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Cited by 357Quantum Information and CryptographyQuantum Mechanics and ApplicationsAdvanced Thermodynamics and Statistical Mechanics

Reversible Framework for Quantum Resource Theories

Open access
Brandão, Fernando G. S. L., Gour, Gilad·2015·Physical Review Letters
DOI 10.1103/physrevlett.115.070503
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Abstract

In recent years it has been recognized that properties of physical systems such as entanglement, athermality, and asymmetry, can be viewed as resources for important tasks in quantum information, thermodynamics, and other areas of physics. This recognition was followed by the development of specific quantum resource theories (QRTs), such as entanglement theory, determining how quantum states that cannot be prepared under certain restrictions may be manipulated and used to circumvent the restrictions. Here we discuss the general structure of QRTs, and show that under a few assumptions (such as convexity of the set of free states), a QRT is asymptotically reversible if its set of allowed operations is maximal, that is, if the allowed operations are the set of all operations that do not generate (asymptotically) a resource. In this case, the asymptotic conversion rate is given in terms of the regularized relative entropy of a resource which is the unique measure or quantifier of the resource in the asymptotic limit of many copies of the state. This measure also equals the smoothed version of the logarithmic robustness of the resource.

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Written from the paper’s full text.

TL;DR

This Letter proves a general theorem for quantum resource theories: under five physically motivated assumptions on the free states, a resource theory is asymptotically reversible if its allowed operations are maximal, i.e., resource non-generating. In that case the asymptotic conversion rate is governed by the regularized relative entropy of a resource, which is the unique asymptotic resource measure.

Key findings

  • If the set of free states \(F\) satisfies five postulates—closure under tensor products, partial trace of spatially separated subsystems, permutations of spatially separated subsystems, closedness, and convexity—then the regularized relative entropy of a resource can be expressed as in the paper’s Eq. (4).
  • For every resource state \(\sigma\) with \(E_\infty(\sigma)>0\), the optimal asymptotic conversion rate from \(\rho\) to \(\sigma\) under asymptotically resource non-generating operations is \(E_\infty(\rho)/E_\infty(\sigma)\).
  • The regularized relative entropy is identified as the “unique” measure of a resource in the asymptotic limit of many copies, determining reversible conversion rates.
  • The abstract states that this measure also equals the smoothed version of the logarithmic robustness of the resource.

Methods

  • Theoretical proof in a general quantum resource theory framework, with finite-dimensional systems, free states \(F\), and free operations constrained by the free operations postulate (FOP): free operations cannot generate a resource.
  • The paper imposes five postulates on \(F\): tensor-product closure, partial-trace closure for spatially separated subsystems, permutation closure, closedness of each \(F_m\), and convexity of each \(F_m\).
  • The proof defines \(\varepsilon\)-resource non-generating operations and asymptotically resource non-generating operations, and partly uses a recent generalization of the quantum Stein’s Lemma and the exponential distinguishability property (EDP). No empirical data or sample are used.

Limitations

  • The paper states that it considers only finite-dimensional Hilbert spaces; extension to infinite-dimensional systems under a suitable energy constraint is left to future work.
  • The paper states that the result cannot be applied directly to the resource theory of non-Gaussianity, partly because the set of Gaussian states is not convex, nor straightforwardly to the resource theory of asymmetry because the regularized relative entropy of asymmetry is zero.
  • The paper notes that general single-shot results are far from trivial and left for further study; it also states that convexity may fail in some QRTs, in which case the main result cannot be applied.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevlett.115.070503

Cites

46 of 46 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 14 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § A Reversible Framework for Resource Theories · p. 4

    The sequence of maps constructed in the proof is asymptotically non-resource generating.

    “we find that indeed lim n →∞ max ω n ∈F n R (Λ n ( ω n )) = 0.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The set of free states F is closed under tensor products.
    • The authors state this as an assumption: The set of free states F is closed under the partial trace of spatially separated subsystems.
    • The authors state this as an assumption: The set of free states F is closed under permutations of spatially separated subsystems.
    • The authors state this as an assumption: Each F_m is a closed set.
    • The authors state this as an assumption: Each F_m is a convex set.
  • Findingreasoning
    § A Reversible Framework for Resource Theories · p. 3

    Under the five postulates, the regularized relative entropy of a resource can be expressed as Eq. (4).

