pith. sign in

arxiv: 2207.11059 · v3 · submitted 2022-07-22 · ❄️ cond-mat.stat-mech

Majorization requires infinitely many second laws

Pedro Hack , Daniel A. Braun , Sebastian Gottwald This is my paper

Pith reviewed 2026-05-24 11:03 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords majorizationsecond lawentropy functionsthermo-majorizationresource theorycatalytic majorizationmolecular diffusion
0
0 comments X

The pith

Any complete second law for majorization requires a countably infinite family of entropy-like functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that in the majorization model of uncertainty, which underpins resource theories in thermodynamics and quantum physics, the second law cannot be expressed using only a finite number of entropy-like functions when the state space is large. A countably infinite family is necessary to fully characterize the preorder that determines allowed state transformations. The result applies similarly to thermo-majorization without size limits if the equilibrium is nonuniform, and to a majorization variant for molecular diffusion where finite families fail entirely. It also shows that characterizations of catalytic majorization require infinite families of real-valued functions. A reader would care because it shows the thermodynamic constraints in these models have an inherently infinite structure.

Core claim

For a sufficiently large state space, any family of entropy-like functions constituting a second law must be countably infinite. An analogous result holds for thermo-majorization provided the equilibrium distribution is not uniform. In a variation of majorization used to model molecular diffusion, no finite family of entropy-like functions constituting a second law exists. Characterizations of trumping require an infinite family of real-valued functions.

What carries the argument

Families of entropy-like functions that together completely characterize the majorization preorder and thereby serve as a second law.

If this is right

  • No finite collection of entropy functions can serve as a complete second law for majorization in sufficiently large systems.
  • Thermo-majorization requires an infinite family of functions unless the equilibrium distribution is uniform.
  • Models of molecular diffusion based on certain majorization variants admit no finite second-law family.
  • Any complete characterization of catalytic majorization by real-valued functions must use an infinite family.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Practical work with large but finite state spaces may still rely on increasingly large finite families as approximations.
  • The result suggests that resource theories built on majorization involve an infinite hierarchy of constraints rather than a closed finite set.
  • Analogous infinitude arguments could apply to other preorders arising in physical resource theories.

Load-bearing premise

A second law consists exactly of a family of functions that together give a complete characterization of the majorization preorder on the state space.

What would settle it

Identifying a finite family of entropy-like functions that fully determines the majorization relation on state spaces of arbitrary size would falsify the claim.

Figures

Figures reproduced from arXiv: 2207.11059 by Daniel A. Braun, Pedro Hack, Sebastian Gottwald.

Figure 1
Figure 1. Figure 1: Simple system of two molecular states for a gas in a box wher [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of the difference between a fu [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Representation of the simple ordered set defined in ( [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representation of pS, ĺSq inside the 2-simplex. We introduced pS, ĺSq in Theorem 1 to show that no finite family of second laws of disorder exists if |Ω| ě 3. Again, we represent ĺS as a directed graph, where p ĺS q whenever there exists a path following the arrows from p to q, and 1 2 ă x ă y ă z ă 1 2 ` ε ´ γ. From this representation, we can see immediately that pS, ĺSq is order isomorphic to the partia… view at source ↗
Figure 5
Figure 5. Figure 5: Representation of pPΩ, ĺdq when both |Ω| “ 2 and d is not a uniform distribution. We include d and six elements in Sd0 , which we defined in (7) and used in Theorem 3 to show that no finite family of second laws of d-disorder exists provided |Ω| “ 2 and d is not a uniform distribution. The elements in Sd0 that we include are qr “ pr, 1´rq and pr “ pp1´rqd0{p1´d0q, 1´ p1´rqd0{p1´ d0qq for r P tx, y, zu. Mor… view at source ↗
Figure 6
Figure 6. Figure 6: Representation of the majorization ordering for [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Representation of the molecular diffusion ordering for [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
read the original abstract

