Classical Channel Entropy: A Primer

• Classical channel entropy is an entropy-like measure defined on stochastic channels based solely on their conditional output distributions.
• It captures the minimum conditional entropy over all possible inputs, establishing a worst-case uncertainty measure for the channel.
• The framework connects to majorization theory and maximum-entropy principles, with examples including binary-symmetric and erasure channels.

Searching arXiv for the cited channel-entropy papers and closely related work to ground the article. arXiv search query: "(Sanna et al., 2018) channel entropy", max_results=5 Classical channel entropy is an entropy-like quantifier assigned to a channel rather than to a single probability distribution. For a classical stochastic channel with transition law $P(y|x)$, one explicit formula is

$$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$

so the quantity is the minimum, over all inputs, of the average conditional output entropy, which collapses to the minimum row entropy of the stochastic matrix (Faist et al., 6 Aug 2025). A complementary order-theoretic line of work studies the uncertainty inherent in classical channels through majorization, relative majorization, and conditional majorization; within that framework, classical channel entropy is defined as an additive monotone with respect to the relevant majorization preorder, and well-known state entropies are uniquely extended to channels via optimal extensions (Nateeboon, 16 Jul 2025).

1. Formal setting

In the purely classical setting, a channel is a stochastic map $P(y|x)$ between finite alphabets $A=\{1,2,\dots,d_A\}$ and $B=\{1,2,\dots,d_B\}$. In matrix form, the corresponding channel $\Phi$ acts on computational-basis operators as

$$\Phi\bigl(|x\rangle\langle x'|\bigr) = \delta_{x,x'}\sum_y P(y|x)\,|y\rangle\langle y|.$$

If a pure input is written as

$$|\psi\rangle_{AR}=\sum_x \sqrt{p_x}\,|x\rangle_A\otimes |x\rangle_R,$$

then the joint output is

$$\rho_{BR} = \Phi(\psi_{AR}) = \sum_{x,y} p_x P(y|x)\,|y\rangle\langle y|_B\otimes |x\rangle\langle x|_R.$$

For such classical channels, the conditional von Neumann entropy of the output is exactly the classical conditional Shannon entropy,

$$S(B|R)_\rho = H(Y|X)_p = \sum_x p_x\left[-\sum_y P(y|x)\log P(y|x)\right],$$

which makes the channel-entropy construction a direct specialization of a more general channel-entropy formalism (Faist et al., 6 Aug 2025).

This specialization is important because it identifies classical channel entropy with an intrinsic property of the channel’s conditional distributions $$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$0, rather than with an entropy of an externally chosen source. In this form, the quantity is defined entirely by the stochastic matrix.

2. Majorization-based uncertainty order on channels

A distinct but closely related development treats classical channel entropy as part of a broader theory of uncertainty for channels. The relevant background includes probability vector majorization and its variants, relative majorization and conditional majorization. Within that program, three conceptually distinct approaches are introduced to formalize the notion of uncertainty inherent in classical channels, and these three approaches define the same preordering on the domain of classical channels (Nateeboon, 16 Jul 2025).

With that preorder in place, classical channel entropy is defined to be an additive monotone with respect to the majorization relation. The same thesis states that the well-known entropies in the domain of classical states are uniquely extended to the domain of channels via the optimal extensions (Nateeboon, 16 Jul 2025). This places channel entropy in an order-theoretic setting: the quantity is not only a formula attached to a stochastic matrix, but also a monotone compatible with a channel comparison relation.

This suggests a structural analogy with ordinary majorization theory for probability vectors. In the state domain, entropy functions quantify uncertainty while respecting the majorization preorder; in the channel domain, the thesis asserts an analogous role for channel entropies, now over stochastic maps rather than single distributions.

3. General channel-entropy definition and classical reduction

A general quantum-channel entropy is defined for a completely positive, trace-preserving map $$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$1 by

$$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$2

where $$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$3 is the completely depolarizing map in non-normalized form. An equivalent expression is

$$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$4

The definition of these channel entropy measures appeared in (Sanna et al., 2018), and the explicit classical specialization is summarized in (Faist et al., 6 Aug 2025).

For a classical channel, the specialization yields

$$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$5

Equivalently, in explicit classical notation,

$$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$6

and therefore

$$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$7

The reduction from $$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$8 to $$H_{\rm class}(P) =\min_{p(x)\ge 0,\;\sum_x p(x)=1} H_p(Y|X) =\min_x\left[-\sum_y P(y|x)\ln P(y|x)\right],$$9 follows from the fact that the minimum of an average is attained by concentrating the input distribution on a single input symbol $P(y|x)$0 with minimal conditional entropy. In this sense, the classical channel entropy is a worst-case conditional entropy over channel inputs (Faist et al., 6 Aug 2025).

