This paper shows how a class of quantum stochastic models can carry the same symmetry and topological information that physicists use to classify one‑dimensional quantum phases. The authors work with hidden quantum Markov models (HQMMs). These are quantum versions of hidden Markov models in which a “hidden” virtual system influences a sequence of observed quantum systems through completely positive maps. The new element is a careful description of how a symmetry group acts on the hidden and observed parts at the same time.

Concretely, the authors let a symmetry group G act projectively on the hidden Hilbert space H and act linearly on the observable Hilbert space K. A projective action means the group may only be represented up to an overall phase, and that ambiguity is captured by a mathematical object called a 2‑cocycle. The paper shows that these symmetry actions on HQMMs are classified by a class [ω] in group cohomology H2(G,U(1)). In plain terms, the same cohomology class that distinguishes symmetry‑protected topological (SPT) phases in one dimension appears naturally in the HQMM description.

The authors develop an operator‑algebraic framework and give precise conditions for invariance under symmetry. The HQMM is built from an initial hidden state and two kinds of completely positive unital maps: one that advances the hidden state and one that emits the observable degrees of freedom. A key structural condition is that the emission map must intertwine the projective action on the hidden space with the linear action on the observable space. This “intertwining” absorbs the projective anomaly at the interface between hidden and observed sectors. The paper treats two causal orderings — a measurement‑then‑evolution ordering and an evolution‑then‑measurement ordering — and proves (Theorem 3.3) that under the symmetry conditions the global HQMM states for both orders are invariant under the combined projective‑linear action of G. The hidden marginal is invariant under the projective action and the observed marginal under the linear action.