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Uncertainty and entropies of classical channels

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The three approaches to classical channel uncertainty define the same preorder, so channel entropy can be grounded in a canonical order.

desk verdict The channel majorization equivalences are real and the proofs I checked hold, but the abstract's 'unique extension' claim only holds for min- and max-entropy, so the package needs a toned-down abstract and a cleanup pass. read the letter →

arxiv 2507.12310 v1 pith:IWESOGRR submitted 2025-07-16 quant-ph

Takla Nateeboon This is my paper
classification quant-ph MSC 81P4594A17
keywords classicalchannelschannelmajorizationuncertaintypreorderentropyrandom-permutationsuperchannelcompletelyuniformity-preservingoperationalt-gameminimaloutput
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 thesis establishes that the uncertainty of a classical channel—a device that turns inputs into probability distributions over outputs—can be compared by a single preorder. It offers three independent definitions: constructive (one channel is more uncertain if the other can be produced by randomly permuting its output), axiomatic (the transformation must preserve all channels that are maximally random on their output), and operational (the more certain channel never gives a better winning chance in any guessing game). The paper proves these three definitions coincide, and then defines classical channel entropy as an additive measure that is monotone under the order. As a result, the standard entropies of probability distributions extend to channels in a principled way, with minimal output entropy singled out as the maximal extension of any quasiconcave entropy.

What carries the argument

The load-bearing object is the classical superchannel in standard form: every classical superchannel decomposes uniquely as a sum of pre-processing maps and post-processing channels, and it is a random-permutation superchannel exactly when all of its post-processings are doubly stochastic. The axiomatic definition is carried by conditionally uniform channels—channels whose output marginal on Y is maximally mixed—and Theorem 3.2.8 proves that a classical superchannel is completely uniformity-preserving exactly when it is a random-permutation superchannel. Theorem 3.2.14 then equates that convertibility with dominance of winning chances in all t-games, via a Farkas-lemma linear-programming argument.

What would settle it

Enumerate all classical superchannels with two input and two output states in standard form and check whether every completely uniformity-preserving one is a random-permutation superchannel; a single counterexample would falsify Theorem 3.2.8. Equivalently, linear programming can test any channel pair for predictability dominance for all sorted vectors while checking whether no random-permutation superchannel exists, which would falsify Theorem 3.2.14.

Watch

Extended reading notes

Core claim

The paper's central claim is that uncertainty in a classical channel is not a matter of taste: the relation 'N is at least as uncertain as M' is the same preorder whether defined constructively (M is obtained from N by a random-permutation superchannel), axiomatically (a completely uniformity-preserving superchannel sends N to M), or operationally (N gives no worse winning chances than M in every t-game). The two load-bearing theorems are Theorem 3.2.8, which identifies completely uniformity-preserving superchannels with random-permutation superchannels, and Theorem 3.2.14, which equates this convertibility with dominance of t-game winning probabilities. On this preorder the paper defines classical channel entropy as an additive antitone and proves that the familiar state entropies extend to the channel domain, with minimal output entropy as the maximal extension of any quasiconcave entropy.

Load-bearing premise

The three definitions coincide only under the chosen axiom that legitimate operations must preserve not just the most random channel but every channel that is uniform on its output; this axiom is a modeling choice, not forced by the other two approaches.

Editorial extensions

If this is right

  • Channel majorization is a single well-defined preorder, so any comparison of classical channels by uncertainty is invariant across the constructive, axiomatic, and operational perspectives.
  • Replacement channels embed state majorization into channel majorization, meaning the new order directly generalizes the classical uncertainty order for probability vectors.
  • Minimal output entropy is the maximal extension of any quasiconcave state entropy to classical channels, giving it a distinguished role among additive channel uncertainty measures.
  • Max-entropy and min-entropy have unique extensions to the channel domain, so those two endpoint measures of state uncertainty lift to channels without ambiguity.
  • The operational t-game characterization gives a finite test of channel majorization: a channel is at least as uncertain as another exactly when it never offers a higher winning chance in any guessing game.

