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Uncertainty relations from graph theory

Carlos de Gois*, Kiara Hansenne*,†, and Otfried Gühne

  • Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany

  • *These authors contributed equally to this work.
  • †Corresponding author: kiara.hansenne@uni-siegen.de
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Phys. Rev. A 107, 062211 – Published 22 June, 2023

DOI: https://doi.org/10.1103/PhysRevA.107.062211

Abstract

Quantum measurements are inherently probabilistic and quantum theory often forbids precisely predicting the outcomes of simultaneous measurements. This phenomenon is captured and quantified through uncertainty relations. Although studied since the inception of quantum theory, the problem of determining the possible expectation values of a collection of quantum measurements remains, in general, unsolved. By constructing a close connection between observables and graph theory, we derive uncertainty relations valid for any set of dichotomic observables. For arbitrary observables, we obtain an upper bound on the sum of the square of their expectation values. We furthermore show how this bound behaves when the observables are imprecisely calibrated. As applications, our results can be straightforwardly used to formulate entropic uncertainty relations, separability criteria, and entanglement witnesses.

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Information

Phys. Rev. A 107, 062211– Published 22 June, 2023
  • Received 22 August 2022
  • Revised 28 December 2022
  • Accepted 5 June 2023
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