Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states

Coles, Patrick Joseph; Cerezo, Marco Vinicio Sebastain; Cincio, Lukasz

doi:10.1103/PhysRevA.100.022103

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:

^[1];

^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:: Tue Aug 06 00:00:00 EDT 2019

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):: 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.

Copy to clipboard


                    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

Copy to clipboard


                    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.

Copy to clipboard


                    
@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         = {Tue Aug 06 00:00:00 EDT 2019},

  month        = {Tue Aug 06 00:00:00 EDT 2019}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (Publisher)

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1103/PhysRevA.100.022103

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 17 works

Citation information provided by
Web of Science

Figures / Tables:

FIG. 1: Bounds for Q(ρ_i, σ_i) = D(ρ_i, σ_i)²/D_HS(ρ_i, σ_i) for d = 16 (top) and d = 32 (bottom). The gray dots correspond to Q(ρ_i, σ_i) for random states σ_i with uniformly distributed rank and purity and random states ρ_i with bounded rank (1 ≤ rank(ρ) ≤ d/4). Resultsmore »

All figures and tables (1 total)

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Variational quantum state diagonalization
journal, June 2019

LaRose, Ryan; Tikku, Arkin; O’Neel-Judy, Étude
npj Quantum Information, Vol. 5, Issue 1
DOI: 10.1038/s41534-019-0167-6

Entanglement measures and the Hilbert–Schmidt distance
journal, April 2000

Ozawa, Masanao
Physics Letters A, Vol. 268, Issue 3
DOI: 10.1016/S0375-9601(00)00171-7

Quantum Computation and Quantum Information
book, January 2011

Nielsen, Michael A.; Chuang, Isaac L.
Cambridge University Press
DOI: 10.1017/CBO9780511976667

swap test and Hong-Ou-Mandel effect are equivalent
journal, May 2013

Garcia-Escartin, Juan Carlos; Chamorro-Posada, Pedro
Physical Review A, Vol. 87, Issue 5
DOI: 10.1103/PhysRevA.87.052330

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
DOI: 10.1088/1367-2630/aae94a

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
DOI: 10.1038/s41467-019-11417-0

Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
journal, July 2008

Li, Hui; Haldane, F. D. M.
Physical Review Letters, Vol. 101, Issue 1
DOI: 10.1103/PhysRevLett.101.010504

Quantum detection and estimation theory
journal, January 1969

Helstrom, Carl W.
Journal of Statistical Physics, Vol. 1, Issue 2
DOI: 10.1007/BF01007479

Quantum-assisted quantum compiling
journal, May 2019

Khatri, Sumeet; LaRose, Ryan; Poremba, Alexander
Quantum, Vol. 3
DOI: 10.22331/q-2019-05-13-140

Unification of different views of decoherence and discord
journal, April 2012

Coles, Patrick J.
Physical Review A, Vol. 85, Issue 4
DOI: 10.1103/PhysRevA.85.042103

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
DOI: 10.1007/BF01456804

Quantum detection and estimation theory
journal, January 1978

Yuen, H. P.
Proceedings of the IEEE, Vol. 66, Issue 2
DOI: 10.1109/proc.1978.10894

Quantum Information Theory
book, January 2017

Hayashi, Masahito
Graduate Texts in Physics
DOI: 10.1007/978-3-662-49725-8

Quantum information theory
journal, August 1976

Ingarden, Roman S.
Reports on Mathematical Physics, Vol. 10, Issue 1
DOI: 10.1016/0034-4877(76)90005-7

Quantum computation and quantum information†
journal, February 2006

Nagy, Marius; Akl, Selim G.
International Journal of Parallel, Emergent and Distributed Systems, Vol. 21, Issue 1
DOI: 10.1080/17445760500355678

Learning the quantum algorithm for state overlap
text, January 2018

Cincio, Lukasz; Subaşı, Yiğit; Sornborger, Andrew T.
arXiv
DOI: 10.48550/arxiv.1803.04114

Variational Consistent Histories as a Hybrid Algorithm for Quantum Foundations
text, January 2018

Arrasmith, Andrew; Cincio, Lukasz; Sornborger, Andrew T.
arXiv
DOI: 10.48550/arxiv.1812.10759

Figures / Tables found in this record:

FIG. 1 (p. 5)

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

Similar Records in DOE PAGES and OSTI.GOV collections:

Contractivity of the Hilbert-Schmidt distance under open-system dynamics

Journal Article Xiaoting, Wang ; Schirmer, S G - Physical Review. A

It is shown that the Hilbert-Schmidt (HS) norm and distance, unlike the trace norm and distance, are generally not contractive for open quantum systems under Lindblad dynamics. Necessary and sufficient conditions for contractivity of the HS norm and distance are given, and explicit criteria in terms of the Lindblad operators are derived. It is also shown that the requirements for contractivity of the HS distance are strictly weaker than those for the HS norm, although simulations suggest that noncontractivity is a typical case, i.e., that systems for which the HS distance between quantum states is monotonically decreasing are exceptional formore »« less
https://doi.org/10.1103/PHYSREVA.79.052326
On the dimension of subspaces with bounded Schmidt rank

