pith. sign in

arxiv: 2410.16362 · v2 · submitted 2024-10-21 · 🪐 quant-ph · cs.IT· math.IT

Semidefinite optimization of the quantum relative entropy of channels

Gereon Ko{\ss}mann , Mark M. Wilde This is my paper

Pith reviewed 2026-05-23 18:24 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords quantum relative entropyquantum channelssemidefinite programmingchannel discriminationintegral representationconvex optimizationquantum information theory
0
0 comments X

The pith

Discretized linearization converts maximization of quantum relative entropy over channels into semidefinite programs that produce arbitrarily tight upper and lower bounds.

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

The paper develops a numerical method to compute the quantum relative entropy of two channels, a quantity used in channel discrimination and resource theories. It adapts a prior integral representation and discretization technique that worked for minimizing relative entropy over states, now handling the harder case of maximization over channels. The resulting semidefinite programs yield efficiently computable bounds that converge to the exact value with finer discretization. This supplies a practical algorithm for an otherwise intractable optimization problem in quantum information.

Core claim

By discretizing and linearizing the integral representation of the relative entropy of states and applying it to channel optimization, one obtains a hierarchy of semidefinite programs whose optimal values sandwich the quantum relative entropy of channels from above and below, with the gap controllable to any desired precision.

What carries the argument

Discretized linearization of the integral representation, which converts the channel maximization into a sequence of semidefinite programs.

If this is right

  • The relative entropy of any two quantum channels becomes numerically computable to arbitrary accuracy.
  • Upper and lower bounds on channel discrimination error probabilities become available via semidefinite programming.
  • Resource-theoretic measures that rely on the relative entropy of channels can now be evaluated in practice.
  • The method extends the scope of semidefinite optimization techniques from states to channels for this divergence.

Where Pith is reading between the lines

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

  • The same discretization strategy may apply to other channel divergences whose state versions already possess integral representations.
  • Numerical values obtained this way could be used to test conjectures about additivity or multiplicativity of the channel relative entropy.
  • Implementation in existing semidefinite solvers would immediately make the quantity accessible for small-dimensional channels.

Load-bearing premise

The same discretization and linearization that sandwich the relative entropy for state minimization continue to sandwich it without uncontrolled error when the optimization is switched to maximization over channels.

What would settle it

A pair of explicit channels for which the gap between the computed upper and lower bounds fails to shrink below a fixed positive constant no matter how fine the discretization grid becomes.

read the original abstract

This paper introduces a method for calculating the quantum relative entropy of channels, an essential quantity in quantum channel discrimination and resource theories of quantum channels. By building on recent developments in the optimization of relative entropy for quantum states [Ko{\ss}mann and Schwonnek, arXiv:2404.17016], we introduce a discretized linearization of the integral representation for the relative entropy of states, enabling us to handle maximization tasks for the relative entropy of channels. Our approach here extends previous work on minimizing relative entropy to the more complicated domain of maximization. It also provides efficiently computable upper and lower bounds that sandwich the true value with any desired precision, leading to a practical method for computing the relative entropy of channels.

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

2 major / 2 minor

Summary. The paper claims to compute the quantum relative entropy of channels via semidefinite programming. It discretizes an integral representation of relative entropy (previously developed for states) to convert the channel problem into a maximization task that can be bounded from above and below by SDPs, with the gap controllable to arbitrary precision by refining the discretization.

Significance. A rigorous, computable sandwiching method for channel relative entropy would be useful in quantum channel discrimination and resource theories. The extension from state minimization to channel maximization is the novel step; if the error bounds carry over without additional assumptions, the result supplies a practical numerical tool that was previously unavailable.

major comments (2)
  1. [Section 3 (extension to channels) and the paragraph following Eq. (integral representation)] The manuscript does not supply a self-contained argument that the quadrature error remains uniformly controlled once the outer optimization is changed from minimization over states to maximization over channels (or equivalently over input states for the channel relative entropy). The state-case analysis in the cited prior work does not automatically guarantee that the same nodes and weights produce rigorous, controllable bounds when the optimizing state can depend on the discretization parameter.
  2. [Abstract and Section 4 (numerical method)] No explicit theorem or lemma is stated that the discretized linearization preserves the sandwiching property (upper and lower bounds converging to the true value) for the channel quantity; the abstract asserts this but the derivation details, error analysis, and verification that discretization errors do not introduce uncontrolled dependence on the optimizing input are absent.
minor comments (2)
  1. [Introduction] Notation for the channel relative entropy D(Φ‖Ψ) should be defined explicitly at first use and distinguished from the state case.
  2. [Section 4] The dependence of the SDP size on the number of quadrature points should be stated clearly, together with any scaling limitations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the thorough review. The comments correctly identify that the extension from state minimization to channel maximization requires explicit justification of uniform quadrature error control, which is not fully detailed in the current manuscript. We will revise to supply the missing self-contained arguments and theorem.

