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Title: Strong bound between trace distance and Hilbert-Schmidt distance for low-rank 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 Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt 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 low-rank states but also for low-entropy states. Here, our results have relevance to quantum information theory, quantum algorithm design, and quantum complexity theory.

Authors:
ORCiD logo [1]; ORCiD logo [1]; ORCiD logo [1]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (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):
LA-UR-19-22724
Journal ID: ISSN 2469-9926; 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 2469-9926
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 Hilbert-Schmidt distance for low-rank 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 Hilbert-Schmidt distance for low-rank 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 Hilbert-Schmidt distance for low-rank 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 Hilbert-Schmidt distance for low-rank 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 Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt 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 low-rank states but also for low-entropy 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}
}

Journal Article:

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Figures / Tables:

FIG. 1 FIG. 1: Bounds for Q(ρi, σi) = D(ρi, σi)2/DHS(ρi, σi) for d = 16 (top) and d = 32 (bottom). The gray dots correspond to Q(ρi, σi) for random states σi with uniformly distributed rank and purity and random states ρi with bounded rank (1 ≤ rank(ρ) ≤ d/4). Resultsmore » were ordered by increasing value of R, with the value of R shown as the solid red curve. The dashed blue line is the alternative upper bound Q(ρ, σ)≤ d/4, obtained from the norm equivalence, while the dashed red line is the lower bound 1/2 ≤ Q(ρ, σ) from (6). Over this range of R values, our bound Q(ρ, σ) ≤ R can be saturated and is tighter than the bound from norm equivalence.« less

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Figures / Tables found in this record:

    Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.