Pith. sign in

REVIEW 3 major objections 4 minor 60 references

Classical pair of states as optimal pair for quantum distinguishability quantifiers

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For any bounded quantum distinguishability quantifier that is contractive under completely positive trace-preserving maps and satisfies a saturation-implies-reversibility condition, the state pairs attaining the maximum are exactly the…

desk verdict A genuinely new unifying theorem with a repairable gap in the necessity proof; the abstract overstates the scope. read the letter →

arxiv 2506.02575 v1 pith:67WEBXKR submitted 2025-06-03 quant-ph math-phmath.MP

Bassano Vacchini , Andrea Smirne , Nina Megier This is my paper
classification quant-phmath-phmath.MP
keywords quantumdistinguishabilitycontractivitydataprocessinginequalityoptimalstatepairsorthogonalstatesf-divergencessufficiencyofchannelstracedistance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper identifies a structural reason why many quantum distinguishability measures reach their maximum on classical state pairs. Its central theorem shows that any bounded distinguishability quantifier that is contractive under completely positive trace-preserving maps—the quantum data-processing inequality—and treats a saturated inequality on commuting states as implying reversibility of the channel attains its global maximum exactly on pairs of states with orthogonal supports. Orthogonal supports imply the two states share an eigenbasis, so the pair is described by classical probability distributions. The result unifies known optimal-pair facts for trace distance, quantum skew divergence, and Holevo skew divergence, and it contrasts with non-contractive quantifiers such as the Hilbert-Schmidt distance, where orthogonality is necessary but not sufficient.

What carries the argument

The load-bearing mechanism is the saturation condition on the data-processing inequality: for a contractive quantifier, equality S(Φ[ρ],Φ[σ])=S(ρ,σ) on a commuting pair is assumed to imply that Φ is sufficient, meaning it has a completely positive trace-preserving inverse on that pair, mirroring strict convexity of f in classical f-divergences. The other ingredients are a lemma stating that orthogonality of pure states is sufficient, and with an orthogonal target also necessary, for the existence of a completely positive trace-preserving map sending one pure pair to another, together with purification and assignment-map steps that convert arbitrary mixed-state comparisons into comparisons of pure orthogonal states. These ingredients jointly force any maximally distinguishable pair to have orthogonal supports.

What would settle it

Compute the Holevo skew divergence Kμ with μ≠1/2 on a pair of non-orthogonal states in dimension three and check whether it reaches its maximum value 1; since Kμ satisfies the theorem's hypotheses, any such pair would violate the necessity claim. Alternatively, search among bounded contractive quantifiers with the saturation-implies-reversibility property for a pair with non-orthogonal supports that attains the global maximum, which the theorem says cannot exist.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: if a quantum distinguishability quantifier S satisfies S(Φ[ρ],Φ[σ])≤S(ρ,σ) for every state pair and every completely positive trace-preserving map Φ, and equality on a commuting pair [ρ,σ]=0 forces the existence of a completely positive trace-preserving map that reverses Φ on that pair, then S(ρ1,ρ2)=M, the global maximum, if and only if supp(ρ1)⊥supp(ρ2). The proof purifies arbitrary state pairs, uses the fact that orthogonal pure pairs can be mapped to any target pure pair by a completely positive trace-preserving map, and then invokes the saturation-implies-reversibility property to rule out non-orthogonal maximizers. The paper further shows that every orthogonal pair attains the same maximum value M, so within the covered class of quantifiers, maximal distinguishability and classical, commuting pairs coincide.

Load-bearing premise

The argument assumes that whenever a channel leaves the distinguishability of a commuting pair exactly unchanged, the channel can be undone on that pair by another completely positive trace-preserving map; without this saturation-implies-reversibility property, the theorem's necessity direction need not hold.

