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arXiv · 2605.09605

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

Abdessatar Souissi·Abdessatar Barhoumi
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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.

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Abdessatar Souissi, Abdessatar Barhoumi. 2026-05-10. Cocycle Actions on Hidden Quantum Markov Models: Symmetry Protection and Topological Order. https://arxiv.org/abs/2605.09605

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