Deep thermalization in Gaussian continuous-variable quantum systems

Abstract: We uncover emergent universality arising in the equilibration dynamics of multimode continuous-variable systems. Specifically, we study the ensemble of pure states supported on a small subsystem of a few modes, generated by Gaussian measurements on the remaining modes of a globally pure bosonic Gaussian state. We find that beginning from highly entangled, complex global states, such as random Gaussian states and product squeezed states coupled via a deep array of linear optical elements, the induced ensemble attains a universal form, independent of the choice of measurement basis: it is composed of unsqueezed coherent states whose displacements are distributed normally and isotropically, with variance depending on only the particle-number density of the system. We further show that the emergence of such a universal form is consistent with a generalized maximum entropy principle, which endows the limiting ensemble, which we call the "Gaussian Scrooge distribution", with a special quantum information-theoretic property of having minimal accessible information. Our results represent a conceptual generalization of the recently introduced notion of "deep thermalization" in discrete-variable quantum many-body systems -- a novel form of equilibration going beyond thermalization of local observables -- to the realm of continuous-variable quantum systems. Moreover, it demonstrates how quantum information-theoretic perspectives can unveil new physical phenomena and principles in quantum dynamics and statistical mechanics.

• The paper demonstrates that deep thermalization in Gaussian CV systems leads to universal pure state ensembles emerging from Gaussian measurements on multimode states.
• It employs a phase-space approach by analyzing the first two moments and particle-number density to validate the convergence towards unsqueezed coherent states.
• The study links maximum entropy principles to a Gaussian Scrooge distribution, suggesting broad implications for quantum information theory and experimental validation.

Deep Thermalization in Gaussian Continuous-Variable Quantum Systems

Introduction

The paper presents an exploration of universality within multimode continuous-variable (CV) quantum systems, focusing specifically on the dynamics of equilibration. The authors investigate the conditions under which such systems display universal behavior when beginning from highly entangled global bosonic Gaussian states. The research extends the concept of 'deep thermalization,' initially formulated for discrete-variable quantum systems, to the context of CV quantum systems. It is shown that under particular conditions, the ensemble of pure states emerges in a universal form from Gaussian measurements, independent of the initial measurement basis.

Figure 1: Deep thermalization in multimode CV quantum systems. Gaussian measurements on $n-k$ modes are performed on an $n$-mode global BGS.

Projected Ensemble in CV Systems

In CV systems, the projected ensemble (PE) refers to the collection of pure states that arise from the measurements performed on a subset of modes in a globally entangled state. The researchers focus on Gaussian measurements and examine how the resulting PE approaches a universal distribution composed of unsqueezed coherent states with normally distributed displacements. This transformation is dependent solely on the particle-number density of the system, indicating a measure of depletion-free state convergence.

A standard phase-space representation is utilized, with each quantum state expressed by its first two moments: the displacement vector and the covariance matrix. The paper demonstrates how, under Gaussian measurements of globally entangled states, the PE retains universal characteristics across measurement bases.

Analysis of Random and Physical Model BGS

The authors first analyze the PE arising from random pure bosonic Gaussian states (BGS), establishing that it approaches an unsqueezed coherent state with a displacement vector following an isotropic normal distribution. This phenomenon occurs with probability approaching one as the number of modes $n$ becomes large.

Figure 2: Average KL-divergences of (a) the Wigner functions of a 3-mode projected state from a coherent state of equal displacement, and (b) distribution of displacements.

Subsequently, the study extends to PEs from physically constructed states, specifically states with product squeezing parameters coupled via networks of optical elements like beam-splitters. The evolution of the system through such a network demonstrates the same universal asymptotic behavior, reinforcing the theory's breadth and its applicability to different physical settings.

Maximum Entropy Principle and Gaussian Scrooge Distribution

Deep thermalization's universality is linked to the maximization of entropy, analyzed through quantum information-theoretic frameworks. The authors highlight a 'Gaussian Scrooge distribution' as the ensemble obeying maximum entropy principles, thereby minimizing accessible information. This distribution forms a robust theoretical structure underlying the novel behaviors observed in CV systems. The realization of universality in the projected ensemble is discussed concerning this theoretical backdrop, showcasing the interdisciplinary applications across quantum statistical mechanics and quantum information theory.

Figure 3: KL-divergences over time of (a) Wigner functions of a 3-mode projected state from a coherent state of equal displacement, and (b) distribution of displacements.

Conclusion

The paper demonstrates that in the field of continuous-variable quantum systems, the concept of deep thermalization signifies a new form of equilibration, grounded in the universal attainment of a Gaussian Scrooge distribution. This ensemble becomes instrumental for both theoretical predictions and potential experimental validations, offering substantial relevance for advancements in quantum information science. The consistent emergence of this distribution despite differing initial conditions and system setups provides a robust platform for understanding quantum dynamics in multimode CV systems.

Future investigations may include the exploration of varying initial states, the growth of particle-number densities, and the extension of the concepts to non-Gaussian measurements, potentially expanding the theoretical foundation and practical scope of deep thermalization.