Editorial extensions

If this is right

  • Trace distance, quantum skew divergence, and Holevo skew divergence all have exactly the orthogonal state pairs as optimal pairs, and this now follows from one structural argument rather than separate calculations.
  • The theorem supplies a unified criterion for when a bounded contractive distinguishability quantifier can certify perfect distinguishability: the pair must be described by disjoint classical probability distributions.
  • Non-contractive quantifiers, such as the Hilbert-Schmidt distance and the state-maximization distance D∞, lose the theorem's conclusion: orthogonality remains necessary for optimality, but purity is also required, so mixed orthogonal states are not maximizers.
  • Under joint convexity or contractivity under partial trace, contractivity under all completely positive trace-preserving maps is equivalent to invariance under unitary transformations together with assignment maps, tying the optimal-pair theorem to a small set of testable invariances.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The theorem suggests that any attempt to design a bounded distinguishability quantifier whose maximally distinguishable pairs are intentionally non-classical must sacrifice either contractivity or the saturation-implies-reversibility property; the paper's counterexamples show the first route is viable.
  • An implicit consequence is that contractive quantifiers satisfying the theorem's hypotheses are all maximized by the same geometric structure, so their differences for discriminating states appear only below the maximum; comparisons between such measures should therefore focus on sub-maximal pairs.
  • In the memory-effect and non-Markovianity applications that motivate the paper, the result gives a clean diagnostic: a contractive quantifier saturates its bound exactly when the two states under comparison have become perfectly distinguishable in a classical, orthogonal sense.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies quantum distinguishability quantifiers, i.e., non-negative functions on pairs of quantum states that are contractive under completely positive trace-preserving (CPTP) maps. The main result (Theorem 1) claims that if such a quantifier is bounded and satisfies a sufficiency condition—namely, that saturation of the data processing inequality on a commuting pair of states implies the existence of a CPTP map inverting the channel on that pair—then the pairs of states achieving the maximum value are exactly those with orthogonal supports. Such pairs are called classical because they share a common eigenbasis. The paper proves this theorem, discusses examples (quantum relative entropy, skew divergences, Holevo quantity) and counterexamples (Hilbert-Schmidt distance, operator-norm-based distance), and shows that, under additional assumptions such as joint convexity or contractivity under partial trace, contractivity of a quantifier is equivalent to invariance under unitaries and assignment maps.

Significance. If the proof of Theorem 1 is completed as required, the result provides a clean and fairly general link between data-processing contractivity and the classical nature of maximally distinguishable states. The unified perspective on several well-known quantifiers, the explicit counterexamples, and the self-contained proofs of classical f-divergence properties are valuable. The paper also gives a careful discussion of sufficiency and reversibility, connecting to Petz's theorem. However, the necessity part of the main theorem currently contains a logical gap, and the abstract overstates the result by omitting the sufficiency condition. These issues must be fixed before the paper can be accepted.

major comments (3)
  1. [III.A (Theorem 1, after Eq. (54))] The necessity direction of Theorem 1 contains an invalid inference. After establishing that the purified states Pσ1 and Pσ2 are orthogonal, the authors claim that this implies 'in consequence' the orthogonality of the reduced states σ1 and σ2. This implication is false in general: the orthogonal pure states (|00⟩+|11⟩)/√2 and (|00⟩−|11⟩)/√2 both have reduced state I/2 on the first subsystem. This gap is load-bearing because the necessity of orthogonality for optimal pairs is the central claim of the theorem. The gap is repairable using the paper's own sufficiency condition: since S(σ1,σ2)=S(Pσ1,Pσ2)=M and now [Pσ1,Pσ2]=0, applying the sufficiency condition to the partial trace map Tr_A yields a CPTP map Λ with Λ[σi]=Pσi; trace-distance contractivity then gives σ1⊥σ2. This repair should be incorporated into the proof.
  2. [Abstract and Introduction] The abstract states that contractivity under CPTP maps alone 'warrants that the pairs on which these quantifiers attain their maximal value are pairs of orthogonal states'. This is stronger than the theorem, which requires an additional sufficiency condition (stated in Theorem 1 and in the introduction). The trace distance, for instance, does not satisfy this sufficiency condition, yet the paper lists it among the quantifiers covered in Section III.B. The abstract and conclusion should be reworded to include the sufficiency assumption, or the theorem should be presented as applying only to a class of quantifiers with that reversibility feature.
  3. [Theorem 1 statement and proof] The theorem does not explicitly state that S is bounded, yet the proof uses a maximum value M with 'M ∈ {1, +∞}' according to a proper choice of normalization. If S is unbounded, no finite maximum exists; if S is bounded, M is finite and the notation {1, +∞} is confusing. The statement should be clarified to say 'bounded quantum distinguishability quantifier' and to define M as the finite maximum, or the unbounded case should be treated separately.
minor comments (4)
  1. [Lemma 1 (Section III.A)] The statement 'The condition becomes necessary if the target pair is also orthogonal' is ambiguous; it should be rewritten as 'if the target pair is orthogonal, then the source pair must also be orthogonal' to avoid misreading.
  2. [Section III.B] The claim that Theorem 1 provides a 'unified proof of general validity' for the trace distance is misleading, since the trace distance does not satisfy the sufficiency condition of Theorem 1. The direct proof for trace distance is fine, but the connection to Theorem 1 should be softened or clarified.
  3. [Section II.E (Eq. (26))] The notation for the assignment map Aτ[ρ]=ρ⊗τ is used later in Section V without explicit reintroduction; consider adding a cross-reference.
  4. [Various] There are several minor typographical and formatting issues: in Eq. (86) the subscript 'TrS⊗E' is awkward, and the DOI in reference [65] appears to contain a typo ('-16' vs. an appropriate chapter DOI).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is a conditional derivation from explicitly stated axioms and external lemmas, with no fitted inputs or load-bearing self-citations.

