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Title: Entropic Energy-Time Uncertainty Relation

Abstract

We report that 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. Additionally, 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.

Authors:
ORCiD logo [1];  [2];  [3];  [4];  [2]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Louisiana State Univ., Baton Rouge, LA (United States)
  3. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
  4. Duke Univ., Durham, NC (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1511603
Report Number(s):
LA-UR-18-24312
Journal ID: ISSN 0031-9007; PRLTAO
Grant/Contract Number:  
89233218CNA000001
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review Letters
Additional Journal Information:
Journal Volume: 122; Journal Issue: 10; Journal ID: ISSN 0031-9007
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING

Citation Formats

Coles, Patrick Joseph, Katariya, Vishal, Lloyd, Seth, Marvian, Iman, and Wilde, Mark M. Entropic Energy-Time Uncertainty Relation. United States: N. p., 2019. Web. doi:10.1103/PhysRevLett.122.100401.
Coles, Patrick Joseph, Katariya, Vishal, Lloyd, Seth, Marvian, Iman, & Wilde, Mark M. Entropic Energy-Time Uncertainty Relation. United States. doi:10.1103/PhysRevLett.122.100401.
Coles, Patrick Joseph, Katariya, Vishal, Lloyd, Seth, Marvian, Iman, and Wilde, Mark M. Tue . "Entropic Energy-Time Uncertainty Relation". United States. doi:10.1103/PhysRevLett.122.100401.
@article{osti_1511603,
title = {Entropic Energy-Time Uncertainty Relation},
author = {Coles, Patrick Joseph and Katariya, Vishal and Lloyd, Seth and Marvian, Iman and Wilde, Mark M.},
abstractNote = {We report that 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. Additionally, 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.},
doi = {10.1103/PhysRevLett.122.100401},
journal = {Physical Review Letters},
number = 10,
volume = 122,
place = {United States},
year = {2019},
month = {3}
}

Journal Article:
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This content will become publicly available on March 12, 2020
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