University of Western Ontario: Researchers Model Quantum Entropy Fluctuations with a Novel Boltzmann Bridge Approach

Sina Kazemian and colleagues at University of Western Ontario, Technische Universit¨at Berli and National Research Council Canada have created a quantum version of the Boltzmann Bridge, a concept detailing entropy histories dependent on both starting and ending macrostates. The research shows that directly quantising classical models does not automatically recreate classical entropy behaviour, but it can arise statistically by including hidden microstates within a closed quantum system. This provides new understanding into how the thermodynamic arrow of time emerges and the link between quantum dynamics and classical entropy conditioning.

Suppression of quantum revivals via increased Hilbert space dimensionality mimics classical entropy

Increasing the internal Hilbert-space dimension of a closed quantum system suppresses revival behaviour, reducing oscillatory entropy fluctuations from approximately 80% to less than 20% as measured by the Random Forest classifier. Previously, minimal models consistently exhibited coherent oscillations dominating entropy profiles, but the introduction of hidden microstates and increased Hilbert-space complexity statistically produces entropy profiles resembling the classical Boltzmann Bridge. A dimension of 16 sharply reduced oscillatory behaviour compared to the minimal two-state model, with the Random Forest classifier indicating a shift from approximately 80% revival-dominated profiles to less than 20% exhibiting classical-like coarse-grained characteristics. This suppression of revivals occurred despite the underlying quantum dynamics remaining entirely unitary, demonstrating that classical entropy behaviour isn’t necessarily reliant on external dissipation.

The phenomenon of ‘quantum revival’ describes the non-monotonic time evolution of wave packet overlap in quantum systems, leading to periodic returns to the initial state. In simpler quantum systems, these revivals manifest as oscillatory behaviour in entropy calculations, hindering the emergence of classical entropy profiles. The researchers addressed this by expanding the system’s Hilbert space, the mathematical space encompassing all possible quantum states. Each additional dimension represents a hidden microstate, effectively increasing the system’s internal degrees of freedom. This isn’t a change to the fundamental unitary evolution of the system, energy is still conserved, and the system remains closed, but rather an increase in the number of available quantum pathways. The Random Forest classifier, a machine learning algorithm, was employed to distinguish between entropy profiles dominated by revivals and those exhibiting characteristics more akin to classical behaviour. The classifier’s ability to accurately categorise profiles was significantly reduced as the Hilbert space dimension increased, indicating a transition towards classical-like entropy evolution. Specifically, the classifier’s accuracy in identifying revival-dominated profiles dropped from approximately 80% in the two-state model to less than 20% with a dimension of 16. This suggests that the increased complexity effectively ‘washes out’ the coherent oscillations, leading to a smoother, more classical entropy profile. The Boltzmann entropy definition used in this work is based on counting the number of accessible microstates corresponding to a given macrostate, a standard approach in statistical mechanics.

Emergent classical entropy and the role of Hilbert space dimensionality

Researchers at University of Western Ontario, National Research Council Canada, and Technische Universität Berlin have successfully demonstrated that classical entropy behaviour can arise within entirely quantum systems, a finding with implications for understanding the thermodynamic arrow of time. The approach relies on progressively increasing the complexity of quantum models via the dimension of their ‘Hilbert space’, a measure of the available states within the system. Doubts may arise regarding a truly fundamental connection to classical thermodynamics, given the need to progressively increase the complexity of these quantum models and, specifically, the dimension of the ‘Hilbert space’ which defines the number of possible states.

The classical Boltzmann Bridge describes entropy histories conditioned on both an initial low-entropy macrostate and a later macrostate. Unlike the traditional formulation of the thermodynamic arrow of time, which focuses solely on past conditioning, the Boltzmann Bridge considers both past and future states. This two-time conditioning allows for entropy profiles that can initially rise above the final entropy value before decreasing towards the imposed endpoint, a behaviour not typically observed in systems governed by past-only entropy considerations. The researchers constructed quantum analogues of this bridge using macro-subspace projectors, which effectively isolate the relevant macrostates, and unitary time evolution, ensuring the system’s quantum coherence. The key innovation lies in the systematic increase of the Hilbert space dimension, effectively introducing a multitude of hidden microstates that influence the system’s dynamics.

This work offers a pathway to reconcile quantum mechanics with our everyday experience of time’s arrow, even if achieving this requires increasingly intricate quantum simulations. Classical entropy behaviour, a measure of disorder, can emerge within entirely quantum systems, as shown by scientists at University of Western Ontario, National Research Council Canada, and Technische Universität Berlin. Their approach involved formulating quantum models mirroring the classical Boltzmann Bridge, a concept used to examine how entropy changes over time when both starting and ending conditions are known. By introducing internal complexity to these closed quantum systems, specifically, by increasing the dimension of the ‘Hilbert space’, effectively the number of possible states, quantum oscillations were suppressed and entropy profiles resembling classical predictions were observed. The sign structure and overall shape of the entropy profiles increasingly aligned with the classical Boltzmann Bridge as the hidden microstate space expanded, suggesting a statistical emergence of classicality. However, these results currently focus on simplified models and do not yet demonstrate how to achieve this behaviour in systems with a comparable level of complexity to those found in biological or materials science applications. Future research will need to explore whether this approach can be scaled to more realistic systems and whether the observed statistical emergence of classicality is robust enough to account for the complexities of the macroscopic world. The implications extend to our understanding of the foundations of thermodynamics and the potential for developing quantum technologies that mimic classical behaviour without succumbing to decoherence.

The research demonstrated that classical entropy behaviour can emerge within closed quantum systems. Scientists achieved this by creating quantum models analogous to the classical Boltzmann Bridge and systematically increasing the complexity of these systems via the dimension of the Hilbert space. This increase suppressed quantum oscillations and resulted in entropy profiles that more closely resembled classical predictions. The findings offer insight into the foundations of thermodynamics and how the arrow of time may be reconciled with quantum mechanics, and future work will explore scaling this approach to more complex systems.