    “Theorem 1. Consider a QRT with a set of free states F . If F satisfies the 5 postulates above then the regularized relative entropy of a resource can be expressed as”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The set of free states F is closed under tensor products.
    • The authors state this as an assumption: The set of free states F is closed under the partial trace of spatially separated subsystems.
    • The authors state this as an assumption: The set of free states F is closed under permutations of spatially separated subsystems.
    • The authors state this as an assumption: Each F_m is a closed set.
    • The authors state this as an assumption: Each F_m is a convex set.
  • Findingreasoning
    § A Reversible Framework for Resource Theories · p. 1

    Under the five postulates on the set of free states, a quantum resource theory is asymptotically reversible if its set of allowed operations is maximal (the set of all operations that do not generate a resource asymptotically).

    “Here we discuss the general structure of QRTs, and show that under a few assumptions (such as convexity of the set of free states), a QRT is asymptotically reversible if its set of allowed operations is maximal ; that is, if the allowed operations are the set of all operations that do not generate (asymptotically) a resource.”
    • Also stated in: § A Reversible Framework for Resource Theories · p. 4 · boosted
      “To summarize, we have shown that under 5 very reasonable assumptions on the set of free states F , a QRT is asymptotically reversible if the set of free operations is maximal.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The set of free states F is closed under tensor products.
    • The authors state this as an assumption: The set of free states F is closed under the partial trace of spatially separated subsystems.
    • The authors state this as an assumption: The set of free states F is closed under permutations of spatially separated subsystems.
    • The authors state this as an assumption: Each F_m is a closed set.
    • The authors state this as an assumption: Each F_m is a convex set.
  • Findingreasoning
    § A Reversible Framework for Resource Theories · p. 2

    Since F is closed, the set of resource states is open.

    “Since F is closed, the set of resource states is open.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Each F_m is a closed set.
  • Findingnone given
    § A Reversible Framework for Resource Theories · p. 1

    The regularized relative entropy of a resource equals the smoothed version of the logarithmic robustness of the resource.

    “This measure also equals the smoothed version of the logarithmic robustness of the resource.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The set of free states F is closed under tensor products.
    • The authors state this as an assumption: The set of free states F is closed under the partial trace of spatially separated subsystems.
    • The authors state this as an assumption: The set of free states F is closed under permutations of spatially separated subsystems.
    • The authors state this as an assumption: Each F_m is a closed set.
    • The authors state this as an assumption: Each F_m is a convex set.
  • May combine two claims. 7
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 4

  • May combine two claims.

    Findingreasoning
    § A Reversible Framework for Resource Theories · p. 1

    In an asymptotically reversible QRT with maximal allowed operations, the asymptotic conversion rate is given by the regularized relative entropy of a resource, which is the unique measure/quantifier of the resource in the asymptotic limit of many copies.

    “In this case, the asymptotic conversion rate is given in terms of the regularized relative entropy of a resource which is the unique measure/quantifier of the resource in the asymptotic limit of many copies of the state.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The set of free states F is closed under tensor products.
    • The authors state this as an assumption: The set of free states F is closed under the partial trace of spatially separated subsystems.
    • The authors state this as an assumption: The set of free states F is closed under permutations of spatially separated subsystems.
    • The authors state this as an assumption: Each F_m is a closed set.
    • The authors state this as an assumption: Each F_m is a convex set.

  • May combine two claims.

    Findingreasoning
    § A Reversible Framework for Resource Theories · p. 3

    The relative entropy of a resource is a monotone because S(ρ||σ) is contractive.

    “Since S ( ρ ‖ σ ) is contractive, the relative entropy of a resource is a monotone.”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § A Reversible Framework for Resource Theories · p. 4

    For every σ with E∞(σ)>0, the optimal rate of converting n copies of ρ into m copies of σ under asymptotically resource non-generating operations is E∞(ρ)/E∞(σ).

