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Alternating Projections Methods for Discrete-time Stabilization of Quantum States

Francesco Ticozzi, Luca Zuccato, Peter D. Johnson, Lorenza Viola

Abstract

We study sequences (both cyclic and randomized) of idempotent completely-positive trace-preserving quantum maps, and show how they asymptotically converge to the intersection of their fixed point sets via alternating projection methods. We characterize the robustness features of the protocol against randomization and provide basic bounds on its convergence speed. The general results are then specialized to stabilizing en- tangled states in finite-dimensional multipartite quantum systems subject to a resource constraint, a problem of key interest for quantum information applications. We conclude by suggesting further developments, including techniques to enlarge the set of stabilizable states and ensure efficient, finite-time preparation.

Alternating Projections Methods for Discrete-time Stabilization of Quantum States

Abstract

We study sequences (both cyclic and randomized) of idempotent completely-positive trace-preserving quantum maps, and show how they asymptotically converge to the intersection of their fixed point sets via alternating projection methods. We characterize the robustness features of the protocol against randomization and provide basic bounds on its convergence speed. The general results are then specialized to stabilizing en- tangled states in finite-dimensional multipartite quantum systems subject to a resource constraint, a problem of key interest for quantum information applications. We conclude by suggesting further developments, including techniques to enlarge the set of stabilizable states and ensure efficient, finite-time preparation.

Paper Structure

This paper contains 18 sections, 23 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminary material
  3. Models and stability notions
  4. Fixed points of CPTP maps
  5. Alternating projection methods
  6. von Neumann-Halperin Theorem
  7. CPTP projections and orthogonality
  8. Convergence rate
  9. Robustness with respect to randomization
  10. Quasi-Local State Stabilization
  11. Locality notion
  12. Invariance conditions and minimal fixed point sets
  13. Stabilizability under quasi-locality constraints
  14. Physical interpretation: discrete-time quasi-local stabilizability is equivalent to cooling without frustration
  15. Classes of discrete-time QLS states
  16. ...and 3 more sections

Key Result

Proposition 1

If a set $\mathcal{S}$ is invariant for the dynamics $\{\mathcal{E}_{t,s}\}_{t,s\geq 0}$, then it is simply stable.

Theorems & Definitions (28)

  • Definition 1
  • Proposition 1
  • Theorem 1: Fixed-point sets, generic case
  • Theorem 2
  • Theorem 3: Fixed-point sets, general case
  • Theorem 4: von Neumann-Halperin alternating projections
  • Proposition 2
  • Definition 2
  • Lemma 1
  • Proposition 3
  • ...and 18 more