Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariant measures of randomized quantum trajectories

Tristan Benoist, Sascha Lill, Cornelia Vogel

Abstract

Quantum trajectories are Markov chains modeling quantum systems subjected to repeated indirect measurements. Their stationary regime depends on what observables are measured on the probes used to indirectly measure the system. In this article we explore the properties of quantum trajectories when the choice of probe observable is randomized. The randomization induces some regularization of the quantum trajectories. We show that non-singular randomization ensures that quantum trajectories purify and therefore accept a unique invariant probability measure. We furthermore study the regularity of that invariant measure. In that endeavour, we introduce a new notion of ergodicity for quantum channels, which we call multiplicative primitivity. It is a priory stronger than primitivity but weaker than positivity improving. Finally, we compute some invariant measures for canonical quantum channels and explore the limits of our assumptions with several examples.

Invariant measures of randomized quantum trajectories

Abstract

Quantum trajectories are Markov chains modeling quantum systems subjected to repeated indirect measurements. Their stationary regime depends on what observables are measured on the probes used to indirectly measure the system. In this article we explore the properties of quantum trajectories when the choice of probe observable is randomized. The randomization induces some regularization of the quantum trajectories. We show that non-singular randomization ensures that quantum trajectories purify and therefore accept a unique invariant probability measure. We furthermore study the regularity of that invariant measure. In that endeavour, we introduce a new notion of ergodicity for quantum channels, which we call multiplicative primitivity. It is a priory stronger than primitivity but weaker than positivity improving. Finally, we compute some invariant measures for canonical quantum channels and explore the limits of our assumptions with several examples.

Paper Structure

This paper contains 26 sections, 23 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Setup and assumptions
  3. Main results
  4. Uniqueness of the invariant measure
  5. $\nu_{\textnormal{unif}}$-irreducibility
  6. Uniformly randomized measurement and symmetries
  7. On multiplicative primitivity
  8. Proof of the main results
  9. Purification: Proof of \ref{['thm:purification']}
  10. Regularity: Proof of \ref{['thm:phi-irred', 'thm:absolute continuity']}
  11. Symmetries: Proof of \ref{['thm:symmetries']}
  12. Examples
  13. Invariant measure non-equivalent to uniform
  14. Uniformly randomized quantum trajectories with explicit $\nu_{\textnormal{inv}}$
  15. Uniformly randomized in dimension $2$
  16. ...and 11 more sections

Key Result

Theorem 3.1

Assume $\Phi$ is irreducible and $\mu$ is non-singular, see def:irrdef:non singular. Then, $\Pi$ accepts a unique invariant measure $\nu_{\textnormal{inv}}$ on ${{\mathrm P}({\mathbb C}^{d}) }$, and there exist $m\in \{1,\dotsc,d\}$ and two constants $C>0$ and $\gamma\in[0,1)$ such that for any pro

Theorems & Definitions (56)

  • Definition 2.1: Non-singular $\mu$
  • Definition 2.2: Irreducible channels
  • Definition 2.3: Primitive channels
  • Definition 2.4: Multiplicative primitivity
  • Theorem 3.1
  • Definition 3.2: Purification
  • Theorem 3.3
  • Definition 3.4: Proposition 4.2.1 in MT
  • Theorem 3.5
  • Theorem 3.6
  • ...and 46 more