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

Abdessatar Souissi, Amenallah Andolsi

TL;DR

This work presents a HQMM formulation of the AKLT chain, separating hidden virtual spin-½ degrees of freedom from the observable spin-1 chain. It provides an explicit HQMM decomposition of the AKLT state through a triplet (phi0, EH, EO_H) and shows that the hidden dynamics encode the chain's entanglement via a Stinespring dilation, while the emission channel carries the AKLT tensors in a G-equivariant manner. The analysis demonstrates that the encoding channel is maximally entangled (1 ebit) through its Choi state, and that SPT order is embedded as covariance of EO_H under the symmetry group. The framework links tensor-network descriptions with quantum stochastic processes, offering new perspectives for topological order in quantum information and potential extensions to higher-dimensional SPTs and MBQC.

Abstract

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.

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

TL;DR

This work presents a HQMM formulation of the AKLT chain, separating hidden virtual spin-½ degrees of freedom from the observable spin-1 chain. It provides an explicit HQMM decomposition of the AKLT state through a triplet (phi0, EH, EO_H) and shows that the hidden dynamics encode the chain's entanglement via a Stinespring dilation, while the emission channel carries the AKLT tensors in a G-equivariant manner. The analysis demonstrates that the encoding channel is maximally entangled (1 ebit) through its Choi state, and that SPT order is embedded as covariance of EO_H under the symmetry group. The framework links tensor-network descriptions with quantum stochastic processes, offering new perspectives for topological order in quantum information and potential extensions to higher-dimensional SPTs and MBQC.

Abstract

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.

Paper Structure

This paper contains 8 sections, 6 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hidden Quantum Markov Models
  4. The AKLT State and its SPT Order
  5. HQMM Structure of the AKLT State
  6. Entanglement Dynamics in Hidden Markov Channel
  7. SPT order on the AKLT,s HQMMs
  8. Conclusion

Key Result

Lemma 3.1

The map $\mathcal{E}_H: \mathcal{B}(\mathcal{H}_{\frac{1}{2}}) \otimes \mathcal{B}(\mathcal{H}_{\frac{1}{2}}) \to \mathcal{B}(\mathcal{H}_{\frac{1}{2}})$ defined by the Stinespring dilation where $V: \mathcal{H}_{\frac{1}{2}} \to \mathcal{H}_{\frac{1}{2}} \otimes \mathcal{H}_{\frac{1}{2}}$ is the partial isometry from (eq:V), constitutes a hidden transition expectation CPIP. The dual quantum chan

Figures (1)

  • Figure 1: Schematic diagram with eight qubits on the top row, grouped pairwise into singlet states (dashed ellipses), and projected downward into three spin-$1$ triplet nodes on the lower row. The increased vertical spacing highlights the correspondence between the two levels.

Theorems & Definitions (14)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • proof
  • Theorem 4.1
  • ...and 4 more