Majorization is a fundamental model of uncertainty with several applications in areas ranging from thermodynamics to entanglement theory, and constitutes one of the pillars of the resource-theoretic approach to physics. Here, we improve on its relation to measurement apparatuses. In particular, after discussing what the proper notion of second law in this scenario is, we show that, for a sufficiently large state space, any family of entropy-like functions constituting a second law must be countably infinite. Moreover, we provide an analogous result for a variation of majorization known as thermo-majorization which, in fact, does not require any constraint on the state space provided the equilibrium distribution is not uniform. Lastly, we discuss the applicability of our results to molecular diffusion and catalytic majorization. In this regard, we consider a variation of majorization used in plasma physics as a model of molecular diffusion and show that no finite family of entropy-like functions constituting a second law of molecular diffusion exists. Moreover, we show how our results are useful when dealing with a conjecture regarding catalytic majorization (i.e. trumping). In particular, we show that the sort of characterizations of trumping that have been considered before require an infinite family of real-valued functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims that, after defining the appropriate notion of a second law for majorization (a complete characterization of the preorder via monotonicity of a family of entropy-like functions), any such family must be countably infinite for sufficiently large state spaces. Analogous infinitude results are shown for thermo-majorization (without state-space size constraints when the equilibrium distribution is non-uniform), molecular diffusion (no finite family exists), and catalytic majorization/trumping (characterizations require an infinite family of real-valued functions).

Significance. If the result holds under the stated definition, it establishes a structural limitation on finite sets of Schur-concave or entropy-like functions in resource theories: they cannot jointly separate the majorization preorder exactly when the state space is large enough. This has direct implications for how second laws are formulated in statistical mechanics, quantum information, and applications such as diffusion models and catalysis. The work supplies explicit mathematical arguments and discusses physical applicability.

major comments (1)
  1. [Introduction / discussion of second-law notion] The load-bearing modeling choice is the definition of 'second law' as a family of functions satisfying x ≺ y ⇔ f(x) ≤ f(y) for all f in the family (complete preorder characterization). The abstract and title present the infinitude result as applying to second laws in the majorization setting, but under the weaker reading of second laws as providing only necessary conditions (monotonicity without completeness), finite families of Schur-concave functions remain useful. The paper discusses the 'proper notion' but the completeness requirement needs explicit justification or a clearer statement of scope to support the generality of the claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the importance of clearly justifying the modeling choice for the notion of a second law. We address the comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Introduction / discussion of second-law notion] The load-bearing modeling choice is the definition of 'second law' as a family of functions satisfying x ≺ y ⇔ f(x) ≤ f(y) for all f in the family (complete preorder characterization). The abstract and title present the infinitude result as applying to second laws in the majorization setting, but under the weaker reading of second laws as providing only necessary conditions (monotonicity without completeness), finite families of Schur-concave functions remain useful. The paper discusses the 'proper notion' but the completeness requirement needs explicit justification or a clearer statement of scope to support the generality of the claims.

    Authors: The manuscript already motivates the complete characterization (x ≺ y if and only if f(x) ≤ f(y) for all f in the family) as the appropriate notion of second law in the introduction, precisely because a family providing only necessary conditions cannot fully determine which transitions are allowed under majorization. The weaker notion is acknowledged as useful for deriving constraints but insufficient for the claim that majorization 'requires' infinitely many second laws to be completely characterized. Nevertheless, we agree that an expanded paragraph explicitly contrasting the two readings and reiterating why completeness is required will strengthen the presentation. We will add this justification in the revised introduction and adjust the abstract wording if needed to delineate scope. revision: yes