full rationale

The derivation chain is self-contained as a mathematical argument. Theorem 1 is explicitly conditional: it assumes contractivity and a reversibility property (equality of S on a commuting pair implies a CPTP inverse), and the proof uses that assumption exactly where stated, after Eq. (53), to obtain Eq. (54); it does not derive the assumption from the conclusion. The sufficiency direction relies on external results (Chefles' Lemma 1, purification maps of Kleinmann et al.) together with contractivity, and the necessity direction uses the theorem's own hypothesis plus Lemma 1. There are no fitted parameters, no quantity is defined in terms of the target result, and no load-bearing claim is justified solely by the authors' prior work: the self-citations (e.g., refs. [23], [42]) appear only as background on skewed divergences and non-Markovianity and do not carry the proof. The one substantive concern in the manuscript is a correctness gap, not circularity: immediately after Eq. (54), the inference that orthogonality of the purifications P_sigma1, P_sigma2 implies orthogonality of the reduced states sigma1, sigma2 is false in general (orthogonal Bell-state purifications can have identical marginal I/2). This affects the validity of the written necessity proof, but it is an internal proof gap or a repairable step, not a reduction of the theorem to its own inputs. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; the results are mathematical theorems. The proof depends on the stated sufficiency property (Theorem 1 hypothesis) and on standard results: Chefles' lemma on pure-state transformations, Petz's sufficiency theorem for relative entropy, and Stinespring/Haar-averaging tools in Section V. No new entities are postulated.

assumptions (5)
  • domain assumption Boundedness of S and existence of maximum M (Section III).
    The theorem is stated for bounded quantifiers; the proof uses finiteness of M and attainment of the maximum, which is not guaranteed by contractivity alone.
  • domain assumption Sufficiency property: equality on commuting pairs implies CPTP invertibility on the pair.
    This is the key hypothesis of Theorem 1, used in the necessity direction to recover the orthogonal pure states after saturation of the data processing inequality. It fails for many contractive quantifiers (e.g., trace distance), so the theorem is conditional on it.
  • standard math Lemma 1 (Chefles 2000): orthogonal pure pairs can be mapped to any target pure pair; a map sending a pure pair to an orthogonal pure pair forces the source pair to be orthogonal.
    Used repeatedly in the proof of Theorem 1 to move between purifications and to conclude orthogonality of the source pure states from the existence of an inverse map.
  • standard math Petz theorem: saturation of the data processing inequality for Umegaki relative entropy implies sufficiency of the map.
    Used to argue that skew divergences and Holevo skew divergence satisfy the sufficiency assumption of Theorem 1.
  • standard math Stinespring representation Phi = Tr_E o U o A_tau and Haar averaging over unitaries (Section V).
    Used in Section V to prove equivalence between contractivity and invariance properties under joint convexity or partial-trace contractivity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Classical pair of states as optimal pair for quantum distinguishability quantifiers." pith.science (2026). https://pith.science/paper/67WEBXKR

@misc{pith2026250602575,
  author       = {Pith},
  title        = {Pith review of: Classical pair of states as optimal pair for quantum distinguishability quantifiers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/67WEBXKR}},
  note         = {Machine review of arXiv:2506.02575}
}
read the original abstract

The capability to quantitatively distinguish quantum states is of great importance for a variety of tasks, and has recently played an important role in the study of quantum reduced dynamics and their characterization in terms of memory effects. A crucial property of quantum distinguishability quantifiers considered in the latter framework is the contractivity under the action of completely positive trace-preserving maps. We show that this requirement warrants that the pairs on which these quantifiers attain their maximal value are pairs of orthogonal, and in this sense classical, states.

Figures

Figures reproduced from arXiv: 2506.02575 by the authors.