Journal Article Cubitt, Toby ; Montanaro, Ashley ; Winter, Andreas ; ... - Journal of Mathematical Physics

We consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states. This is motivated in part by results of Hayden et al. [e-print arXiv:quant-ph/0407049; Commun. Math. Phys., 265, 95 (2006)], which show that in large dxd-dimensional systems there exist random subspaces of dimension almost d{sup 2}, all of whose states have entropy of entanglement at least log d-O(1). It is also a generalization of results on the dimension of completely entangled subspaces, which have connections with the construction of unextendible product bases. Here we take asmore »« less
https://doi.org/10.1063/1.2862998
Schmidt numbers of low-rank bipartite mixed states

Journal Article Hao, Chen - Physical Review. A

Schmidt numbers of bipartite mixed states [B. M. Torhal and P. Horodecki, Phys. Rev. A 61, R040301 (2001)] characterize the minimum Schmidt ranks of pure states that are needed to construct such mixed states. Schmidt number is the minimum number of entangled degrees of freedom of a bipartite mixed state. We give a lower bound for the Schmidt numbers of low-rank bipartite mixed states and conclude that generic low-rank mixed states have relatively high Schmidt numbers and thus entangled.
https://doi.org/10.1103/PhysRevA.67.062301
Variational Quantum Fidelity Estimation

Journal Article Cerezo de la Roca, Marco Vinicio Sebastian ; Poremba, Alexander ; Cincio, Lukasz ; ... - Quantum

Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity F ( ρ , σ ) based on the ``truncated fidelity'' F ( ρ m , σ ) , which is evaluated for a state ρ m obtained by projecting more » ρ onto its m -largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with m . To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalize ρ , (2) compute matrix elements of σ in the eigenbasis of ρ , and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where σ is arbitrary and ρ is low rank, which we call low-rank fidelity estimation, and we prove that no classical algorithm can efficiently solve this problem under reasonable assumptions. Finally, we demonstrate that our bounds can detect quantum phase transitions and are often tighter than previously known computable bounds for realistic situations.« less
https://doi.org/10.22331/q-2020-03-26-248

Full Text Available
On the Schmidt-rank-three bipartite and multipartite unitary operator

Journal Article Chen, Lin ; Yu, Li - Annals of Physics (New York)

Unitary operations are physically implementable. We further the understanding of such operators by studying the possible forms of nonlocal unitary operators, which are bipartite or multipartite unitary operators that are not tensor product operators. They are of broad relevance in quantum information processing. We prove that any nonlocal unitary operator of Schmidt rank three on a d{sub A}×d{sub B} bipartite system is locally equivalent to a controlled unitary. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank min(d{sub A}{sup 2},d{sub B}) when d{sub A}≤d{sub B}. We further show that any multipartite unitary operator Umore »« less
https://doi.org/10.1016/J.AOP.2014.09.026

Similar Records

Title: Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states

Abstract

Citation Formats

Figures / Tables:

Variational quantum state diagonalization journal, June 2019

Entanglement measures and the Hilbert–Schmidt distance journal, April 2000

Quantum Computation and Quantum Information book, January 2011

swap test and Hong-Ou-Mandel effect are equivalent journal, May 2013

Learning the quantum algorithm for state overlap journal, November 2018

Variational consistent histories as a hybrid algorithm for quantum foundations journal, July 2019

Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States journal, July 2008

Quantum detection and estimation theory journal, January 1969

Quantum-assisted quantum compiling journal, May 2019

Unification of different views of decoherence and discord journal, April 2012

Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung) journal, December 1912

Quantum detection and estimation theory journal, January 1978

Quantum Information Theory book, January 2017

Quantum information theory journal, August 1976

Quantum computation and quantum information† journal, February 2006

Learning the quantum algorithm for state overlap text, January 2018

Variational Consistent Histories as a Hybrid Algorithm for Quantum Foundations text, January 2018

Variational quantum state diagonalization
journal, June 2019

Entanglement measures and the Hilbert–Schmidt distance
journal, April 2000

Quantum Computation and Quantum Information
book, January 2011

swap test and Hong-Ou-Mandel effect are equivalent
journal, May 2013

Learning the quantum algorithm for state overlap
journal, November 2018

Variational consistent histories as a hybrid algorithm for quantum foundations
journal, July 2019

Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
journal, July 2008

Quantum detection and estimation theory
journal, January 1969

Quantum-assisted quantum compiling
journal, May 2019

Unification of different views of decoherence and discord
journal, April 2012

Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)
journal, December 1912

Quantum detection and estimation theory
journal, January 1978

Quantum Information Theory
book, January 2017

Quantum information theory
journal, August 1976

Quantum computation and quantum information†
journal, February 2006

Learning the quantum algorithm for state overlap
text, January 2018

Variational Consistent Histories as a Hybrid Algorithm for Quantum Foundations
text, January 2018