Circularity Check

0 steps flagged

No significant circularity; result is a direct mathematical theorem on preorder characterization

full rationale

The paper first discusses and adopts an explicit definition of a second law as any family of entropy-like functions that together fully characterize the majorization preorder (i.e., x ≺ y iff f(x) ≤ f(y) for all f in the family). It then proves that no finite such family exists on sufficiently large state spaces. This is a self-contained proof about the separation properties of the majorization relation itself and does not reduce to any fitted input, self-citation chain, or definitional equivalence; the infinitude follows from the preorder structure rather than being presupposed by the chosen definition. No load-bearing step matches the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions of majorization and the notion of entropy-like functions as second laws; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Majorization is the standard partial order on probability vectors.
    Invoked as the fundamental model throughout the abstract.
  • domain assumption A second law consists of a family of entropy-like functions that together characterize the majorization relation.
    Explicitly used to frame the infinitude result.

pith-pipeline@v0.9.0 · 5738 in / 1045 out tokens · 36480 ms · 2026-05-24T11:03:14.967199+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages · 1 internal anchor

  1. [1]

    Alberti and Armin Uhlmann

    Peter M. Alberti and Armin Uhlmann. Dissipative motion in state spaces , volume 33. Teubner, 1981

    work page 1981

  2. [2]

    Alberti and Armin Uhlmann

    Peter M. Alberti and Armin Uhlmann. Stochasticity and partial order . Deutscher Verlag der Wissenschaften Berlin, 1982

    work page 1982

  3. [3]

    Alcantud, Gianni Bosi, and Magal ` ı Zuanon

    Jos´ e Carlos R. Alcantud, Gianni Bosi, and Magal ` ı Zuanon. Richter–Peleg multi-utility representations of preorders. Theory and Decision , 80(3):443– 450, 2016

    work page 2016

  4. [4]

    Arnold and Jos´ e Mar ´ ıa Sarabia

    Barry C. Arnold and Jos´ e Mar ´ ıa Sarabia. Majorization and the Lorenz order with applications in applied mathematics and economi cs, volume 7. Springer, 2018

    work page 2018

  5. [5]

    Luca Bombelli, Joohan Lee, David Meyer, and Rafael D. Sorkin. Sp ace- time as a causal set. Physical review letters , 59(5):521, 1987

    work page 1987

  6. [6]

    The second laws of quantum thermodynamic s

    Fernando Brandao, Micha/suppress l Horodecki, Nelly Ng, Jonathan Oppenheim, and Stephanie Wehner. The second laws of quantum thermodynamic s. Proceedings of the National Academy of Sciences , 112(11):3275–3279, 2015

    work page 2015

  7. [7]

    What is ent ropy? A perspective from games of chance

    Sarah Brandsen, Isabelle Jianing Geng, and Gilad Gour. What is ent ropy? A perspective from games of chance. Physical Review E , 105(2):024117, 2022

    work page 2022

  8. [8]

    Untersuchungen ¨ uber die grundlagen der ther- modynamik

    Constantin Carath´ eodory. Untersuchungen ¨ uber die grundlagen der ther- modynamik. Mathematische Annalen , 67(3):355–386, 1909

    work page 1909

  9. [9]

    Jacob L. B. Cooper. The foundations of thermodynamics. Journal of Mathematical Analysis and Applications , 17(1):172–193, 1967

    work page 1967

  10. [10]

    Elements of information theory

    Thomas M Cover. Elements of information theory . John Wiley & Sons, 1999

    work page 1999

  11. [11]

    Mathematical structure o f entan- glement catalysis

    Sumit Daftuar and Matthew Klimesh. Mathematical structure o f entan- glement catalysis. Physical Review A , 64(4):042314, 2001

    work page 2001

  12. [12]

    The measurement of the inequality of incomes

    Hugh Dalton. The measurement of the inequality of incomes. The Economic Journal, 30(119):348–361, 1920

    work page 1920

  13. [13]

    On the multi-utility representation of pr efer- ence relations

    ¨Ozg¨ ur Evren and Efe A Ok. On the multi-utility representation of pr efer- ence relations. Journal of Mathematical Economics , 47(4-5):554–563, 2011

    work page 2011

  14. [14]