read point-by-point responses

  1. Referee: [Section 3 (extension to channels) and the paragraph following Eq. (integral representation)] The manuscript does not supply a self-contained argument that the quadrature error remains uniformly controlled once the outer optimization is changed from minimization over states to maximization over channels (or equivalently over input states for the channel relative entropy). The state-case analysis in the cited prior work does not automatically guarantee that the same nodes and weights produce rigorous, controllable bounds when the optimizing state can depend on the discretization parameter.

    Authors: We agree that the manuscript lacks a fully self-contained argument for uniform error control under maximization. The state-case bounds hold pointwise, but the optimizing input may in principle depend on the discretization. In the revision we will insert a new lemma (in Section 3) proving that the quadrature error is bounded uniformly over the compact set of density operators, using the joint continuity of the integrand in the state variable and the fact that the discretization nodes and weights are chosen independently of the state. This establishes that the sandwiching property carries over without additional assumptions on the channel. revision: yes

  2. Referee: [Abstract and Section 4 (numerical method)] No explicit theorem or lemma is stated that the discretized linearization preserves the sandwiching property (upper and lower bounds converging to the true value) for the channel quantity; the abstract asserts this but the derivation details, error analysis, and verification that discretization errors do not introduce uncontrolled dependence on the optimizing input are absent.

    Authors: We acknowledge that an explicit theorem stating and proving preservation of the sandwiching property for the channel relative entropy is missing. The abstract summarizes the outcome, but the derivation relies on the state-case analysis without spelling out the channel extension. In the revised manuscript we will add a dedicated theorem (placed after the integral representation in Section 3) that states the convergence result for channels, includes the full error analysis, and verifies that the discretization error remains independent of the optimizing input state. This will make the claim rigorous and self-contained. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior state result; channel extension presented as independent

full rationale

The paper cites Koßmann and Schwonnek (arXiv:2404.17016) for the state-case integral discretization and explicitly frames the lift to maximization over channels as new work that 'extends previous work on minimizing relative entropy to the more complicated domain of maximization.' No equation or bound in the provided text reduces by construction to a fitted input, self-definition, or unverified self-citation chain. The central claim of controllable sandwiching bounds for the channel quantity is presented as a fresh derivation, not a renaming or forced consequence of the cited state result. This is a normal, non-circular use of prior independent work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the state-based discretization technique transfers to channels while preserving bound tightness; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The integral representation of quantum relative entropy admits a discretization that yields a linear semidefinite program for both states and channels.
    Invoked when extending the prior state method to channels (abstract).

pith-pipeline@v0.9.0 · 5648 in / 1138 out tokens · 37351 ms · 2026-05-23T18:24:17.694510+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 5 internal anchors

  1. [1]

    Normalization: Ω( N ) = 0 if N ∈F(A→B) and Ω(N )≥0 for all channels

    work page

  2. [2]

    Faithfulness: Ω( N ) = 0 if and only if N∈F(A→ B) and Ω(N )> 0 otherwise

    work page

  3. [3]

    Monotonicity under composition with a free chan- nelM∈F(A→B) • left composition: Ω( M◦N)≤Ω(N ) • right composition: Ω( N◦M)≤Ω(N ) • tensoring: Ω(M⊗N)≤Ω(N )

    work page

  4. [4]

    black- box

    Convexity: ∑ ipiΩ(Ni)≥Ω(∑ ipiNi) for a proba- bility distribution (pi)i over channels (Ni)i. The axioms for resource monotones have a deep opera- tional meaning. The requirements are in a way such that measuring a resource can be identified with a “cost” of applying, e.g., the considered channel. As shown in [25], the global robustness of a channel N is d...

    work page 2024

  5. [5]

    Tomamichel, Quantum Information Processing with Finite Resources (Springer International Publishing, 2016)

    M. Tomamichel, Quantum Information Processing with Finite Resources (Springer International Publishing, 2016)

    work page 2016

  6. [6]

