Products of weak values: Uncertainty relations, complementarity, and incompatibility

Michael J. W. Hall, Arun Kumar Pati, and Junde Wu
Phys. Rev. A 93, 052118 – Published 27 May 2016

Abstract

The products of weak values of quantum observables are shown to be of value in deriving quantum uncertainty and complementarity relations, for both weak and strong measurement statistics. First, a “product representation formula” allows the standard Heisenberg uncertainty relation to be derived from a classical uncertainty relation for complex random variables. We show this formula also leads to strong uncertainty relations for unitary operators and underlies an interpretation of weak values as optimal (complex) estimates of quantum observables. Furthermore, we show that two incompatible observables that are weakly and strongly measured in a weak measurement context obey a complementarity relation under the interchange of these observables, in the form of an upper bound on the product of the corresponding weak values. Moreover, general tradeoff relations between weak purity, quantum purity, and quantum incompatibility, and also between weak and strong joint probability distributions, are obtained based on products of real and imaginary components of weak values, where these relations quantify the degree to which weak probabilities can take anomalous values in a given context.

  • Figure
  • Received 18 March 2016

DOI:https://doi.org/10.1103/PhysRevA.93.052118

©2016 American Physical Society

Physics Subject Headings (PhySH)

General Physics

Authors & Affiliations

Michael J. W. Hall1, Arun Kumar Pati2,3,*, and Junde Wu3

  • 1Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia
  • 2Quantum Information and Computation Group, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
  • 3Department of Mathematics, Zhejiang University, Hangzhou 310027, People's Republic of China

  • *akpati@hri.res.in

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Issue

Vol. 93, Iss. 5 — May 2016

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