    Mathematical foundations of thermodynamics: Internation al series of monographs on pure and applied mathematics , volume 53

    Robin Giles. Mathematical foundations of thermodynamics: Internation al series of monographs on pure and applied mathematics , volume 53. Elsevier, 2016. 26

    work page 2016

  15. [15]

    The role of quantum information in thermodynamics—a topical review

    John Goold, Marcus Huber, Arnau Riera, L ´ ıdia Del Rio, and Paul Skrzypczyk. The role of quantum information in thermodynamics—a topical review. Journal of Physics A: Mathematical and Theoretical , 49(14):143001, 2016

    work page 2016

  16. [16]

    Bounded rational decis ion- making from elementary computations that reduce uncertainty

    Sebastian Gottwald and Daniel A Braun. Bounded rational decis ion- making from elementary computations that reduce uncertainty. Entropy, 21(4):375, 2019

    work page 2019

  17. [17]

    Quantum majorization and a complete set of entropic cond itions for quantum thermodynamics

    Gilad Gour, David Jennings, Francesco Buscemi, Runyao Duan, a nd Iman Marvian. Quantum majorization and a complete set of entropic cond itions for quantum thermodynamics. Nature communications, 9(1):1–9, 2018

    work page 2018

  18. [18]

    M¨ uller, Varun Narasimhachar, Robert W.Spekkens, and Nicole Yunger Halpern

    Gilad Gour, Markus P. M¨ uller, Varun Narasimhachar, Robert W.Spekkens, and Nicole Yunger Halpern. The resource theory of informational n onequi- librium in thermodynamics. Physics Reports, 583:1–58, 2015

    work page 2015

  19. [19]

    Braun, and Sebastian Gottwald

    Pedro Hack, Daniel A. Braun, and Sebastian Gottwald. On a geo - metrical notion of dimension for partially ordered sets. arXiv preprint arXiv:2203.16272v3, 2022

    work page arXiv 2022

  20. [20]

    Represe nting pre- orders with injective monotones

    Pedro Hack, Daniel A Braun, and Sebastian Gottwald. Represe nting pre- orders with injective monotones. Theory and Decision, 93(4):663–690, 2022

    work page 2022

  21. [21]

    The class ification of preordered spaces in terms of monotones: complexity and optim ization

    Pedro Hack, Daniel A Braun, and Sebastian Gottwald. The class ification of preordered spaces in terms of monotones: complexity and optim ization. Theory and Decision , 94(4):693–720, 2023

    work page 2023

  22. [22]

    Universality and classification o f ele- mentary thermal operations

    Pedro Hack and Christian B Mendl. Universality and classification o f ele- mentary thermal operations. arXiv preprint arXiv:2312.11223 , 2023

    work page arXiv 2023

  23. [23]

    On extreme points of the diffusion p olytope

    MJ Hay, J Schiff, and NJ Fisch. On extreme points of the diffusion p olytope. Physica A: Statistical Mechanics and its Applications , 473:225–236, 2017

    work page 2017

  24. [24]

    Fundamental limitations for quantum and nanoscale thermodynamics

    Micha/suppress l Horodecki and Jonathan Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics. Nature communications, 4(1):1– 6, 2013

    work page 2013

  25. [25]

    Information theory and statistical mechanics

    Edwin T Jaynes. Information theory and statistical mechanics . Physical review, 106(4):620, 1957

    work page 1957

  26. [26]

    Majorization and divergence

    Harry Joe. Majorization and divergence. Journal of mathematical analysis and applications, 148(2):287–305, 1990

    work page 1990

  27. [27]

    Daniel Jonathan and Martin B. Plenio. Entanglement-assisted lo cal ma- nipulation of pure quantum states. Physical Review Letters , 83(17):3566, 1999

    work page 1999

  28. [28]

