A Hidden Quantum Markov model framework for Entanglement and Topological Order in the AKLT Chain

This paper introduces a hidden quantum Markov models (HQMMs) framework to the Affleck-Kennedy-Lieb-Tasaki (AKLT) state-a cornerstone example of a symmetry-protected topological (SPT) phase. The model’s observation system is the physical spin-1 chain, which emerges from a hidden spin-1/2 layer through well-defined quantum emission operation. We show that the underlying Markov dynamics caputure maximal entanglement through the use of significant channels relevant to the AKLT state. We also show that SPT order induces a covariance on the observation decoding channels. This establishes an additional bridge between the quantum Machine learning and many-body physics, with promising implication in topological order and quantum information.

💡 Research Summary

**
The paper presents a novel framework that casts the Affleck‑Kennedy‑Lieb‑Tasaki (AKLT) spin‑1 chain—a paradigmatic example of a symmetry‑protected topological (SPT) phase—into the language of Hidden Quantum Markov Models (HQMMs). An HQMM consists of a hidden quantum system and an observable output system linked by completely positive, trace‑preserving (CPTP) maps: an initial state ϕ₀ on the hidden algebra, a hidden transition channel E_H, and an emission channel E_{O,H} that couples the hidden and observable degrees of freedom.

In this work the hidden system is a virtual spin‑½ space (ℂ²) and the observable system is the physical spin‑1 space (ℂ³). The authors construct a generating triplet Ξ_AKLT = (ϕ₀, E_H, E_{O,H}) such that the sequential application of the combined map
E_{X,Y}(·) = E_H ∘