Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact model reduction for discrete-time conditional quantum dynamics

Tommaso Grigoletto, Francesco Ticozzi

TL;DR

This work presents an exact dimension-reduction framework for discrete-time conditional quantum dynamics that preserves measurement statistics and observable expectations. By combining minimal realizations with conditional expectations in finite-dimensional operator algebras, it constructs reduced conditional evolutions and output maps that reproduce outputs and trajectory probabilities while maintaining physical constraints. The key contributions include defining the non-observable subspace, achieving linear reductions when CPTP structure is not required, and developing a full algebraic reduction pipeline using $*$-algebras, Wedderburn decompositions, and conditional expectations. Through two detailed applications—measured quantum walks and an Ising spin chain—the authors demonstrate that conditioning can yield substantial reductions and reveal a classical-like core within quantum trajectories, with broad implications for efficient quantum filtering and feedback control.

Abstract

Leveraging an algebraic approach built on minimal realizations and conditional expectations in quantum probability, we propose a method to reduce the dimension of quantum filters in discrete-time, while maintaining the correct distributions on the measurement outcomes and the expectations of some relevant observable. The method is presented for general quantum systems whose dynamics depend on measurement outcomes, hinges on a system-theoretic observability analysis, and is tested on prototypical examples.

Exact model reduction for discrete-time conditional quantum dynamics

TL;DR

This work presents an exact dimension-reduction framework for discrete-time conditional quantum dynamics that preserves measurement statistics and observable expectations. By combining minimal realizations with conditional expectations in finite-dimensional operator algebras, it constructs reduced conditional evolutions and output maps that reproduce outputs and trajectory probabilities while maintaining physical constraints. The key contributions include defining the non-observable subspace, achieving linear reductions when CPTP structure is not required, and developing a full algebraic reduction pipeline using -algebras, Wedderburn decompositions, and conditional expectations. Through two detailed applications—measured quantum walks and an Ising spin chain—the authors demonstrate that conditioning can yield substantial reductions and reveal a classical-like core within quantum trajectories, with broad implications for efficient quantum filtering and feedback control.

Abstract

Leveraging an algebraic approach built on minimal realizations and conditional expectations in quantum probability, we propose a method to reduce the dimension of quantum filters in discrete-time, while maintaining the correct distributions on the measurement outcomes and the expectations of some relevant observable. The method is presented for general quantum systems whose dynamics depend on measurement outcomes, hinges on a system-theoretic observability analysis, and is tested on prototypical examples.
Paper Structure (14 sections, 4 theorems, 24 equations, 1 algorithm)

This paper contains 14 sections, 4 theorems, 24 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model and problem definition
  3. Dynamics of interest
  4. Quantum Conditional Evolutions
  5. Observability and Linear Reduction
  6. Linear model reduction
  7. Quantum model reduction
  8. $*$-Algebras and conditional expectations
  9. Quantum reduction
  10. Reduction of measurements and dynamics, separately
  11. Applications
  12. Measured quantum walks
  13. Measured Ising spin chain
  14. Conclusions

Key Result

Proposition 1

$\mathscr{N}$ is $\mathcal{M}_{k}$-invariant for all $k\in\Omega$, and it is contained in $\ker\mathcal{C}$.

Theorems & Definitions (11)

  • Definition 1
  • Remark 1
  • Definition 2: Non-observable subspace
  • Proposition 1
  • proof
  • Proposition 2
  • Definition 3: Output algebra
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 1 more