4. Maximum-channel-entropy principle and thermal channels

A variational formulation asks for the channel of largest entropy within a constrained family. For classical channels $P(y|x)$1 obeying linear constraints

$P(y|x)$2

the problem is to maximize $P(y|x)$3. By standard Lagrange multipliers in the classical maximum-entropy method, the maximizing channel has exponential form,

$P(y|x)$4

or equivalently,

$P(y|x)$5

with the multipliers $P(y|x)$6 chosen so that the expectations attain the prescribed values $P(y|x)$7 (Faist et al., 6 Aug 2025).

The broader maximum-channel-entropy principle defines a thermal channel as one that maximizes a channel entropy measure subject to linear constraints, and proves that thermal channels exhibit an exponential form reminiscent of thermal states. The paper reporting this principle studies examples including thermalizing channels that conserve a state's average energy, as well as Pauli-covariant and classical channels (Faist et al., 6 Aug 2025).

In the classical reduction, the result is the exact analogue of a maximum-entropy construction for stochastic maps. A plausible implication is that the familiar Jaynesian logic of constrained entropy maximization extends from probability distributions to transition laws.

5. Canonical examples

Two elementary channels make the definition explicit. For the binary-symmetric channel,

$P(y|x)$8

the conditional distribution on $P(y|x)$9 is Bernoulli for each input $A=\{1,2,\dots,d_A\}$0, with entropy

$A=\{1,2,\dots,d_A\}$1

Because this conditional entropy is independent of $A=\{1,2,\dots,d_A\}$2, the worst-case and best-case values coincide, and

$A=\{1,2,\dots,d_A\}$3

For the erasure channel,

$A=\{1,2,\dots,d_A\}$4

the conditional entropy for each $A=\{1,2,\dots,d_A\}$5 is

$A=\{1,2,\dots,d_A\}$6

so again the value does not depend on $A=\{1,2,\dots,d_A\}$7, and

$A=\{1,2,\dots,d_A\}$8

(Faist et al., 6 Aug 2025).

Channel	Transition law	$A=\{1,2,\dots,d_A\}$9
Binary-symmetric channel	$B=\{1,2,\dots,d_B\}$0 for $B=\{1,2,\dots,d_B\}$1, $B=\{1,2,\dots,d_B\}$2 for $B=\{1,2,\dots,d_B\}$3	$B=\{1,2,\dots,d_B\}$4
Erasure channel	$B=\{1,2,\dots,d_B\}$5 for $B=\{1,2,\dots,d_B\}$6, $B=\{1,2,\dots,d_B\}$7 for $B=\{1,2,\dots,d_B\}$8	$B=\{1,2,\dots,d_B\}$9

These examples show that when every row of the stochastic matrix has the same entropy, the minimization over inputs becomes trivial. The channel entropy is then simply that common row entropy.

6. Operational role and terminological variation

The phrase “classical channel entropy” also appears in an operationally different setting: the study of classical information transmission through certain quantum channels. For a quantum channel $\Phi$0, the minimal output entropy is

$\Phi$1

In the analysis of Weyl channels, the classical capacity is

$\Phi$2

with

$\Phi$3

For covariant channels one has

$\Phi$4

For the deformed Weyl channel considered in (Amosov, 2020), the main additivity theorem states

$\Phi$5

and the capacity becomes

$\Phi$6

The result holds for finite-dimensional $\Phi$7 Weyl channels obtained by deformation of a $\Phi$8-$\Phi$9 Weyl channel, with the finer weights $$\Phi\bigl(|x\rangle\langle x'|\bigr) = \delta_{x,x'}\sum_y P(y|x)\,|y\rangle\langle y|.$$0 satisfying the ordering condition

$$\Phi\bigl(|x\rangle\langle x'|\bigr) = \delta_{x,x'}\sum_y P(y|x)\,|y\rangle\langle y|.$$1

and with covariance under the Weyl group playing a crucial role in the single-letter capacity formula (Amosov, 2020).

In that specific class of channels, the “classical channel entropy” is captured by the minimal output von Neumann entropy, and the least entropy that $$\Phi\bigl(|x\rangle\langle x'|\bigr) = \delta_{x,x'}\sum_y P(y|x)\,|y\rangle\langle y|.$$2 can produce on any pure input directly determines the maximal rate at which classical information can be sent reliably (Amosov, 2020). A plausible implication is that the term is not uniform across the literature: in one line of work it denotes an uncertainty monotone on classical stochastic maps, while in another it denotes the entropy quantity governing classical communication through a covariant quantum channel.