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REVIEW 2 major objections 4 minor 59 references

Classical capacity of the generalized Pauli channels

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For a generalized Pauli channel, the classical capacity is bracketed by two explicit entropy formulas, and when the bounds coincide the capacity equals their common value; qubit Pauli channels always fall in this case.

desk verdict Useful lower-bound results and a clean specialization, but the capacity equalities rest on an unproven factorization in the imported upper bound, and the P-divisibility remark has a sign error. read the letter →

arxiv 1908.03917 v1 pith:PTD3ISNK submitted 2019-08-11 quant-ph math-phmath.MP

Katarzyna Siudzińska This is my paper · ORCID
classification quant-phmath-phmath.MP MSC 81P4594A40
keywords generalizedPaulichannelsclassicalcapacityHolevomutuallyunbiasedbasesWeyl-covariantweakadditivityquantumchannel
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

This paper calculates usable bounds on the Holevo capacity of every generalized Pauli channel, and shows when those bounds determine the channel's classical capacity exactly. The lower bound is obtained by feeding the channel basis states from mutually unbiased bases; the paper proves this lower bound is weakly additive, so it cannot be improved by using the channel many times. The upper bound is adapted from a known result for Weyl-covariant channels. Whenever the two bounds coincide, the classical capacity equals their common value, and for qubit Pauli channels this always happens, giving a single closed formula in terms of the largest absolute eigenvalue.

What carries the argument

The workhorse is the generalized Pauli channel itself, a random unitary channel built from $d+1$ mutually unbiased bases and specified by $d+1$ eigenvalues $\lambda_\alpha$. The lower bound in Eq. (23) is the capacity of the classical symmetric channel $T^{(\alpha)}$ induced on each basis, and its weak additivity follows because the tensor product of two generalized Pauli channels tensors these classical maps, adding their row entropies. The upper bound in Eq. (32) is the entropy of the vector $\zeta(p)$ of the $d$ largest sums of Kraus probabilities; the proof that this entropy dominates the true output entropy relies on a majorization relation and on a pure optimizer $\rho_*$ inherited from the Weyl-channel setting.

What would settle it

Numerically maximize the Holevo capacity over input ensembles for a $d=4$ generalized Pauli channel with mixed-sign eigenvalues, such as $\lambda=(1/2,-1/4,-1/4,-1/4,-1/4)$; if the maximum exceeds $\ln d - H[\zeta(p)]$ from Eq. (32), or if any channel has a lower bound from Eq. (23) larger than its upper bound, the central theorem pair fails.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that for a generalized Pauli channel with eigenvalues $\lambda_1,\dots,\lambda_{d+1}$, the Holevo capacity $\chi$ satisfies the lower estimate of Eq. (23) and the upper estimate of Eq. (32). Because the lower bound is weakly additive, any channel for which the two estimates agree has classical capacity $C(\Lambda_{\rm GP}) = \chi(\Lambda_{\rm GP})$ equal to that common value. The paper further claims that for $d=2$ the estimates always agree, so every Pauli channel has $C(\Lambda_P) = \frac{1+\lambda_*}{2}\ln(1+\lambda_*) + \frac{1-\lambda_*}{2}\ln(1-\lambda_*)$ with $\lambda_* = \max\{|\lambda_{\min}|,\lambda_{\max}\}$.

Load-bearing premise

The result stands on an imported upper-bound lemma that assumes, for every generalized Pauli channel, an optimal input $\rho_*$ exists and that its channel output can be written as a mixture of unitary rotations of one fixed state; the paper states explicitly that it does not check whether $\rho_*$ exists.

Editorial extensions

If this is right

  • Every Pauli channel has an exact classical capacity, determined by $\lambda_*=\max\{|\lambda_{\min}|,\lambda_{\max}\}$ rather than by the full noise structure.
  • Whenever the lower and upper Holevo bounds meet, the asymptotic classical capacity equals the single-use Holevo capacity, so no regularization over many channel uses is needed.
  • Highly symmetric generalized Pauli channels with all eigenvalues of one sign and at most two distinct values have closed-form capacities, Eqs. (41)-(42), with the depolarizing channel as the all-equal case.
  • For P-divisible Pauli dynamics, the classical capacity is non-increasing in time; the converse fails, so capacity monotonicity is only a one-way witness of divisibility.