    Hayashi, Quantum Information Theory: Mathemat- ical Foundation , Graduate Texts in Physics (Springer Berlin Heidelberg, Berlin, Heidelberg, 2017)

    M. Hayashi, Quantum Information Theory: Mathemat- ical Foundation , Graduate Texts in Physics (Springer Berlin Heidelberg, Berlin, Heidelberg, 2017)

    work page 2017

  7. [7]

    M. M. Wilde, Quantum Information Theory , 2nd ed. (Cambridge University Press, Cambridge, UK; New York, 2017)

    work page 2017

  8. [8]

    Watrous, The Theory of Quantum Information , 1st ed

    J. Watrous, The Theory of Quantum Information , 1st ed. (Cambridge University Press, 2018)

    work page 2018

  9. [9]

    A. S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction , 2nd ed., Texts and Mono- graphs in Theoretical Physics (De Gruyter, Berlin, Ger- many, 2019)

    work page 2019

  10. [10]

    Khatri and M

    S. Khatri and M. M. Wilde, Principles of Quantum Communication Theory: A Modern Approach (arXiv,

    work page

  11. [11]

    arXiv:2011.04672v2 [cond-mat, physics:hep-th, physics:math-ph, physics:quant-ph]

    work page arXiv 2011

  12. [12]

    Hiai and D

    F. Hiai and D. Petz, The proper formula for relative en- tropy and its asymptotics in quantum probability, Com- munications in Mathematical Physics 143, 99 (1991)

    work page 1991

  13. [13]

    Nagaoka and T

    H. Nagaoka and T. Ogawa, Strong converse and Stein’s lemma in quantum hypothesis testing, IEEE Transac- tions on Information Theory 46, 2428 (2000)

    work page 2000

  14. [14]

    Umegaki, Conditional expectation in an operator alge- bra, IV (entropy and information), Kodai Mathematical Journal 14, 59 (1962)

    H. Umegaki, Conditional expectation in an operator alge- bra, IV (entropy and information), Kodai Mathematical Journal 14, 59 (1962)

    work page 1962

  15. [15]

    Vedral, M

    V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying entanglement, Physical Review Letters 78, 2275 (1997)

    work page 1997

  16. [16]

    Vedral and M

    V. Vedral and M. B. Plenio, Entanglement measures and purification procedures, Physical Review A 57, 1619 (1998)

    work page 1998

  17. [17]

    E. M. Rains, Bound on distillable entanglement, Physical Review A 60, 179 (1999), arXiv:quant-ph/9809082

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  18. [18]

    E. M. Rains, A semidefinite program for distillable en- tanglement, IEEE Transactions on Information Theory 47, 2921 (2001), arXiv:quant-ph/0008047

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  19. [19]

    Chitambar and G

    E. Chitambar and G. Gour, Quantum resource theories, Reviews of Modern Physics 91, 025001 (2019)

    work page 2019

  20. [20]

    Fawzi and J

    H. Fawzi and J. Saunderson, Lieb’s concavity theorem, matrix geometric means, and semidefinite optimization, Linear Algebra and its Applications 513, 240 (2017), arXiv:1512.03401

    work page arXiv 2017

  21. [21]

    Fawzi and O

    H. Fawzi and O. Fawzi, Efficient optimization of the quantum relative entropy, Journal of Physics A: Math- ematical and Theoretical 51, 154003 (2018)

    work page 2018

  22. [22]

    Fawzi, J

    H. Fawzi, J. Saunderson, and P. A. Parrilo, Semidefinite approximations of the matrix logarithm, Foundations of Computational Mathematics 19, 259 (2019), package cvxquad at https://github.com/hfawzi/cvxquad

    work page 2019

  23. [23]

    Fawzi and J

    H. Fawzi and J. Saunderson, Optimal self-concordant barriers for quantum relative entropies (2023), arXiv:2205.04581 [math.OC]

    work page arXiv 2023

  24. [24]

    Faust and H

    O. Faust and H. Fawzi, Rational approximations of op- erator monotone and operator convex functions (2023), arXiv:2305.12405, arXiv:2305.12405 [math.OC]

    work page arXiv 2023

  25. [25]

    Koßmann and R

    G. Koßmann and R. Schwonnek, Optimising the relative entropy under semi definite constraints – a new tool for estimating key rates in QKD (2024), arXiv:2404.17016v1 [quant-ph]

    work page arXiv 2024

  26. [26]