Reading between the lines

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

  • The three-way equivalence leans on a specific modeling choice: 'uncertainty non-decreasing' means preserving every conditionally uniform channel, not just the maximally randomizing one; a weaker axiom would leave the constructive and operational orders intact but could break the triality.
  • Because the classical channel preorder is the incoherent slice of the quantum channel domain, the same axiomatic template could be used to define a majorization preorder for general quantum channels, with the classical result as the dephased special case.
  • The t-game characterization suggests a decision-theoretic reading: channel entropy is exactly an additive utility consistent with dominance in every finite guessing game, so new candidate channel entropies can be tested by checking monotonicity under random-permutation superchannels.
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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. This manuscript, an MSc thesis, develops a majorization preorder for classical channels. It reviews majorization, relative majorization, and conditional majorization, then proposes three ways of defining channel majorization: constructively via random-permutation superchannels, axiomatically via completely uniformity-preserving superchannels, and operationally via t-game winning probabilities. The central result is that these three approaches coincide, so classical channel majorization is well defined. The paper then defines a channel entropy as an additive antitone of this preorder and studies the optimal extensions of state entropies to the channel domain, claiming that the usual state entropies are uniquely extended.

Significance. The main preorder results are coherent and valuable. The Farkas/linear-programming reduction in Theorem 3.2.14, the convex-combination argument in Theorem 3.2.4, and the standard-form partial-order argument in Theorem 3.3.6 are convincing, so the equivalence of the constructive, axiomatic, and operational definitions gives a genuine foundation for comparing channel uncertainty. The sandwich theorem for entropy extensions and the proof that min- and max-entropy have unique extensions are useful contributions. The manuscript is also transparent that some results and figures originate from the collaboration reported in [1].

major comments (2)
  1. [§3.2.2, Definition 3.2.7 and Theorem 3.2.8] The abstract's claim that the well-known state entropies are uniquely extended to channels is not supported by the text. Theorem 4.2.2 only establishes the sandwich L ≤ H' ≤ U for every extension H' of a state entropy H, where L = sup{H(q) : q ≻ N} and U = inf{H(q) : N ≻ q}; uniqueness follows only when L = U, and Theorem 4.2.6 proves this coincidence for min- and max-entropy only. For Shannon entropy the two optimal-extension bounds differ. Consider the standard-form channel N with columns p1 = (0.8, 0.1, 0.1)^T and p2 = (0.6, 0.4, 0)^T. A state q with q ≻ N must satisfy q1 ≥ 0.8 and q1 + q2 ≥ 1, so L = H(0.8, 0.2) ≈ 0.722. On the other side, the admissibility condition N ≻ q contains q = p1, and since the minimum of the concave function H over the convex hull of the columns is attained at an extreme point, U = H(p1) ≈ 0.922. Thus L < U, and the claim of a unique optimal extension is false for Shannon entropy; the abstract and the discussion in §4.2 should be revised to state that uniqueness holds for H_min and H_max, while for other entropies the optimal extensions only provide a sandwich.
  2. [Theorem 3.2.4, proof] The axiomatic approach rests on a postulate that is introduced only after plain uniformity preservation fails. The counterexample in §3.2.2 motivating the definition is legitimate, but the paper does not derive complete uniformity preservation from an independent operational principle; it is chosen so that the class of allowable superchannels matches the constructive random-permutation class. Consequently Theorem 3.2.8, although internally sound, partly builds the desired equivalence into the definition. For the advertised conceptual independence of the three approaches, the manuscript should either justify the complete-uniformity-preservation axiom from a prior principle, or explicitly present it as a normative modeling choice and discuss how sensitive the equivalence is to weakening it.
minor comments (4)
  1. [§2.3.1, Definition 2.3.4] The sum in the Ky Fan k-norm definition starts at i = 0 but should start at i = 1; also the norm is defined on Rn using absolute values, while the subsequent majorization definition for probability vectors does not require this, so the relationship should be made explicit.
  2. [Theorem 1.4.6] The domain notation for an entropy is ill-formed: the expression S_{n∈N} Prob(n) × Prob(n) is ambiguous, and the additivity condition under tensor product requires functions defined on probability vectors of varying dimension; this should be written as a union of product domains.
  3. [Theorem 3.2.4] The 'Krause representation' should be the 'Kraus representation'. There are also several minor typos elsewhere, such as 'purposed' for 'proposed', that should be corrected in a final pass.
  4. [Theorem 3.2.4, proof] The proof begins with 'N = N^↓', but the theorem is stated for arbitrary N; since every channel is equivalent to its non-increasing-column version by random-permutation superchannels, the reduction should be stated explicitly before invoking this assumption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the three majorization characterizations are proven from stated definitions, and the only load-bearing use of prior work is non-circular; the abstract's 'uniquely extended' claim is an overgeneralization but not a circular step.