Reading between the lines

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

  • If the upper bound in Eq. (32) were also weakly additive, exact capacities would follow for every channel that saturates it, not only for channels where the two bounds meet; the paper leaves this route open.
  • The paper's explicit caveat that an optimal $\rho_*$ is not shown to exist suggests a numerical search over $d=4$ and $d=8$ generalized Pauli channels with mixed-sign eigenvalues; a channel whose true capacity lies strictly below Eq. (32) would confine the exact-capacity formulas to the symmetric cases.
  • Because the Pauli-channel formula depends only on the extreme channel fidelities, a natural testable conjecture is that generalized Pauli channel capacity is controlled by the extreme fidelities on mutually unbiased basis projectors.
  • Monitoring the classical capacity of a time-dependent generalized Pauli channel could give an information-theoretic probe of divisibility: a transient increase in $C$ would signal a break of P-divisibility, while a decrease alone would not certify divisibility.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies generalized Pauli channels on prime-power-dimensional Hilbert spaces and derives lower and upper bounds on their Holevo capacity. The lower bound is obtained from projectors onto mutually unbiased bases, and the upper bound is imported from known results for Weyl channels via a lemma. The paper proves that the lower bound is weakly additive, so that whenever the lower and upper bounds coincide, the classical capacity equals the common value. Explicit capacity formulas are given for depolarizing-type limits and for two-qubit Pauli channels, and a claim is made about monotonicity of Pauli channel capacity under P-divisible dynamics.

Significance. If the main results are correct, the paper provides explicit classical capacities for a natural family of covariant quantum channels, complementing earlier additivity results. The proof of weak additivity of the lower bound (Proposition 1) is self-contained and appears sound, and the specialization to Pauli channels is compact and potentially useful. The main caveat is that the upper bound and the consequent equality results depend on Lemma 1, whose proof is only recapitulated from Refs. [34,35] and contains an unjustified factorization step; in addition, the paper contains a false statement about P-divisibility and monotonicity of Pauli channel capacity. The analytical style is clear, but the central claim currently rests on an unverified imported lemma and a demonstrably incorrect side claim.

major comments (2)
  1. [Appendix, Proof to Lemma 1; used in Theorem 2] The proof of Lemma 1, specifically Eqs. (60)-(61), states that an admissible choice is s_j q_k = p_jk. This forces the Kraus coefficient matrix p_jk of an arbitrary (multipartite) Weyl channel to factor as the outer product of two probability vectors, which is not true for generic p_jk. No argument is given that linear dependencies among the orbit matrices W_jk ρ* W_jk† make the factorization unnecessary. Since the main text explicitly says 'we do not check whether ρ* exists' (after Theorem 2), the existence of a pure optimal state realizing both the majorization condition and the factorization is not established. The upper bound in Theorem 2, Eq. (32), and consequently the coinciding-bound capacity formulas in Eqs. (41), (42), and (44), all depend on Lemma 1; this gap is therefore load-bearing and must be closed.
  2. [Special case: Pauli channels, Eq. (52); Conclusions] The claim that P-divisibility of a Pauli dynamical map implies monotone decrease of its classical capacity is false. Equation (52) differentiates only λ_max(t), but the capacity formula in Eq. (44) depends on λ* = max{|λ_min|, λ_max}. If the most negative eigenvalue becomes more negative, |λ_min| increases even though all eigenvalues are non-increasing. A concrete counterexample is λ1 = 0.3, λ2 = 0.2, and λ3(t) = -0.2 - 0.3t for t ∈ [0,1]; all eigenvalues are non-increasing, so the map is P-divisible by the criterion cited in the paper, and the Fujiwara-Algoet conditions remain satisfied throughout. Yet λ* increases from 0.3 to 0.5, so C(Λ_P(t)) increases. The corresponding sentence in the Conclusions must be corrected or removed.
minor comments (4)
  1. [Classical capacity, paragraph after Eq. (42)] The text says 'eqs. (42) and (42) recover the classical capacity of the depolarizing channel'; this should read 'eqs. (41) and (42)'.
  2. [Special case: Pauli channels, Eqs. (43)-(44)] The symbols λ_max and λ_min are used before being defined; please define them explicitly, for example as λ_max = max_α λ_α and λ_min = min_α λ_α.
  3. [Introduction, paragraph after Eq. (3)] There is a typo: 'quantim channel' should be 'quantum channel'.
  4. [Appendix, Proof to Theorem 2] The set J defined by Eq. (64) as {p0, p1/(d-1), ..., p_{d+1}/(d-1)} has d+1 distinct entries only if multiplicities are ignored; since the text states |J| = d^2, it should be described as a multiset with p_α/(d-1) appearing d-1 times.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; lower bound is derived from first principles, upper bound is imported from external references, and self-citations are not load-bearing.

