Strong bound between trace distance and HilbertSchmidt distance for lowrank states
Abstract
The trace distance between two quantum states, ρ and σ, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of ρ–σ. In contrast, the HilbertSchmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the HilbertSchmidt distance with a bound that is particularly strong when either ρ or σ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace norms. We also consider bounds that are useful not only for lowrank states but also for lowentropy states. Here, our results have relevance to quantum information theory, quantum algorithm design, and quantum complexity theory.
 Authors:

 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE Laboratory Directed Research and Development (LDRD) Program
 OSTI Identifier:
 1565902
 Alternate Identifier(s):
 OSTI ID: 1547973
 Report Number(s):
 LAUR1922724
Journal ID: ISSN 24699926; PLRAAN; TRN: US2000938
 Grant/Contract Number:
 89233218CNA000001
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review A
 Additional Journal Information:
 Journal Volume: 100; Journal Issue: 2; Journal ID: ISSN 24699926
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Information Science; Mathematics
Citation Formats
Coles, Patrick Joseph, Cerezo, Marco Vinicio Sebastain, and Cincio, Lukasz. Strong bound between trace distance and HilbertSchmidt distance for lowrank states. United States: N. p., 2019.
Web. doi:10.1103/PhysRevA.100.022103.
Coles, Patrick Joseph, Cerezo, Marco Vinicio Sebastain, & Cincio, Lukasz. Strong bound between trace distance and HilbertSchmidt distance for lowrank states. United States. https://doi.org/10.1103/PhysRevA.100.022103
Coles, Patrick Joseph, Cerezo, Marco Vinicio Sebastain, and Cincio, Lukasz. Tue .
"Strong bound between trace distance and HilbertSchmidt distance for lowrank states". United States. https://doi.org/10.1103/PhysRevA.100.022103. https://www.osti.gov/servlets/purl/1565902.
@article{osti_1565902,
title = {Strong bound between trace distance and HilbertSchmidt distance for lowrank states},
author = {Coles, Patrick Joseph and Cerezo, Marco Vinicio Sebastain and Cincio, Lukasz},
abstractNote = {The trace distance between two quantum states, ρ and σ, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of ρ–σ. In contrast, the HilbertSchmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the HilbertSchmidt distance with a bound that is particularly strong when either ρ or σ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace norms. We also consider bounds that are useful not only for lowrank states but also for lowentropy states. Here, our results have relevance to quantum information theory, quantum algorithm design, and quantum complexity theory.},
doi = {10.1103/PhysRevA.100.022103},
journal = {Physical Review A},
number = 2,
volume = 100,
place = {United States},
year = {2019},
month = {8}
}
Web of Science
Figures / Tables:
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