    P. E. Frenkel, Integral formula for quantum relative en- tropy implies data processing inequality, Quantum 7, 1102 (2023)

    work page 2023

  27. [27]

    Jenˇ cov´ a, Recoverability of quantum channels via hy- pothesis testing (2023), arXiv:2303.11707 [quant-ph]

    A. Jenˇ cov´ a, Recoverability of quantum channels via hy- pothesis testing (2023), arXiv:2303.11707 [quant-ph]

    work page arXiv 2023

  28. [28]

    Coecke, T

    B. Coecke, T. Fritz, and R. W. Spekkens, A mathematical theory of resources, Information and Computation 250, 59 (2016), Quantum Physics and Logic

    work page 2016

  29. [29]

    Theurer, D

    T. Theurer, D. Egloff, L. Zhang, and M. B. Plenio, Quantifying operations with an application to coherence, Physical Review Letters 122, 190405 (2019)

    work page 2019

  30. [30]

    Resource theories of quantum channels and the universal role of resource erasure

    Z.-W. Liu and A. Winter, Resource theories of quantum channels and the universal role of resource erasure (2019), arXiv:1904.04201 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  31. [31]

    Liu and X

    Y. Liu and X. Yuan, Operational resource theory of quantum channels, Physical Review Research 2, 012035 (2020)

    work page 2020

  32. [32]

    Cooney, M

    T. Cooney, M. Mosonyi, and M. M. Wilde, Strong con- verse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication, Communications in Mathematical Physics 344, 797–829 (2016)

    work page 2016

  33. [33]

    Gour and M

    G. Gour and M. M. Wilde, Entropy of a quantum chan- nel, Physical Review Research 3, 023096 (2021)

    work page 2021

  34. [34]

    Leditzky, E

    F. Leditzky, E. Kaur, N. Datta, and M. M. Wilde, Ap- proaches for approximate additivity of the Holevo in- formation of quantum channels, Physical Review A 97, 012332 (2018)

    work page 2018

  35. [35]

    M. M. Wilde, M. Berta, C. Hirche, and E. Kaur, Amor- tized channel divergence for asymptotic quantum chan- nel discrimination, Letters in Mathematical Physics 110, 2277–2336 (2020)

    work page 2020

  36. [36]

    Wang and M

    X. Wang and M. M. Wilde, Resource theory of asym- metric distinguishability for quantum channels, Physical Review Research 1, 033169 (2019)

    work page 2019

  37. [37]

    Metger, O

    T. Metger, O. Fawzi, D. Sutter, and R. Renner, Gen- eralised entropy accumulation, in 2022 IEEE 63rd An- nual Symposium on Foundations of Computer Science (FOCS) (IEEE, 2022) pp. 844–850

    work page 2022

  38. [38]

    Boyd and L

    S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004)

    work page 2004

  39. [39]

    Skrzypczyk and D

    P. Skrzypczyk and D. Cavalcanti, Semidefinite Program- ming in Quantum Information Science (IOP Publishing, 2023)

    work page 2023

  40. [40]

    Fang and H

    K. Fang and H. Fawzi, Geometric R´ enyi divergence and its applications in quantum channel capacities, Com- munications in Mathematical Physics 384, 1615–1677 (2021)

    work page 2021

  41. [41]

    Fawzi and O

    H. Fawzi and O. Fawzi, Semidefinite programming lower bounds on the squashed entanglement (2022), arXiv:2203.03394 [quant-ph]

    work page arXiv 2022

  42. [42]

    Brown, H

    P. Brown, H. Fawzi, and O. Fawzi, Device-independent 12 lower bounds on the conditional von Neumann entropy (2023), arXiv:2106.13692 [quant-ph]

    work page arXiv 2023

  43. [43]

    Huang and M

    Z. Huang and M. M. Wilde, Semi-definite optimization of the measured relative entropies of quantum states and channels (2024), arXiv:2406.19060 [quant-ph]

    work page arXiv 2024

  44. [44]

    Hirche and M

    C. Hirche and M. Tomamichel, Quantum R´ enyi and f- divergences from integral representations, Communica- tions in Mathematical Physics 405, 208 (2024)

    work page 2024

  45. [45]

    Datta, Min- and max-relative entropies and a new entanglement monotone, IEEE Transactions on Informa- tion Theory 55, 2816 (2009)