    Doppelg¨ anger entropies

    AD Kirwan and William Seitz. Doppelg¨ anger entropies. Journal of Math- ematical Chemistry , 54:1942–1951, 2016. 27

    work page 1942

  29. [29]

    Entropy measures and catalysis of bipartite q uantum state transformations

    Matthew Klimesh. Entropy measures and catalysis of bipartite q uantum state transformations. In International Symposium onInformation Theory,

    work page

  30. [30]

    Proceedings

    ISIT 2004. Proceedings. , pages 357–357. IEEE, 2004

    work page 2004

  31. [31]

    Inequalities that Collectively Completely Characterize the Catalytic Majorization Relation

    Matthew Klimesh. Inequalities that collectively completely charac terize the catalytic majorization relation. arXiv preprint arXiv:0709.3680 , 2007

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  32. [32]

    Recovering gardner restacking with pur ely dif- fusive operations

    EJ Kolmes and NJ Fisch. Recovering gardner restacking with pur ely dif- fusive operations. Physical Review E , 102(6):063209, 2020

    work page 2020

  33. [33]

    Optimizing thermalization

    Kamil Korzekwa and Matteo Lostaglio. Optimizing thermalization. Phys- ical review letters , 129(4):040602, 2022

    work page 2022

  34. [34]

    Statistical Physics: Volume 5

    Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz. Statistical Physics: Volume 5 . Pergamon Press, 1980

    work page 1980

  35. [35]

    Landsberg

    Peter T. Landsberg. Main ideas in the axiomatics of thermodyna mics. Pure and Applied Chemistry , 22(3-4):215–228, 1970

    work page 1970

  36. [36]

    Lieb and Jakob Yngvason

    Elliott H. Lieb and Jakob Yngvason. The physics and mathematics of the second law of thermodynamics. Physics Reports, 310(1):1–96, 1999

    work page 1999

  37. [37]

    The entropy concept for non - equilibrium states

    Elliott H Lieb and Jakob Yngvason. The entropy concept for non - equilibrium states. Proceedings of the Royal Society A: Mathematical, Phys- ical and Engineering Sciences , 469(2158):20130408, 2013

    work page 2013

  38. [38]

    Entropy meters and the entr opy of non-extensive systems

    Elliott H Lieb and Jakob Yngvason. Entropy meters and the entr opy of non-extensive systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 470(2167):20140192, 2014

    work page 2014

  39. [39]

    Methods of measuring the concentration of wea lth

    Max O Lorenz. Methods of measuring the concentration of wea lth. Publi- cations of the American statistical association , 9(70):209–219, 1905

    work page 1905

  40. [40]

    An introductory review of the resource the ory approach to thermodynamics

    Matteo Lostaglio. An introductory review of the resource the ory approach to thermodynamics. Reports on Progress in Physics , 82(11):114001, 2019

    work page 2019

  41. [41]

    Alhambra, and Christopher Perry

    Matteo Lostaglio, ´Alvaro M. Alhambra, and Christopher Perry. Elementary thermal operations. Quantum, 2:52, 2018

    work page 2018

  42. [42]

    Continuous thermomajo rization and a complete set of laws for Markovian thermal processes

    Matteo Lostaglio and Kamil Korzekwa. Continuous thermomajo rization and a complete set of laws for Markovian thermal processes. Physical Re- view A , 106(1):012426, 2022

    work page 2022

  43. [43]

    Marshall, Ingram Olkin, and Barry C

    Albert W. Marshall, Ingram Olkin, and Barry C. Arnold. Inequalities: theory of majorization and its applications , volume 143. Springer, 1979

    work page 1979

  44. [44]

    Mixing character and its application to irreversible pr ocesses in macroscopic systems

    Alden Mead. Mixing character and its application to irreversible pr ocesses in macroscopic systems. The Journal of Chemical Physics , 66(2):459–467, 1977. 28

    work page 1977

  45. [45]