full rationale

The derivation chain is not circular. Theorem 1's lower bound (Eq. 23) is obtained by evaluating the Holevo expression on projectors onto mutually unbiased bases, using only the covariance relation (Eq. 21) and the explicit channel action (Eq. 54). Theorem 2's upper bound (Eq. 32) is inherited from Lemma 1, which is quoted from Refs. [34,35] (not authored by the present author); the appendix merely recaps that proof. Proposition 1 proves weak additivity of the lower bound from the product structure of the associated classical maps (Eqs. 69-71). The Pauli-channel capacity formula (Eq. 44) follows by identifying when the lower and upper bounds coincide for d=2, not by fitting any parameter. The self-citations in the paper (Refs. [24,25,31,32,33,49]) are used for definitions, covariance constructions, and the time-local evolution setup; they do not carry the central capacity argument. The main rigor concern flagged in the manuscript — 'we do not check whether rho* exists' and the appendix's 'admissible choice s_j q_k = p_jk' in Eq. (61) — is an unproved existence/factorization assumption in the imported upper-bound proof. That is a correctness or completeness gap, not a case where a prediction is equivalent to its input by construction. No fitted parameter is renamed as a prediction, and no load-bearing conclusion depends on the author's own prior work. Therefore the circularity score is 0.

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

The central claims rest on standard quantum information theorems and one imported lemma from Refs. [34,35]; no fitted parameters or invented entities are introduced.

assumptions (5)
  • standard math Holevo-Schumacher-Westmoreland theorem: C(Λ) = lim (1/n) χ(Λ⊗n) and C ≥ χ.
    Used in the Classical Capacity section to relate classical capacity to Holevo capacity.
  • domain assumption For unitarily covariant channels, χ(Λ) = ln d - min_ρ S(Λ[ρ]) (Ref. [11]).
    Invoked in the proof of Theorem 1 to connect Holevo capacity to minimal output entropy.
  • domain assumption Lemma 1 (upper bound for multipartite Weyl channels) from Refs. [34,35].
    Imported result used as the basis for Theorem 2; the paper recaps the proof but does not derive it independently.
  • domain assumption King's additivity of minimal output entropy for unital qubit channels [13].
    Implied when the paper states that for Pauli channels the lower bound is weakly additive and thus C = χ_low.
  • domain assumption Existence and properties of maximal sets of mutually unbiased bases in prime power dimensions.
    The generalized Pauli channel construction relies on d+1 MUBs, which only exist for powers of a prime.

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Cite this review

Pith. "Pith review of Classical capacity of the generalized Pauli channels." pith.science (2026). https://pith.science/paper/PTD3ISNK

@misc{pith2026190803917,
  author       = {Pith},
  title        = {Pith review of: Classical capacity of the generalized Pauli channels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTD3ISNK}},
  note         = {Machine review of arXiv:1908.03917}
}
read the original abstract

We calculate and analyze the bounds of the Holevo capacity and classical capacity for the generalized Pauli channels. In particular, we obtain the lower and upper bounds of the Holevo capacity and show that if these bounds coincide, the Holevo capacity is weakly additive. We find the classical capacity for the Pauli channels and two-parameter generalized Pauli channels.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

59 extracted references · 55 canonical work pages

  1. [1]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information , Cambridge University Press, Cambridge 2010

    work page 2010

  2. [2]

    (34) in Theorem 2 allows us to calculate χ up(Λ P ) = 3 4 ln 3 − ln 2

    (37) Eq. (34) in Theorem 2 allows us to calculate χ up(Λ P ) = 3 4 ln 3 − ln 2. (38) Now, let us construct the bipartite channel Λ P ⊗Λ P . The corresponding vector ζ(p) of increasingly ordered pαpβ reads ζ(p) = 1 16 (4, 2, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0). (39) Using Lemma 1, we see that χ up(Λ P ⊗ Λ P ) = 15 16 ln 5 − 11 8 ln 2 ⁄= 2χ up(Λ P ). ...

    work page 2018

  3. [3]

    Lloyd, Phys

    S. Lloyd, Phys. Rev. A 55, 1613–1622 (1997)

    work page 1997

  4. [4]

    Shor, MSRI Workshop on Quantum Computation Lec- ture Notes (2002)

    P. Shor, MSRI Workshop on Quantum Computation Lec- ture Notes (2002)

    work page 2002

  5. [5]