    N. Datta, Min- and max-relative entropies and a new entanglement monotone, IEEE Transactions on Informa- tion Theory 55, 2816 (2009)

    work page 2009

  46. [46]

    Sion, On general minimax theorems, Pacific Journal of Mathematics 8, 171 (1958)

    M. Sion, On general minimax theorems, Pacific Journal of Mathematics 8, 171 (1958)

    work page 1958

  47. [47]

    Tomamichel, R

    M. Tomamichel, R. Colbeck, and R. Renner, A fully quantum asymptotic equipartition property, IEEE Transactions on Information Theory 55, 5840–5847 (2009)

    work page 2009

  48. [48]

    Entanglement distillation by extendible maps

    L. Pankowski, F. G. S. L. Brand˜ ao, M. Horodecki, and G. Smith, Entanglement distillation by extendible maps, Quantum Information and Computation 13, 751–770 (2013), arXiv:1109.1779

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  49. [49]

    E. Kaur, S. Das, M. M. Wilde, and A. Winter, Extendibil- ity limits the performance of quantum processors, Physi- cal Review Letters 123, 070502 (2019), arxiv:2108.03137

    work page arXiv 2019

  50. [50]

    E. Kaur, S. Das, M. M. Wilde, and A. Winter, Re- source theory of unextendibility and nonasymptotic quantum capacity, Physical Review A 104, 022401 (2021), arxiv:1803.10710

    work page arXiv 2021

  51. [51]

    Berta, F

    M. Berta, F. Borderi, O. Fawzi, and V. B. Scholz, Semidefinite programming hierarchies for constrained bi- linear optimization, Mathematical Programming 194, 781 (2021), arXiv:1810.12197

    work page arXiv 2021

  52. [52]

    Horodecki, P

    M. Horodecki, P. W. Shor, and M. B. Ruskai, Entan- glement breaking channels, Reviews in Mathematical Physics 15, 629–641 (2003)

    work page 2003

  53. [53]

    Barnum, M

    H. Barnum, M. A. Nielsen, and B. Schumacher, In- formation transmission through a noisy quantum chan- nel, Physical Review A 57, 4153 (1998), arXiv:quant- ph/9702049

    work page arXiv 1998

  54. [54]

    Fawzi, The set of separable states has no fi- nite semidefinite representation except in dimension 3 × 2, Communications in Mathematical Physics 386, 1319–1335 (2021)

    H. Fawzi, The set of separable states has no fi- nite semidefinite representation except in dimension 3 × 2, Communications in Mathematical Physics 386, 1319–1335 (2021)

    work page 2021

  55. [55]

    A. S. Holevo, Entanglement-assisted capacity of con- strained channels, in First International Symposium on Quantum Informatics , Vol. 5128, edited by Y. I. Ozhigov, International Society for Optics and Photonics (SPIE,

    work page

  56. [56]

    Entanglement-assisted capacity of constrained quantum channel

    pp. 62 – 69, arXiv:quant-ph/0211170

    work page internal anchor Pith review Pith/arXiv arXiv

  57. [57]

    Ramakrishnan, R

    N. Ramakrishnan, R. Iten, V. Scholz, and M. Berta, Quantum Blahut-Arimoto algorithms, in 2020 IEEE In- ternational Symposium on Information Theory (ISIT) (2020) pp. 1909–1914

    work page 2020

  58. [58]

    Ramakrishnan, R

    N. Ramakrishnan, R. Iten, V. B. Scholz, and M. Berta, Computing quantum channel capacities, IEEE Transac- tions on Information Theory 67, 946 (2021)

    work page 2021

  59. [59]

    Berta, O

    M. Berta, O. Fawzi, and M. Tomamichel, On variational expressions for quantum relative entropies, Letters in Mathematical Physics 107, 2239–2265 (2017)

    work page 2017

  60. [60]

    Appendix A: T echnical lemmas Lemma A.1 ([27])

    NISO, Credit – contributor roles taxonomy, https:// credit.niso.org/, Accessed 2024-10-3. Appendix A: T echnical lemmas Lemma A.1 ([27]). For the channel relative entropy (2), we have D(M∥N) = sup ψAA D(MA→B(ψAA)∥NA→B(ψAA)) (A1) where the optimization is over every pure state ψAA∈ S(HA⊗HA) andHA∼=HA. Proof. This was proven in [27, Lemma 7] (see also [29, ...

    work page 2024