Figure 1
Figure 1. We graphically summarize the relationship between [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

60 extracted references · 32 canonical work pages

  1. [1]

    Bengtsson and K

    I. Bengtsson and K. Życzkowski, Geometry of Quan- tum States: An Introduction to Quantum Entanglement, 2nd edn. (Cambridge University Press, 2017)

    work page 2017

  2. [2]

    Quantum Reservoir Computing (QuReCo)

    Note that an example of states that are orthogonal but not pure can only be obtained if the dimension of the Hilbert space is higher than two. The same condition is necessary to show non- contractivity of the Hilbert-Schmidt distance. This is simply due to the fact that in C 2 the trace distance Eq. (36) and the Hilbert-Schmidt distance Eq. (60) coin- cid...

    work page 2022

  3. [3]

    Chruściński, Phys

    D. Chruściński, Phys. Rep. 992, 1 (2022). DOI: 10.1016/j.physrep.2022.09.003

    doi:10.1016/j.physrep.2022.09.003 2022

  4. [4]

    Vacchini,Open Quantum Systems: Foundations and Theory, Graduate Texts in Physics (Springer Nature Switzerland, Cham, 2024)

    B. Vacchini,Open Quantum Systems: Foundations and Theory, Graduate Texts in Physics (Springer Nature Switzerland, Cham, 2024). DOI: 10.1007/978-3-031- 58218-9

    doi:10.1007/978-3-031- 2024

  5. [5]

    Wißmann, A

    S. Wißmann, A. Karlsson, E.-M. Laine, J. Piilo, and H.-P. Breuer, Phys. Rev. A 86, 062108 (2012). DOI: 10.1103/PhysRevA.86.062108

    work page doi:10.1103/physreva.86.062108 2012

  6. [6]

    Breuer, J

    H.-P. Breuer, J. Phys. B 45, 154001 (2012). DOI: 10.1088/0953-4075/45/15/154001

    doi:10.1088/0953-4075/45/15/154001 2012

  7. [7]

    T. M. Osán, D. G. Bussandri, and P. W. Lamberti, Phys- ica A: Statistical Mechanics and its Applications 594, 127001 (2022). DOI: 10.1016/j.physa.2022.127001

    work page arXiv 2022

  8. [8]

    D. G. Bussandri and T. M. Osán, Entropy25 (2023). DOI: 10.3390/e25060912

    doi:10.3390/e25060912 2023

Show all 60 references
  1. [9]

    Lanier, J

    D. Lanier, J. Béguinot, and O. Rioul, From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum $f$-Divergence(2025), 2501.14340

    2025 arXiv

  2. [10]

    De Palma, M

    G. De Palma, M. Marvian, D. Trevisan, and S. Lloyd, IEEE Transactions on Information Theory pp. 1–1 (2021). DOI: 10.1109/TIT.2021.3076442

    2021

  3. [11]

    F. Hiai, M. Mosonyi, D. Petz, and C. Bény, Re- views in Mathematical Physics 23, 691 (2011). DOI: 10.1142/S0129055X11004412 , https://doi.org/10. 1142/S0129055X11004412

    2011 doi

  4. [12]

    Crooks,Fisher Information and Statistical Me- chanics

    Gavin E. Crooks,Fisher Information and Statistical Me- chanics

  5. [13]

    Bhattacharyya, A., in Bulletin of the Calcutta Mathe- matical Society, Vol-35(1943), pp. 99–109

    1943

  6. [14]

    Bhattacharyya, Sankhya: The Indian Journal of Statistics (1933-1960)7, 401 (1946),25047882

    A. Bhattacharyya, Sankhya: The Indian Journal of Statistics (1933-1960)7, 401 (1946),25047882

    1946

  7. [15]

    Jozsa, Journal of Modern Optics 41, 2315 (1994)

    R. Jozsa, Journal of Modern Optics 41, 2315 (1994). DOI: 10.1080/09500349414552171

    1994 doi

  8. [16]

    Uhlmann, Reports on Mathematical Physics9, 273 (1976)

    A. Uhlmann, Reports on Mathematical Physics9, 273 (1976). DOI: 10.1016/0034-4877(76)90060-4

    1976 doi

  9. [17]

    Csiszár, Publ

    I. Csiszár, Publ. Math. Inst. Hun- gar. Acad. Sci.8, 85 13 (1963)

    1963

  10. [18]

    Csiszár, Studia scientiarum Mathematicarum Hungar- ica 2, 229 (1967)