    Devetak, IEEE Trans

    I. Devetak, IEEE Trans. Inf. Theor. 51, 44–55 (2005)

    work page 2005

  6. [6]

    A. S. Holevo, IEEE Trans. Info. Theor. 44, 269–273 (1998)

    1998

  7. [7]

    Schumacher and M

    B. Schumacher and M. D. Westmoreland, Phys. Rev. A 56, 131–138 (1997)

    work page 1997

  8. [8]

    Devetak, IEEE Trans

    I. Devetak, IEEE Trans. Inf. Theor. 51, 44–55 (2005). 6

    work page 2005

Show all 59 references
  1. [9]

    Bennett, P

    C. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thap- liyal, IEEE Trans. Inf. Theor. 48, 2637–2655 (2002)

    2002

  2. [10]

    Gyongyosi, S

    L. Gyongyosi, S. Imre, and H. V. Nguyen, IEEE Com- mun. Surv. Tut. 20, 1149–1205 (2018)

    2018

  3. [11]

    Smith, IEEE Information Theory Workshop (2010)

    G. Smith, IEEE Information Theory Workshop (2010)

    2010

  4. [12]

    A. S. Holevo, Remarks on the classical capacity of quan- tum channel (2000), arXiv:quant-ph/0212025

    2000 arXiv

  5. [13]

    A. S. Holevo, Int. J. Quant. Inform. 3(1), 41–48 (2005)

    2005

  6. [14]

    King and M

    C. King and M. B. Ruskai, IEEE Trans. Info. Theory 47, 192–209 (2001)

    2001

  7. [15]

    King, IEEE Trans

    C. King, IEEE Trans. Info. Theor. 49, 221–229 (2003)

    2003

  8. [16]

    A. S. Holevo, Rep. Math. Phys. 32(2), 211–216 (1993)

    1993

  9. [17]

    A. S. Holevo, J. Math. Phys. 37, 1812 (1996)

    1996

  10. [18]

    M. A. Nuwairan, SU (2)-Irreducibly covariant and EPOSIC channels (2013), arXiv:1306.5321

    2013 arXiv

  11. [19]

    Jen˘ cová and M

    A. Jen˘ cová and M. Plávala, J. Math. Phys. 57(12), 122203 (2016)

    2016

  12. [20]

    Mozrzymas, M

    M. Mozrzymas, M. Studziński, and N. Datta, J. Math. Phys. 58, 052204 (2017)

    2017

  13. [21]

    G. G. Amosov, J. Math. Phys. 48, 2104–2117 (2007)

    2007

  14. [22]

    Datta, M

    N. Datta, M. Fukuda, and A. S. Holevo, Quant. Inf. Proc. 5(2), 179–207 (2006)

    2006

  15. [23]

    C. King, K. Matsumoto, M. Nathanson, and M. B. Ruskai, Markov Process Relat. 13, 391–423 (2007)

    2007

  16. [24]

    Fukuda and G

    M. Fukuda and G. Gour, IEEE Transactions on Informa- tion Theory 63(3), 1818–1828 (2017)

    2017

  17. [25]

    Siudzińska and D

    K. Siudzińska and D. Chruściński, Quantum channels irreducibly covariant with respect to the finite group generated by the Weyl operators (2018), arXiv:math- ph/1711.10823

    2018 arXiv

  18. [26]

    Siudzińska, Evolution of open quantum systems gov- erned by unitarily covariant quantum channels , Ph.D

    K. Siudzińska, Evolution of open quantum systems gov- erned by unitarily covariant quantum channels , Ph.D. thesis, Nicolaus Copernicus University (2019)

    2019

  19. [27]

    Nathanson and M

    M. Nathanson and M. B. Ruskai, J. Phys. A: Math. Theor. 40, 8171 (2007)

    2007

  20. [28]

    W. K. Wootters and B. D. Fields, Ann. Phys. 191, 363 (1989)

    1989

  21. [29]

    Ruppert and A

    L. Ruppert and A. Magyar, Martingale approach in quantum state estimation using indirect measurements , in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems , Izola, Slovenia 2010

    2010

  22. [30]

    Ruppert, D

    L. Ruppert, D. Virosztek, and K. Hangos, J. Phys. A: Math. Theor. 45, 265305 (2012)

    2012

  23. [31]

    Siudzińska, Rep

    K. Siudzińska, Rep. Math. Phys. 80, 361–372 (2017)

    2017

  24. [32]

