Finite-dimensional quantum-to-classical learnability crossover

Determine how to properly generalize the quantum-to-classical crossover in learnability from bosonic Gaussian systems to finite-dimensional quantum systems.

Background

The paper establishes a temperature-dependent crossover in the learnability of bosonic Gaussian states: cold passive Gaussian states require substantially more samples under non-entangled measurements, whereas sufficiently warm states can be learned with classical-like sample complexity. The authors note that an analogous crossover has not yet been explored for finite-dimensional quantum systems.

A direct extension is obstructed by the known hard instances for full tomography of finite-dimensional states. Those instances are perturbations of the maximally mixed state, which already corresponds to an infinite-temperature state, leaving no immediately warmer family whose learnability could be compared. The unresolved problem is therefore to formulate an appropriate finite-dimensional notion of temperature or classicality and characterize the corresponding learnability crossover.

References

As quantum devices surpass this frontier, which ensemble properties remain efficiently learnable? Conversely, are there any features that are provably hard to learn?

— Deep thermalization and Hilbert space ergodicity  (2609.30248 - Mark et al., 24 Sep 2026) in Discussion and outlook, subsection “Experiments, postselection, and learnability”

How to properly generalize the crossover to finite-dimensional systems is thus an interesting open question.

— When are bosonic Gaussian states classical to learn?  (2609.26705 - Chen et al., 22 Sep 2026) in Section 1, Related works, paragraph “Quantum learning theory in finite-dimensional systems”