full rationale

The thesis proves its central equivalence theorems directly from the definitions it introduces. The constructive random-permutation superchannels, the axiomatic completely-uniformity-preserving superchannels, and the operational t-game winning chances are not identified with each other by definition; Theorem 3.2.8 and Theorem 3.2.14 supply actual arguments (including a Farkas-lemma duality step) for the equivalences. The choice of the 'completely uniformity-preserving' axiom is motivated by an explicit counterexample, and the equivalence with random-permutation superchannels is a substantive theorem rather than a restatement of the definition. No parameter is fitted to data and then renamed as a prediction. The self-citation to the author's own article [1] appears in the Preface and in figure credits only; the main theorems do not rest on an unverified result imported from that citation. The abstract's claim that 'the well-known entropies are uniquely extended' exceeds what Theorem 4.2.6 proves, since uniqueness is established only for min- and max-entropy while Shannon and other Renyi entropies generally have distinct optimal lower and upper extensions. That is a correctness or scope-of-claim problem, not a circularity: the theorem does not define the conclusion into its assumptions. Overall, no circular step in the derivation chain was found.

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

Pure mathematics paper, so no fitted parameters exist and all parameters (Renyi alpha, game sizes, dimensions) are structural. The non-standard assumption is the complete uniformity preservation axiom (Definition 3.2.7), which is reverse-engineered in Section 3.2.2 to make the axiomatic approach match the constructive one. Standard results used without full proof include Birkhoff-von Neumann, Farkas lemma, and the superchannel realization theorem.

assumptions (5)
  • domain assumption Finite-dimensional Hilbert space framework
    Chapter 1 states 'we work exclusively on the finite dimensional Hilbert space'; all majorization and channel results are stated for finite dimensions, and the Choi and superchannel realization theorems rely on this.
  • standard math Birkhoff-von Neumann theorem (doubly stochastic matrices are convex combinations of permutations)
    Invoked in Theorem 2.2.3 and used throughout Chapter 3 wherever the constructive random-permutation superchannel is related to doubly stochastic post-processing.
  • standard math Farkas lemma and strong duality of linear programs
    Used in Theorem 2.6.13 for the conditional majorization LP characterization and in Theorem 3.2.14 to prove the equivalence of the axiomatic/constructive and operational definitions.
  • ad hoc to paper Complete uniformity preservation axiom
    Definition 3.2.7 requires superchannels to preserve conditionally uniform channels. Section 3.2.2 shows that plain uniformity preservation admits a counterexample superchannel, so the axiom is added specifically to make the axiomatic order coincide with the constructive order.
  • standard math Classical superchannel realization in standard form
    Theorem 3.1.2 and Corollary 3.1.3, proven from the quantum realization theorem (Theorem 1.6.2), give every classical superchannel the standard form sum used for all characterizations in Chapter 3.

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Cite this review

Pith. "Pith review of Uncertainty and entropies of classical channels." pith.science (2026). https://pith.science/paper/IWESOGRR

@misc{pith2026250712310,
  author       = {Pith},
  title        = {Pith review of: Uncertainty and entropies of classical channels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWESOGRR}},
  note         = {Machine review of arXiv:2507.12310}
}
read the original abstract

In this thesis, I studied a mathematical development to define and quantify the uncertainty inherent in classical channels. This thesis starts with the introduction and background on how to formally think about uncertainty in the domain of classical states. The concept of probability vector majorization and its variants, relative majorization and conditional majorization, are reviewed. This thesis introduces three conceptually distinct approaches to formalize the notion of uncertainty inherent in classical channels. These three approaches define the same preordering on the domain of classical channels, leading to characterizations from many perspectives. With the solid foundation of uncertainty comparison, classical channel entropy is then defined to be an additive monotone with respect to the majorization relation. The well-known entropies in the domain of classical states are uniquely extended to the domain of channels via the optimal extensions, providing not only a solid foundation but also the quantifiers of uncertainty inherent in classical channels.

