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.

• The paper establishes that randomized quantum measurement schemes yield a unique invariant measure under irreducibility, ensuring purification and exponential convergence.
• It introduces multiplicative primitivity as a refined ergodicity criterion, guaranteeing absolute continuity when Kraus operators are almost surely invertible.
• Explicit GAP measure formulas and SageMath examples illustrate practical computation of invariant measures, while open questions in higher dimensions remain.

Invariant Measures of Randomized Quantum Trajectories: A Technical Overview

Introduction

The study of quantum trajectories—Markovian stochastic processes describing the evolution of quantum systems subjected to repeated indirect measurements—constitutes a crucial aspect of modern quantum probability and quantum information theory. The stationary or invariant properties of these trajectories depend sensitively on the measurement scheme, specifically on the observables measured on the quantum probes interacting with the system. The paper "Invariant measures of randomized quantum trajectories" (2603.28664) rigorously investigates the effect of randomizing the choice of the probe observable, focusing on the existence, uniqueness, and regularity properties of invariant measures under various ergodic assumptions on the system's quantum channel.

Mathematical and Physical Framework

The state space is modeled as the complex projective space, with the system's dynamics induced by a quantum channel $\Phi$, represented via Kraus decompositions. Unlike classical Markov chains, the trajectory evolution depends not only on the channel but also on the choice of Kraus operators, isomorphic to a choice of basis or measurement observable on the probe. The freedom afforded by the Stinespring/Kraus theorem allows for randomized selections of these observables, implemented by random sampling over the unitary group $U(k)$ with respect to the Haar measure or more general measures.

Key ergodicity assumptions include:

• Irreducibility of $\Phi$: ensuring a unique full-rank invariant density.
• Primitivity: strict positivity after a finite number of steps.
• Multiplicative primitivity: a newly introduced, intermediate notion strengthening primitivity in terms of the action of repeated Kraus operator applications on arbitrary projective states.

The Markov process for quantum trajectories is formally defined via a Markov kernel on projective space, parameterized by the induced measurement policy and the associated measure on $U(k)$.

Main Results

Uniqueness and Existence of the Invariant Measure

It is shown that for irreducible quantum channels and any non-singular randomization (i.e., the associated measure is not singular with respect to Haar measure on $U(k)$), quantum trajectories admit a unique invariant probability measure. Purification—the property that the trajectory asymptotically approaches pure states—holds under these conditions, implying uniqueness of the invariant measure via the machinery developed in earlier works ([BenFra]). This extends previous analyses beyond deterministic measurement selections.

Quantitative contraction in the Wasserstein-1 metric is proved, ensuring exponential mixing and convergence of the occupation measure, a feature not generally obtainable for quantum trajectories without sufficient randomization.

Ergodicity and Regularity

The introduction of multiplicative primitivity as an ergodicity property enables the authors to establish $\varphi$-irreducibility with respect to the uniform probability measure on projective space for sufficiently randomized trajectories. If, in addition, the Kraus operators are almost surely invertible, the invariant measure is not only unique but also absolutely continuous and equivalent to the uniform (Haar) measure.

Contrastingly, the lack of invertibility for a set of positive measure leads to scenarios where the invariant measure may be singular or have pure point components, as exemplified in constructed counterexamples.

Symmetry and Structure of Invariant Measures

Specializing to uniform randomization (i.e., the Haar measure), the symmetries of the quantum channel $\Phi$ induce corresponding symmetries of the invariant measure. If the channel is $U$-covariant, the invariant measure is invariant under the group of symmetries, and, for maximally symmetric channels (e.g., depolarizing channels), the invariant measure coincides with the uniform measure on the projective space.

Utilizing the theory of GAP (Gaussian Adjusted Projected) measures, the paper provides explicit density formulas for the invariant measures in low-dimensional cases, leveraging properties from quantum information and statistical mechanics. In dimension two, closed-form integral equations characterize the invariant density.

Examples and Limitations

The paper delivers explicit calculation and algebraic criteria for multiplicative primitivity—using tools from algebraic independence and explicit SageMath code—for non-trivial examples in dimensions two and three, including cases where all Kraus operators are non-invertible but multiplicative primitivity still holds.

A critical open question highlighted is the precise relationship between multiplicative primitivity and primitivity in higher dimensions. In $d=2$, these notions coincide, but in $d \ge 3$, the equivalence has not been established nor have counterexamples been found.

Implications and Future Outlook

This work formalizes and extends the ergodic theory of quantum trajectories under randomized measurement schemes, bridging techniques from quantum Markov processes, information theory, and random matrix theory. The results admit the following implications:

• Regularization by Randomization: Systematic randomization of the measurement observable acts as a regularizer, ensuring purification, uniqueness, and often absolute continuity of invariant measures, thereby enabling robust statistical analysis.
• Foundations for Quantum Tomography: The connection to informationally complete instruments reveals that randomized trajectories can be harnessed for complete state tomography and related quantum statistical estimation problems.
• Theory of Ergodicity for Quantum Channels: The new multiplicative primitivity criterion invites further research, potentially influencing the classification of quantum channels arising in open quantum systems, quantum memory, and quantum feedback processes.
• Explicit Computation of Stationary Distributions: For tractable cases, explicit formulas via GAP measures facilitate the study of equilibrium properties and fluctuation relations, crucial for quantum thermodynamics and statistical mechanics.

Theoretical advances stemming from these results could impact the design and analysis of quantum experiments employing randomized measurement protocols, enhancing the controllability and observability of quantum devices.

Conclusion

The systematic investigation of invariant measures for quantum trajectories under randomized measurement schemes exposes a rich structure governed by ergodic properties of the underlying quantum channel and measurement randomization. The work establishes strong uniqueness and regularity results, introduces multiplicative primitivity as a key technical tool, and opens several avenues for future research, notably concerning the classification of ergodicity notions for quantum channels and the explicit construction of invariant measures in higher dimensions (2603.28664).