Universal Uncertainty Relations

Shmuel Friedland, Vlad Gheorghiu, and Gilad Gour
Phys. Rev. Lett. 111, 230401 – Published 3 December 2013; Erratum Phys. Rev. Lett. 112, 119905 (2014)
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Abstract

Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring noncommuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabeling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, significantly extending a majorization-based uncertainty relation first introduced in M. H. Partovi, Phys. Rev. A 84, 052117 (2011). Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory.

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  • Received 10 May 2013

DOI:https://doi.org/10.1103/PhysRevLett.111.230401

© 2013 American Physical Society

Erratum

Erratum: Universal Uncertainty Relations [Phys. Rev. Lett. 111, 230401 (2013)]

Shmuel Friedland, Vlad Gheorghiu, and Gilad Gour
Phys. Rev. Lett. 112, 119905 (2014)

Authors & Affiliations

Shmuel Friedland1,*, Vlad Gheorghiu2,3,†, and Gilad Gour2,‡

  • 1Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Illinois 60607-7045, USA
  • 2Institute for Quantum Science and Technology and Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada
  • 3Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

  • *friedlan@uic.edu
  • vgheorgh@gmail.com
  • gour@ucalgary.ca

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Issue

Vol. 111, Iss. 23 — 6 December 2013

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Heating up of Superconductors
January 27, 2017

This collection marks the 30th anniversary of the discovery of high-temperature superconductors. The papers selected highlight some of the advances that have been made to date, both in understanding why these compounds behave in the way they do, and in utilizing them in applications. The papers included in the collection have been made free to read.

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