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Causal Architecture in Hidden Quantum Markov Models

Abdessatar Souissi, Abdessatar Barhoumi

Abstract

We introduce a class of causal hidden quantum Markov models (cHQMMs) that refine standard HQMMs by explicitly reversing the order between hidden updates and emissions. Through a minimal qubit model, we show that the conventional "emission-then-transition" and the alternative "transition-then-emission" architectures generally generate nonequivalent quantum processes, with distinct temporal correlation structures and different patterns of entanglement across time. At the same time, we prove that these two classes coincide for entangled lifting of classical hidden Markov models, where they share the same classical reduced process, thereby identifying a sharp boundary between classical and genuinely quantum hidden memory. These features suggest potential utility for modeling and analyzing quantum memory in sequential quantum processes.

Causal Architecture in Hidden Quantum Markov Models

Abstract

We introduce a class of causal hidden quantum Markov models (cHQMMs) that refine standard HQMMs by explicitly reversing the order between hidden updates and emissions. Through a minimal qubit model, we show that the conventional "emission-then-transition" and the alternative "transition-then-emission" architectures generally generate nonequivalent quantum processes, with distinct temporal correlation structures and different patterns of entanglement across time. At the same time, we prove that these two classes coincide for entangled lifting of classical hidden Markov models, where they share the same classical reduced process, thereby identifying a sharp boundary between classical and genuinely quantum hidden memory. These features suggest potential utility for modeling and analyzing quantum memory in sequential quantum processes.
Paper Structure (8 sections, 7 theorems, 98 equations)

This paper contains 8 sections, 7 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Causal hidden quantum Markov models
  4. A qubit HQMM: conventional vs. causal
  5. Distinguishability of conventional and causal HQMMs
  6. Choi entanglement of qubit block maps
  7. Equivalence of Causal and Conventional Entangled Hidden Quantum Markov Models
  8. Discussion and Outlook

Key Result

Lemma 3.3

[Dual block maps in Kraus form] Fix a time step $n$ and suppose that the hidden transition and emission expectations admit minimal Kraus decompositions where $K_{H;\alpha}:\mathcal{H}_n\to\mathcal{H}_n\otimes\mathcal{H}_{n+1}$ and $K_{H,O;\beta}:\mathcal{H}_n\to\mathcal{H}_n\otimes\mathcal{K}_n$ satisfy $\sum_{\alpha}K_{H;\alpha}^{*}K_{H;\alpha} = \sum_{\beta}K_{H,O;\beta}^{*}K_{H,O;\beta} = \mat

Theorems & Definitions (18)

  • Definition 2.1: Diamond distance
  • Definition 2.2: Choi operator
  • Definition 3.1
  • Definition 3.2: Causal hidden quantum Markov model
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 8 more