Entropic Energy-Time Uncertainty Relation

Patrick J. Coles, Vishal Katariya, Seth Lloyd, Iman Marvian, and Mark M. Wilde
Phys. Rev. Lett. 122, 100401 – Published 12 March 2019
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Abstract

Energy-time uncertainty plays an important role in quantum foundations and technologies, and it was even discussed by the founders of quantum mechanics. However, standard approaches (e.g., Robertson’s uncertainty relation) do not apply to energy-time uncertainty because, in general, there is no Hermitian operator associated with time. Following previous approaches, we quantify time uncertainty by how well one can read off the time from a quantum clock. We then use entropy to quantify the information-theoretic distinguishability of the various time states of the clock. Our main result is an entropic energy-time uncertainty relation for general time-independent Hamiltonians, stated for both the discrete-time and continuous-time cases. Our uncertainty relation is strong, in the sense that it allows for a quantum memory to help reduce the uncertainty, and this formulation leads us to reinterpret it as a bound on the relative entropy of asymmetry. Because of the operational relevance of entropy, we anticipate that our uncertainty relation will have information-processing applications.

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  • Received 11 June 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.100401

© 2019 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Patrick J. Coles1, Vishal Katariya2, Seth Lloyd3,4, Iman Marvian5, and Mark M. Wilde2,6

  • 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
  • 2Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA
  • 3Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  • 4Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  • 5Departments of Physics & Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA
  • 6Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA

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Issue

Vol. 122, Iss. 10 — 15 March 2019

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