    I. Csiszár, Studia scientiarum Mathematicarum Hungar- ica 2, 229 (1967)

    1967

  11. [19]

    S. M. Ali and S. D. Silvey, Journal of the Royal Statistical Society: Series B (Methodological)28, 131 (1966). DOI: 10.1111/j.2517-6161.1966.tb00626.x

    1966

  12. [20]

    Nielsen and I

    M. Nielsen and I. Chuang, Quantum Computation and Quantum Information(Cambridge University Press, Cambridge, 2000)

    2000

  13. [21]

    Hiai and M

    F. Hiai and M. Mosonyi, Reviews in Math- ematical Physics 29, 1750023 (2017). DOI: 10.1142/S0129055X17500234

    2017 doi

  14. [22]

    K. M. R. Audenaert, Telescopic Relative Entropy, in Theory of Quantum Computation, Communication, and Cryptography, edited by Bacon, Dave and Martin- Delgado, Miguel and Roetteler, Martin (Springer Berlin Heidelberg, Berlin, Heidelberg, 2014), pp. 39–52. DOI: 10.1007/978-3-642...

    2014 doi

  15. [23]

    K. M. R. Audenaert, J. Math. Phys.55, 112202 (2014). DOI: 10.1063/1.4901039

    2014 doi

  16. [25]

    A. N. Kolmogorov, Sankhya: The Indian Journal of Statistics, Series A (1961-2002) 25, 159 (1963). DOI: http://www.jstor.org/stable/25049284 , 25049261

    1963

  17. [26]

    Csiszar, The Annals of Probability3, 146 (1975)

    I. Csiszar, The Annals of Probability3, 146 (1975). DOI: 10.1214/aop/1176996454

    1975

  18. [27]

    D. M. Endres and J. E. Schindelin, IEEE Transac- tions on Information Theory 49, 1858 (2003). DOI: 10.1109/TIT.2003.813506

    2003

  19. [28]

    Jenčová, Reviews in Mathematical Physics 24, 1250016 (2012)

    A. Jenčová, Reviews in Mathematical Physics 24, 1250016 (2012). DOI: 10.1142/S0129055X1250016X , https://doi.org/10.1142/S0129055X1250016X

    2012 doi

  20. [29]

    A. S. Holevo,Statistical Structure of Quantum Theory, Vol. m 67 ofLect. Not. Phys.(Springer, Berlin, 2001). DOI: 10.1007/3-540-44998-1

    2001 doi

  21. [30]

    Petz, Commun.Math

    D. Petz, Commun.Math. Phys.105, 123 (1986). DOI: 10.1007/BF01212345

    1986 doi

  22. [31]

    M. E. Shirokov, Sb. Math. 204, 1215 (2013). DOI: 10.1070/SM2013v204n08ABEH004337

    2013 doi

  23. [32]

    Jenčová and D

    A. Jenčová and D. Petz, Commun. Math. Phys.263, 259 (2006). DOI: 10.1007/s00220-005-1510-7

    2006 doi

  24. [33]

    Buscemi, Commun

    F. Buscemi, Commun. Math. Phys. 310, 625 (2012). DOI: 10.1007/s00220-012-1421-3

    2012 doi

  25. [35]

    C. K. (ed.) Zachos, D. B. (ed.) Fairlie, and T. L. (ed.) Curtright,Quantum Mechanics in Phase Space. An Overview with Selected Papers.(World Scientific, 2005)

    2005

  26. [36]

    Lindblad, Comm

    G. Lindblad, Comm. Math. Phys.48, 119 (1976). DOI: 10.1007/BF01608499

    1976 doi

  27. [37]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys.17, 821 (1976). DOI: 10.1063/1.522979

    1976 doi

  28. [38]

    Vacchini, Phys

    B. Vacchini, Phys. Rev. Lett.95, 230402 (2005). DOI: 10.1103/PhysRevLett.95.230402

    2005 doi

  29. [39]

    Breuer and B

    H.-P. Breuer and B. Vacchini, Phys. Rev. Lett. 101, 140402 (2008). DOI: 10.1103/PhysRevLett.101.140402

    2008 doi

  30. [40]

    Vacchini, Phys

    B. Vacchini, Phys. Rev. Lett.117, 230401 (2016). DOI: 10.1103/PhysRevLett.117.230401

    2016 doi

  31. [41]

    Vacchini, Sci

    B. Vacchini, Sci. Rep. 10, 5592 (2020). DOI: 10.1038/s41598-020-62260-z

    2020 doi

  32. [44]

