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Our short paper (you want to get straight to the point): arxiv.org/abs/2508.03993 Our technical companion paper (you want a careful mathematical construction with all the details): arxiv.org/abs/2508.03994 Thanks for reading!

The thermal state is so ubiquitous throughout physics, information theory, and algorithms; wouldn't it be fun if the thermal quantum channel had as broad a range of applicability? I think we've only begun to scratch the surface of what it can do.

All of our methods are detailed in our technical paper arxiv.org/abs/2508.03994 . We use lots of Lagrange duality of convex problems, Schur-Weyl duality, typicality techniques for noncommuting operators and channels, and develop an extended postselection theorem for quantum channels.

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Are there any uses of the thermal quantum channel in quantum learning algorithms? We think that state learning algorithms based on maximum entropy arguments can naturally be extended to channels, and they do seem to converge to the correct channel for some simple examples.

Except that now, we can account for arbitrary correlations and constraints the dynamics must obey, which may correlate input and output systems. In this sense, the thermal quantum channel represents thermalizing dynamics with arbitrary “partial information.”

What's the thermal quantum channel good for? Much like how we often replace a complex or unknown quantum microstate by a thermal state in a many-body system [(a)], I'd like to think that we could often replace complex or unknown evolution 𝒰 by the thermal quantum channel [(b)].

Much like how we replace a complex or unknown quantum state by a thermal state in a many-body system [(a)], I'd like to think that we can replace complex or unknown evolution 𝒰 by the thermal quantum channel [(b)]

Therefore, we have two independent arguments that identify the same *thermal quantum channel*.

We find a *microcanonical channel* Ω that represents a global channel on the system and a large environment and which obeys a global conservation law represented by a set of constraints. On a single system, Ω looks the same as the channel defined from the maximum channel entropy principle!

So Jaynes' maximum entropy principle extends naturally to channels. But what happens if we extend other definitions of the thermal state to channels? It would be worrying if different approaches to define of the thermal quantum channel led to different channels.

For *average* energy conservation, the thermal quantum channel outputs, for each input energy eigenstate |E⟩, a thermal state with temperature such that the state has energy E. This channel is thermalizing (it outputs thermal states) but conserves memory of its input (the average energy)!

The expectation values of physically accessible quantities imposed as constraints can be arbitrary general linear constraints on the channel. E.g., for strict energy conservation, the thermal quantum channel is maximally depolarizing within each energy subspace.

The entropy of a channel 𝒩 measures “how thermalizing” 𝒩 is. It quantifies how entropic 𝒩's output is guaranteed to be, for any input state, and even using a reference system as side information. Specifically: if 𝒩 acts on systems A→B and R ≃ A, then S(𝒩) ≡ min_ρ S(B|R), where ρ is any input on AR.

The channel's entropy quantifies how entropic 𝒩's output is guaranteed to be, for any input state, and even using a reference system as side information.

This maximum entropy principle for channels generalizes Jaynes' famous maximum entropy principle for states. But what's the entropy of a channel? Fortunately, we already have an answer thanks to [Devetak et al. quant-ph/0506196], [Wilde and Gour 1808.06980], and [Yuan 1807.05958].

Thermalization with partial information

Sumeet Khatri and I answer this question in arxiv.org/abs/2508.03993 (short) and arxiv.org/abs/2508.03994 (technical). We define the *thermal quantum channel* as the channel with maximal entropy that reproduces expectation values of physically accessible quantities.

arxiv.org

A many-body system, whether in contact with a large environment or evolving under complex dynamics, can typically be modeled as occupying the thermal state singled out by Jaynes' maximum entropy princ...

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Philippe Faist @phfaist.bsky.social · 11/06/2025

We all have a favorite characterization of the thermal state: as an equilibrium state, via a maximum entropy principle, from the resource theory of thermodynamics, as a step in learning algorithms, among others. What is the analogous concept for quantum channels? 🎉 We've got an answer! ⚛️ 🧪 👇

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I’m told the reason they didn’t invite a fifth Nobel prize winner on stage is that they would collapse into a black hole.

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Philippe Faist @phfaist.bsky.social · 30/04/2025

Congratulations, the idea looks beautiful!

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Philippe Faist @phfaist.bsky.social · 01/04/2025

Exciting times ahead for the zoo! More info: errorcorrectionzoo.org/c/surface

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Philippe Faist @phfaist.bsky.social · 25/03/2025

What would you like to parse? Pylatexenc should be able to give you a workable abstract-syntax-tree-like representation of your formula. (You can also transpire a subset of pylatexenc to js, in case it’s useful for your web app.)

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Philippe Faist @phfaist.bsky.social · 26/02/2025

Almost as good as the company! 😀

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Reposted by Philippe Faist

Error Correction Zoo @eczoo.bsky.social · 02/01/2025

Major update: The zoo is not just a repository, but also a taxonomy. Now, every code page has a visualization of the code's primary ancestors. 🧵

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Reposted by Philippe Faist

qctip2025.bsky.social @qctip2025.bsky.social · 05/12/2024

Submissions for QCTiP 2025 are now open.

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Reposted by Philippe Faist

Error Correction Zoo @eczoo.bsky.social · 04/12/2024

You can now BibTeX the thousands of references mentioned in the zoo!

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Philippe Faist @phfaist.bsky.social · 25/11/2024

Hi 👋 would it be possible to add me too? Thanks!

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Philippe Faist @phfaist.bsky.social · 23/11/2024

@physgal.bsky.social could I get added too? Thanks! I’m a researcher in quantum information & computation - phfaist.com