    Correlating thermal machines and the second law at the nanoscale

    Markus P M¨ uller. Correlating thermal machines and the second law at the nanoscale. Physical Review X , 8(4):041051, 2018

    work page 2018

  46. [46]

    A generalization of majoriz ation that characterizes shannon entropy

    Markus P M¨ uller and Michele Pastena. A generalization of majoriz ation that characterizes shannon entropy. IEEE Transactions on Information Theory, 62(4):1711–1720, 2016

    work page 2016

  47. [47]

    Michael A. Nielsen. Conditions for a class of entanglement trans formations. Physical Review Letters , 83(2):436, 1999

    work page 1999

  48. [48]

    Michael A. Nielsen. An introduction to majorization and its applica tions to quantum mechanics. Lecture Notes, Department of Physics, University of Queensland, Australia , 2002

    work page 2002

  49. [49]

    Nielsen and Isaac Chuang

    Michael A. Nielsen and Isaac Chuang. Quantum computation and quantum information, 2002

    work page 2002

  50. [50]

    Wealth and welfare

    Arthur Cecil Pigou. Wealth and welfare. Macmillan and Company, limited, 1912

    work page 1912

  51. [51]

    The diagram lattice as structural principle A

    Ernst Ruch. The diagram lattice as structural principle A. New a spects for representations and group algebra of the symmetric group B. Definition of classification character, mixing character, statistical order, statistical disorder; a general principle for the time evolution of irreversible pr ocesses. Theoretica Chimica Acta, 38(3):167–183, 1975

    work page 1975

  52. [52]

    The principle of increasing mixing char acter and some of its consequences

    Ernst Ruch and Alden Mead. The principle of increasing mixing char acter and some of its consequences. Theoretica chimica acta, 41(2):95–117, 1976

    work page 1976

  53. [53]

    Seligman

    Ernst Ruch, Rudolf Schranner, and Thomas H. Seligman. The mix ing distance. The Journal of Chemical Physics , 69(1):386–392, 1978

    work page 1978

  54. [54]

    Genera lization of a theorem by Hardy, Littlewood, and P´ olya

    Ernst Ruch, Rudolf Schranner, and Thomas H Seligman. Genera lization of a theorem by Hardy, Littlewood, and P´ olya. Journal of Mathematical Analysis and Applications , 76(1):222–229, 1980

    work page 1980

  55. [55]

    Entropy, Divergence, and Majorization in Classical and Quantum Thermodynamics, volume 16

    Takahiro Sagawa. Entropy, Divergence, and Majorization in Classical and Quantum Thermodynamics, volume 16. Springer Nature, 2022

    work page 2022

  56. [56]

    Mixed-up-ness or entropy? Entropy, 24(8):1090, 2022

    W Seitz and AD Kirwan Jr. Mixed-up-ness or entropy? Entropy, 24(8):1090, 2022

    work page 2022

  57. [57]

    The causal set approach to quantum gravity

    Sumati Surya. The causal set approach to quantum gravity. Living Reviews in Relativity , 22:1–75, 2019

    work page 2019

  58. [58]

    Introduction to nonextensive statistical mechanics: Approaching a complex world

    Constantino Tsallis. Introduction to nonextensive statistical mechanics: Approaching a complex world. Springer, 1(1):2–1, 2009

    work page 2009

  59. [59]

    Catalytic transformations for bipartite pure st ates

    Sadi Turgut. Catalytic transformations for bipartite pure st ates. Journal of Physics A: Mathematical and Theoretical , 40(40):12185, 2007. 29

    work page 2007

  60. [60]

    Operational resource theory of coherence

    Andreas Winter and Dong Yang. Operational resource theory of coherence. Physical review letters , 116(12):120404, 2016

    work page 2016

  61. [61]

    A note on the attainability of states by equalizing pro- cesses

    Christian Zylka. A note on the attainability of states by equalizing pro- cesses. Theoretica chimica acta , 68(5):363–377, 1985. 30

    work page 1985