}(d) measurements in mutually unbiased bases. We then show that using more mutually unbiased bases does not always lead to a better locking effect. We prove that the optimal bound for the accessible information using up to {radical}(d) specific mutually unbiased bases is log d/2, which is the same as can be achievedmore » by using only two bases. Our result indicates that merely using mutually unbiased bases is not sufficient to achieve a strong locking effect and we need to look for additional properties.« less