Should also mention concurrent and complementary work tat came out today (scirate.com/arxiv/2509.2...) by @jjmeyer.bsky.social et al.
He also wrote a nice thread ⛓️ about relative entropies und why computational constraints we both consider matter. Check it out

As a fun aside, I am very happy with the continuity bound and its proof. It contains, I think, a very fun and beautiful, but out of context meaningless formula that I want to leave you with. Made me reflect about beauty in maths. And I'd never thought so many different Ms could have real meaning.

⚛️ Computational Quantum Resources Theory:

We introduce complexity-aware resource measures, prove an asymptotic continuity bound, and demonstrate explicit separations from the information-theoretic regime (e.g., entanglement) implying that computational restrictions do matter in practice.

🔎 Computational Hypothesis Testing:

Even with many copies, the asymmetric hypothesis-testing exponent (Steins exponent) achievable by efficient measurements is upper-bounded by the regularized computational measured relative entropy.

✨ We introduce computational versions of the max-divergence (via some beautiful conical structures in QIT) and measured Rényi divergences. We analyze their behavior under efficient operations and show that they from a cohesive framework (for α→∞ they coincide).
Further we consider two applications

In practice, experiments are fundamentally bound to efficiently implementable operations. 🧪

Together with Alvaro Yángüez and Thomas A. Hahn, we formalize quantum state discrimination and resource quantification under these efficiency constraints. 💻

I’ll be giving a talk about this work at Beyond IID this Thursday, which will be recorded and live streamed should you be interested!
(sites.google.com/view/beyondi...)

We present a ‚quantum’ extension of mixed matrix norms showing hardness results for among other the tasks of computing the minimal output Rényi entropy of entanglement breaking (EB) channels (1->p) and the optimal one-shot distinguishability of a difference of EB channels (1->1).

And I am thankful to my coauthors and teachers @angelacapel.bsky.social, @alvalhambra.bsky.social, and Cambyse Rouzé for your guidance and patience along the way, and from whom I learned and continue to learn a lot.

Here’s its SciRate entry: scirate.com/arxiv/2502.0...
ツ

This was a really enjoyable joint work with Omar Fawzi, Cambyse Rouzé, and Thomas van Himbeeck.

arxiv.org/abs/2502.01611

These norms can be defined for arbitrary many indices. In particular for two they give nice expression for certain entropic quantities, which are why most applications restrict to those.

Importantly we give more tractable formulas for 3+ indexed ones opening the way to many more QI-applications

Our main technical tool are norms on so called operator values Schatten spaces. We can these ‚multi-index Schatten norms‘.

Even though they have been knows since ~80, their usefulness is QIT was realized in ~06, yet they still seems somewhat niece in the QI community.

On the applications side do we generalize and give new results that are of interest in quantum cryptography and e.g. for entropy accumulation theorems.

But in particular do we want to highlight the bridge and usefulness of operator space in quantum information theory.
See also [Beigi,Goodarzi 22]

As for non-hypercubic systems, usually if the growth constant or the degree is bounded qualitatively similar results should hold. Ours pretty surely extend.
Otherwise you may need much stronger assumptions to get decay.

Q.random walks are also a tool to prove efficiency state preparation, but I am not an expert on that. I think you also points to what happens to correlations over time (OTOC) which is interesting to look into. They can prob. also yield rapid mixing if you look at the right (prob. entropic) ones.