Cocycle Actions on Hidden Quantum Markov Models: Symmetry Protection and Topological Order

TL;DR

This paper introduces a framework linking symmetry actions, hidden quantum Markov models, and topological order in quantum spin systems, exemplified by the AKLT chain.

Contribution

It develops a symmetry action framework for HQMMs that classifies SPT phases via group cohomology, connecting stochastic processes with topological quantum order.

Findings

Classifies symmetry actions on HQMMs using group cohomology.

Reproduces SPT properties of the AKLT state within the HQMM formalism.

Provides a stochastic, Markovian description of virtual dynamics in SPT phases.

Abstract

We develop a symmetry action framework for hidden quantum Markov models (HQMMs) tailored to one-dimensional quantum spin systems and symmetry-protected topological (SPT) phases. In our setting, a symmetry group $G$ acts projectively on the hidden (virtual) degrees of freedom and linearly on the physical observation space, yielding a global HQMM state that is invariant under the combined action of $G$ for both conventional and causal (input--output) structures. We show that such symmetry actions are naturally classified by a group-cohomology $2$-cocycle $[\omega] \in H^{2}(G,\mathrm{U}(1))$, in direct analogy with the standard cohomological classification of one-dimensional bosonic SPT phases via projective edge representations. As an explicit example, we apply this construction to the Affleck--Kennedy--Lieb--Tasaki (AKLT) chain, where the hidden layer carries a nontrivial class $[\omega] \in H^{2}(\mathrm{SO}(3),\mathrm{U}(1))$ encoding its SPT order. In this case the HQMM formalism reproduces the known SPT properties of the AKLT state while providing a stochastic, Markovian description of the underlying virtual dynamics. Our results establish HQMMs as a natural bridge between quantum stochastic processes, tensor-network descriptions of many-body systems, and symmetry-protected topological order.