    “so that indeed the rate is E ∞ ( ρ ) /E ∞ ( σ ).”
    • Also stated in: § A Reversible Framework for Resource Theories · p. 4 · plain
      “and for every σ such that E ∞ ( σ ) > 0 ,”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The set of free states F is closed under tensor products.
    • The authors state this as an assumption: The set of free states F is closed under the partial trace of spatially separated subsystems.
    • The authors state this as an assumption: The set of free states F is closed under permutations of spatially separated subsystems.
    • The authors state this as an assumption: Each F_m is a closed set.
    • The authors state this as an assumption: Each F_m is a convex set.

  • May combine two claims.

    Premisecitation
    § A Reversible Framework for Resource Theories · p. 3

    Both the robustness and the relative entropy measure are monotones under NR, convex, and faithful (zero iff ρ∈F_m).

    “Both the robustness and the relative entropy measure are monotones under NR . They are also both convex and faithful (see e.g. [4, 5]) in the sense that they are zero iff ρ ∈ F m .”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [4] M. B. Plenio and S. Virmani, Quant. Inf. Comp. 7 , 1 (2007). source

      Not checked in the free run.

    • cites [5] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys. 81 , 865 (2009). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § A Reversible Framework for Resource Theories · p. 4

    Eq. (5) identifies the regularized relative entropy as the unique measure of a resource in the asymptotic limit; a single function E∞ determines the rate of reversible conversion of many copies of ρ to many copies of σ under non-resource generating operations.

    “Eq.(5) in the theorem above identifies the regularized relative entropy as the 'unique' measure of a resource in the asymptotic limit. That is, there is a single function, E ∞ , that determines the rate of (reversible) conversion of many copies of ρ to many copies of σ under non-resource generating operations.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The set of free states F is closed under tensor products.
    • The authors state this as an assumption: The set of free states F is closed under the partial trace of spatially separated subsystems.
    • The authors state this as an assumption: The set of free states F is closed under permutations of spatially separated subsystems.
    • The authors state this as an assumption: Each F_m is a closed set.
    • The authors state this as an assumption: Each F_m is a convex set.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § A Reversible Framework for Resource Theories · p. 4

    In [26] it was shown that the regularized relative entropy of a resource is the unique asymptotic rate of any reversible QRT.

    “In [26] it was shown that the regularized relative entropy of a resource is the unique asymptotic rate of any reversible QRT.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [26] M. Horodecki, J. Oppenheim, and R. Horodecki, Phys. Rev. Lett. 89, 240403 (2002), quant-ph/0207177 source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § A Reversible Framework for Resource Theories · p. 3

    Any measure M defined as the distance from F_m using a contractive metric C is a resource quantifier and is non-increasing under any map in NR_m.

    “where C is a contractive metric, is a resource quantifier and in particular non-increasing under any map in NR m (see e.g. [4, 5]).”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [4] M. B. Plenio and S. Virmani, Quant. Inf. Comp. 7 , 1 (2007). source

      Not checked in the free run.

    • cites [5] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys. 81 , 865 (2009). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationcitation
    § A Reversible Framework for Resource Theories · p. 4

    The results presented here cannot be applied directly to all QRTs because the paper considered only finite dimensional Hilbert spaces, and it is not applicable to the resource theory of Non-Gaussianity (also the set of Gaussian states is not convex [15]).

    “The results presented here, however, can not be applied directly to all QRTs since for example we considered only finite dimensional Hilbert spaces, and especially it is not applicable for the resource theory of Non-Gaussianity (also the set of Gaussian states is not convex [15]).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [15] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77 513, (2005). source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § A Reversible Framework for Resource Theories · p. 4

    In [25] Proposition II.1 showed that in this case the relative entropy of a resource can be expressed as in Eq. (4).

    “Furthermore, it was also shown in Proposition II.1 of [25] that in this case the relative entropy of a resource can be expressed as in Eq. (4).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [25] F. G.S.L. Brandao and M. B. Plenio, Commun. Math. Phys. 295 , 791 (2010). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § A Reversible Framework for Resource Theories · p. 4

    The proof of the main theorem cannot follow directly from its analog in entanglement theory [23].

    “Note however that the proof of the theorem above can not follow directly from its analog in entanglement theory [23].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [23] F. G.S.L. Brand˜ ao and M. B. Plenio, Nature Physics 4 , 873 (2008). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § A Reversible Framework for Resource Theories · p. 3

    The robustness measure in entanglement theory measures the amount of noise that can be added to an entangled state before it becomes unentangled (separable).