    Chruściński and K

    D. Chruściński and K. Siudzińska, Phys. Rev. A 94, 022118 (2016)

    2016

  25. [33]

    Siudzińska and D

    K. Siudzińska and D. Chruściński, Phys. Rev. A 96, 022129 (2017)

    2017

  26. [34]

    Siudzińska and D

    K. Siudzińska and D. Chruściński, Phys. Rev. A 100, 012303 (2019)

    2019

  27. [35]

    ur Rehman, Y

    J. ur Rehman, Y. Jeong, J. S. Kim, and H. Shin, Sci. Rep. 8, 17457 (2018)

    2018

  28. [36]

    ur Rehman, Y

    J. ur Rehman, Y. Jeong, and H. Shin, Phys. Rev. A 99, 042312 (2019)

    2019

  29. [37]

    L. J. Landau and R. F. Streater, Linear Algebra Appl. 193, 107–127 (1993)

    1993

  30. [38]

    K. M. R. Audenaert and S. Scheel, New J. Phys. 10, 023011 (2008)

    2008

  31. [39]

    Watrous, The Theory of Quantum Information , Cam- bridge University Press, Cambridge 2018

    J. Watrous, The Theory of Quantum Information , Cam- bridge University Press, Cambridge 2018

    2018

  32. [40]

    Alicki and K

    R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications , Springer, Berlin 1987

    1987

  33. [41]

    Gregoratti and R

    M. Gregoratti and R. F. Werner, J. Mod. Opt. 50, 915 (2003)

    2003

  34. [42]

    Fujiwara and P

    A. Fujiwara and P. Algoet, Phys. Rev. A 59, 3290 (1999)

    1999

  35. [43]

    M. B. Ruskai, S. Szarek, and E. Werner, Linear Algebra Appl. 347(1-3), 159–187 (2002)

    2002

  36. [44]

    Bandyopadhyay, P

    S. Bandyopadhyay, P. Boykin, V. Roychowdhury, and F. Vatan, Algorithmica 34, 512 (2002)

    2002

  37. [45]

    Grassl, in: Proc

    M. Grassl, in: Proc. ERATO Conf. Quantum Informa- tion Science, Tokyo 60–61 (2004)

    2004

  38. [46]

    Klappenecker and M

    A. Klappenecker and M. Rötteler, Constructions of Mu- tually Unbiased Bases , in Proceedings of the 7th Inter- national Conference on Finite Fields , vol. Fq7, 137–144, Springer LNCS, Toulouse, France 2004

    2004

  39. [47]

    Bengtsson and K

    I. Bengtsson and K. Życzkowski, Geometry of Quan- tum States: An Introduction to Quantum Entanglement , Cambridge University Press, Cambridge 2007

    2007

  40. [48]

    Gibbons, M

    S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A 70, 062101 (2004)

    2004

  41. [49]

    T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, New York 2012

    2012

  42. [50]

    Siudzińska, Phys

    K. Siudzińska, Phys. Rev. A 99, 012340 (2019)

    2019

  43. [51]

    M. F. Ernst and R. Klesse, Phys. Rev. A 96, 062319 (2017)

    2017

  44. [52]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. Sudarshan, J. Math. Phys. 17, 821 (1976)

    1976

  45. [53]

    Lindblad, Comm

    G. Lindblad, Comm. Math. Phys. 48, 119 (1976)

    1976

  46. [54]

    Chruściński and A

    D. Chruściński and A. Kossakowski, J. Phys. B: At. Mol. Opt. Phys. 45, 154002 (2012)

    2012

  47. [55]

    Chruściński and F

    D. Chruściński and F. A. Wudarski, Phys. Lett. A 377, 1425 (2013)

    2013

  48. [56]

    E.-M. L. H.-P. Breue and and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009)

    2009

  49. [57]

    Chruściński and S

    D. Chruściński and S. Maniscalco, Phys. Rev. Lett. 112, 120404 (2014)

    2014

  50. [58]

    M. M. Wilde, Quantum Information Theory , Cambridge University Press, Cambridge 2013

    2013

  51. [59]

    Aniello, J

    P. Aniello, J. Phys.: Conf. Ser. 804, 012003 (2017). Appendix Proof to Theorem 1 The generalized Pauli channels are unitarily covariant with respect to Uα ,α = 1,...,d + 1. Using eq. (21), one arrives at χ (Λ GP ) = lnd − min ρ S(Λ GP [ρ]) ≥ lnd − min α S(Λ GP [P (α ) k ]) =: ...

    2017

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