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 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):
 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. doi: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. doi: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
Works referenced in this record:
Variational quantum state diagonalization
journal, June 2019
 LaRose, Ryan; Tikku, Arkin; O’NeelJudy, Étude
 npj Quantum Information, Vol. 5, Issue 1
Entanglement measures and the Hilbert–Schmidt distance
journal, April 2000
 Ozawa, Masanao
 Physics Letters A, Vol. 268, Issue 3
swap test and HongOuMandel effect are equivalent
journal, May 2013
 GarciaEscartin, Juan Carlos; ChamorroPosada, Pedro
 Physical Review A, Vol. 87, Issue 5
Learning the quantum algorithm for state overlap
journal, November 2018
 Cincio, Lukasz; Subaşı, Yiğit; Sornborger, Andrew T.
 New Journal of Physics, Vol. 20, Issue 11
Variational consistent histories as a hybrid algorithm for quantum foundations
journal, July 2019
 Arrasmith, Andrew; Cincio, Lukasz; Sornborger, Andrew T.
 Nature Communications, Vol. 10, Issue 1
Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in NonAbelian Fractional Quantum Hall Effect States
journal, July 2008
 Li, Hui; Haldane, F. D. M.
 Physical Review Letters, Vol. 101, Issue 1
Quantum detection and estimation theory
journal, January 1969
 Helstrom, Carl W.
 Journal of Statistical Physics, Vol. 1, Issue 2
Quantumassisted quantum compiling
journal, May 2019
 Khatri, Sumeet; LaRose, Ryan; Poremba, Alexander
 Quantum, Vol. 3
Unification of different views of decoherence and discord
journal, April 2012
 Coles, Patrick J.
 Physical Review A, Vol. 85, Issue 4
Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)
journal, December 1912
 Weyl, Hermann
 Mathematische Annalen, Vol. 71, Issue 4