    Reed and B

    M. Reed and B. Simon,I: Functional Analysis(Academic Press, San Diego, 1980)

    1980

  33. [46]

    P. W. Lamberti, A. P. Majtey, A. Borras, M. Casas, and A. Plastino, Phys. Rev. A77, 052311 (2008). DOI: 10.1103/PhysRevA.77.052311

    2008 doi

  34. [47]

    Sra, Linear Algebra Appl

    S. Sra, Linear Algebra Appl. 616, 125 (2021). DOI: 10.1016/j.laa.2020.12.023

    2021 doi

  35. [48]

    Virosztek, Adv

    D. Virosztek, Adv. Math. 380, 107595 (2021). DOI: 10.1016/j.aim.2021.107595

    2021

  36. [49]

    Chefles, Phys

    A. Chefles, Phys. Lett. A 270, 14 (2000). DOI: 10.1016/S0375-9601(00)00291-7

    2000 doi

  37. [51]

    Kleinmann, H

    M. Kleinmann, H. Kampermann, T. Meyer, and D. Bruß, Applied Physics B86, 371 (2007). DOI: 10.1007/s00340- 006-2515-4

    2007 doi

  38. [52]

    Ozawa, Phys

    M. Ozawa, Phys. Lett. A 268, 158 (2000). DOI: 10.1016/S0375-9601(00)00171-7

    2000 doi

  39. [53]

    R. V. Kadison, Ann. Math. 56, 494 (1952). DOI: 10.2307/1969657 , 1969657

    1952 doi

  40. [54]

    Bhatia,Positive Definite Matrices(Princeton Univer- sity Press, 2007),j.ctt7rxv2

    R. Bhatia,Positive Definite Matrices(Princeton Univer- sity Press, 2007),j.ctt7rxv2

    2007

  41. [55]

    Pérez-García, M

    D. Pérez-García, M. M. Wolf, D. Petz, and M. B. Ruskai, J. Math. Phys. 47, 083506 (2006). DOI: 10.1063/1.2218675

    2006 doi

  42. [56]

    D. A. Lidar, P. Zanardi, and K. Khodjasteh, Phys. Rev. A78, 012308 (2008). DOI: 10.1103/PhysRevA.78.012308

    2008 doi

  43. [57]

    A. E. Rastegin, J. Stat. Phys.148, 1040 (2012). DOI: 10.1007/s10955-012-0569-8

    2012 doi

  44. [58]

    A. A. Budini, R. L. d. M. Filho, and M. F. Santos, Phys. Rev. A110, 022228 (2024). DOI: 10.1103/Phys- RevA.110.022228

    2024 doi

  45. [59]

    Gu and I

    B. Gu and I. Franco, The Journal of Chemical Physics 151, 014109 (2019). DOI: 10.1063/1.5099499

    2019 doi

  46. [60]

    Chen, P.-Y

    H.-B. Chen, P.-Y. Lo, C. Gneiting, J. Bae, Y.-N. Chen, and F. Nori, Nat. Commun. 10, 3794 (2019). DOI: 10.1038/s41467-019-11502-4

    2019 doi

  47. [61]

    A. A. Budini, Phys. Rev. A108, 042203 (2023). DOI: 10.1103/PhysRevA.108.042203

    2023 doi

  48. [62]

    M. B. Ruskai, Rev. Math. Phys.6, 1147 (1994). DOI: 10.1142/S0129055X94000407

    1994 doi

  49. [63]

    Müller-Hermes and D

    A. Müller-Hermes and D. Reeb, Ann. Henri Poincaré18, 1777 (2017). DOI: 10.1007/s00023-017-0550-9

    2017 doi

  50. [64]

    40, 147(1975)

    G.Lindblad, Commun.Math.Phys. 40, 147(1975). DOI: 10.1007/BF01609396

    1975 doi

  51. [65]

    Collins and P

    B. Collins and P. Śniady, Commun. Math. Phys.264, 773 (2006). DOI: 10.1007/s00220-006-1554-3

    2006 doi

  52. [66]

    Spehner, F

    D. Spehner, F. Illuminati, M. Orszag, and W. Roga, in Lectures on General Quantum Correlations and Their Applications, edited by Fanchini, Felipe Fernandes and Soares Pinto, Diogo de Oliveira and Adesso, Gerardo (Springer International Publishing, Cham, 2017), pp. 105–157, ISB...

    2017 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.