Figures

Figures reproduced from arXiv: 2507.12310 by the authors.

Figure 2.1
Figure 2.1. The testing region of the pair p = (0.9, 0.1, 0)T , q = (0.1, 0.8, 0.1)T . one repeat ratio, px qx = px+1 qx+1 . Then, we set E ∈ Stoch(n − 1, n) to be a matrix representation of a linear map g : R n → R n−1 such that g(p1, . . . , px, px+1, . . . , pn) = (p1, . . . , px + px+1, . . . , pn). To see that this operation is linear, consider g on αp + βq for any α, β ∈ R, g(αp + βq) = (αp1 + βq1, . . . , α(px + px+1) + … view at source ↗
Figure 2.2
Figure 2.2. Ratios between elements of p and q on a real line. The leftmost point is pn/qn. If pn−1/qn−1 = pn/qn, then adding small number to qn−1 will shift the fraction pn−1/qn−1 to the left while pn/qn is shifted to the right changing the ordering of {px/qx}x∈[n] . Suppose that the pair (p ′ , q ′ ) has that the last n ′ elements with the identical ratio pn qn . Pick ⃗ε = (ε1, . . . , εn) to be positive except at the last n … view at source ↗
Figure 2.3
Figure 2.3. Possible lower Lorenz curves for a pair (p, q) (solid and dashed purple line) and a pair (p ′ , q ′ ) (dashed orange). The dashed purple portion shows the lowest possible lower Lorenz curve for a given Dmax(p ′∥q ′ ) = p ′ 1/q′ 1 . The dashed orange shows the highest possible lower Lorenz curve for a given Dmin(p∥q) = P x∈supp(p) qx. Remark. The converse of the above theorem is not true. There is a pair of probabili… view at source ↗
Figures from the paper (10 more)
Figure 2.4
Figure 2.4. Figure 2.4: A diagramatic definition of semicausal channel. [PITH_FULL_IMAGE:figures/full_fig_p066_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: Diagram visualization of a semilocalizable map [PITH_FULL_IMAGE:figures/full_fig_p068_2_5.png]
Figure 2.6
Figure 2.6. Figure 2.6: A diagram of conditionally doubly stochastic map, where [PITH_FULL_IMAGE:figures/full_fig_p071_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: A diagram for a k-game in which a player is given y from the joint probability vector p XY . player can make. However, one also could devise a bigger family of gambling games which includes the scenario where a player can influence a number of guesses they are given.…
Figure 2.8
Figure 2.8. Figure 2.8: A diagram for a conditional T-game. 69 [PITH_FULL_IMAGE:figures/full_fig_p079_2_8.png]
Figure 3.1
Figure 3.1. Figure 3.1: A realization of classical superchannel is invariant under composition with the dephasing [PITH_FULL_IMAGE:figures/full_fig_p086_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: Random-permutation superchannel. The pre-processing channel [PITH_FULL_IMAGE:figures/full_fig_p091_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: Completely uniformity preserving superchannel. In this figure, the channel [PITH_FULL_IMAGE:figures/full_fig_p096_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: The diagram of how a t-game is played. A player, named Alice, is given a full knowledge of a joint probability distribution t and a transition matrix N associated with the classical channel N . Once the game start, she will be made aware of the hint w and then give a…
Figure 4.1
Figure 4.1. Figure 4.1: The Ky Fan k-norm of p1, p2 and the optimal upper bound pˇ. Relating this back to the extensions of majorization monotones, we have that an optimal upper bound for the image of N has the entropy equal to the maximal extension entropy of N , i.e. H(pˇ) = H(Y |X)N . (4…

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