    “The measure of robustness [23, 33, 34] in entanglement theory measures the amount of noise that can be added to an entangled state before it becomes unentangled (separable).”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [23] F. G.S.L. Brand˜ ao and M. B. Plenio, Nature Physics 4 , 873 (2008). source

      Not checked in the free run.

    • cites [33] G. Vidal and R. Tarrach, Phys.Rev. A 59 141 (1999). source

      Not checked in the free run.

    • cites [34] A. Harrow and M. A.Nielsen, Phys. Rev. A 68 , 012308 (2003). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationcitation
    § A Reversible Framework for Resource Theories · p. 4

    General QRTs may have many inequivalent maximal resource states, unlike entanglement theory which has a unique golden unit such as the Bell/Singlet state.

    “Unlike entanglement theory, that have a unique 'golden' unit such as the Bell/Singlet state, general QRTs may have many such units, and more precisely, can have many inequivalent maximal resource states.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [23] F. G.S.L. Brand˜ ao and M. B. Plenio, Nature Physics 4 , 873 (2008). source

      Not checked in the free run.

12 of 12

  • The authors state this as an assumption.

    The set of free states F is closed under tensor products.

    “Postulate I : The set of free states F is closed under tensor products.”

  • The authors state this as an assumption.

    The set of free states F is closed under the partial trace of spatially separated subsystems.

    “Postulate II : The set of free states F is closed under the partial trace of spatially separated subsystems.”

  • The authors state this as an assumption.

    The set of free states F is closed under permutations of spatially separated subsystems.

    “Postulate III : The set of free states F is closed under permutations of spatially separated subsystems.”

  • The authors state this as an assumption.

    Each F_m is a closed set.

    “Postulate IV : Each F m is a closed set.”

  • The authors state this as an assumption.

    Each F_m is a convex set.

    “Postulate V : Each F m is a convex set.”

  • The authors state this as an assumption.

    The set of free operations cannot generate a resource; they cannot convert free states into resource states.

    “The free operations postulate (FOP): The set of free operations can not generate a resource; they can not convert free states into resource states.”

  • The paper relies on this without support.

    So far there are no known theorems that can be applied to a large class of QRTs.

    “Despite this large body of work, so far there are no known theorems that can be applied to a large class of QRTs.”
7 of 7

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The variance of energy ∆E^2 is nonincreasing for pure-to-pure state transformations achieved by TI quantum operations.

    “Note that all the properties (i), (ii) and (iii) of A min/max are also satisfied by the variance of the energy, ∆ 2 E , and consequently, using essentially the same argument, one can show that the variance is also nonincreasing for pure to pure state transformations that are achieved by TI quantum opertations [28, 30].”

    From Quantum speed limits, coherence, and asymmetry (2016) · cites this paper as [30]

Cites

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Research works printed but not linked: 2

  1. 10.1103/physreva.80.012307 · DOI
  2. 10.1088/1367-2630/14/7/073022 · DOI

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Related

Authors
Brandão, F. G. S. L., & Gour, G.
Affiliations
  • Microsoft (United States) (US)
  • University of Calgary (CA)
ORCID
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
2015-08-14
Volume
115
Issue
7
Pages
070503-070503
ISSN
  • 0031-9007 (ISSN-L)
  • 1079-7114
DOI
10.1103/physrevlett.115.070503
OpenAlex
W189715775
Semantic Scholar
fd17449747d72f2bbbca96b0f40d9753403c81d9
Cited by
357
References
46
Citations by year
2026: 13 · 2025: 18 · 2024: 19 · 2023: 22 · 2022: 30 · 2021: 50 · 2020: 49 · 2019: 43 · 2018: 51 · 2017: 33 · 2016: 24 · 2015: 5
FWCI
36.93
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review Letters (publishedVersion)
  • arXiv (Cornell University) (publishedVersion)
  • PubMed (publishedVersion)
  • CaltechAUTHORS (California Institute of Technology) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Engineering and Physical Sciences Research Council
  • Natural Sciences and Engineering Research Council of Canada
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
Keywords
quantum resource theory, reversibility, entanglement, quantum asymmetry, Free State
MeSH terms
Models, Theoretical, Quantum Theory

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