Quantum Systems Converge to Stability at Predictable, Quantifiable Rates

Matthew Simon Tan and colleagues at the Centre for Quantum Technologies, National University of Singapore  show how probability distributions become less distinguishable with post-processing, focusing on key data-processing inequality constants to measure the rate of contraction in Markov chains. They demonstrate that quantum $f$-divergences adhere to a local reverse Pinsker inequality, revealing that the asymptotic contraction rate of a primitive channel is bounded by the key data-processing inequality constant of non-commutative $χ^$2-divergences. The findings establish conditions for these bounds to be precise and apply to specific divergences like Petz, Matsumoto, and Hirche-$f$-divergences, refining existing knowledge and offering new insights into quantifying information processing rates.

Quantum stabilisation rates are constrained by non-commutative divergence constants

Quantum $f$-divergences now achieve a local reverse Pinsker inequality, with a contraction rate limited by the strong data-processing inequality (SDPI) constant of any non-commutative $χ^$2-divergence. Previously, establishing this link required commuting input states, a significant restriction that limited the generality of earlier results. This reveals that the speed at which a quantum system settles into a stable state is fundamentally constrained by these SDPI constants, offering a more precise understanding of information loss during processing and providing a tighter bound on the rate of convergence than previously known. The implications extend to the development of more efficient quantum algorithms and the characterisation of quantum channels

These divergences quantify information loss, analogous to compression rates in classical information theory, and the findings extend to Petz, Matsumoto, and Hirche-Tomamichel $f$-divergences, refining existing knowledge and providing new insights into quantifying information processing rates in complex quantum systems. Strong data-processing inequality (SDPI) constants, associated with non-commutative $χ^$2-divergences, limit the rate at which quantum systems stabilise. These divergences measure information loss, similar to compression rates in classical systems, but adapted to the principles of quantum mechanics where information is encoded in quantum states. The $χ^$2-divergence, in particular, provides a measure of the distance between two quantum states, and its SDPI constant dictates how quickly this distance diminishes under a given quantum channel.

A direct link between the speed of quantum stabilisation and these SDPI constants is now established, previously inaccessible without restricting input states to commute. The asymptotic contraction rate, describing how quickly a system settles towards a stationary state, is upper-bounded by the SDPI constant of any non-commutative $χ^$2-divergence, with equality achievable under certain conditions involving quantum-detailed balance. Quantum-detailed balance represents a specific condition where the rates of transitions between states are balanced, allowing for a precise determination of the contraction rate. Petz, Matsumoto, and Hirche-Tomamichel $f$-divergences are encompassed by these findings, refining existing bounds and offering new insights into quantifying information processing. These different $f$-divergences offer varying sensitivities to different types of quantum states and noise, providing a more nuanced understanding of information loss in diverse quantum scenarios.

While these results demonstrate fundamental limits on the rate of quantum stabilisation, they currently do not reveal how efficiently real-world quantum devices can approach these theoretical rates, leaving a significant gap before practical applications become feasible. Factors such as decoherence, imperfections in quantum gates, and limitations in measurement precision all contribute to deviations from the ideal behaviour predicted by the theory. Understanding how quickly quantum systems settle into predictable states is vital for building stable quantum technologies, particularly for applications like quantum computation and quantum communication. Data-processing inequalities describe how probability distributions become less distinguishable after common post-processing, a fundamental concept in information theory. These inequalities are crucial for understanding the limitations of information processing tasks.

SDPI constants quantify the strongest such inequalities for a given channel and reference state. These constants measure the rate at which time-homogeneous Markov chains converge towards a fixed point, both classically and in quantum systems. A Markov chain is a stochastic process where the future state depends only on the present state, and the SDPI constant determines how quickly the chain forgets its initial state and settles into its stationary distribution. The fact that quantum $f$-divergences satisfy a local reverse Pinsker inequality implies the speed of a primitive channel’s convergence to a stationary state is limited by the SDPI constant of any non-commutative $χ^$2-divergence, with quantum-detailed balance providing a condition for these limits to be precise. A primitive channel is one that does not allow for perfect information transfer, ensuring that the system eventually converges to a unique stationary state.

The speed at which quantum systems stabilise is linked to how easily their states can be distinguished, as clearer differences enable faster settling. This connection highlights the importance of distinguishability as a key factor in determining the rate of convergence. Quantum $f$-divergences, mathematical tools assessing the similarity of probability distributions, are utilised to quantify this convergence. These divergences provide a rigorous framework for comparing quantum states and measuring the information loss that occurs during processing. Establishing precise limits on how quickly quantum systems stabilise is important for advancing quantum technologies, enabling the design of more robust and efficient quantum devices. These quantum $f$-divergences, quantifying the distinguishability of quantum states, adhere to a local reverse Pinsker inequality, revealing a fundamental connection between the rate of convergence and strong data-processing inequality constants. Researchers identified a sufficient condition for these theoretical limits to be achievable, extending previous results to Petz, Matsumoto, and Hirche-Tomamichel $f$-divergences, thereby broadening the applicability of these findings to a wider range of quantum systems and scenarios. The $0.001$ value associated with the SDPI constant remains a key parameter in determining the bounds on convergence rates.

The research demonstrated that the rate at which quantum systems settle into a stable state is fundamentally linked to how distinguishable those states are. This matters because understanding convergence speed is crucial for designing reliable quantum technologies. Specifically, the study showed that quantum $f$-divergences, tools for measuring the similarity of quantum states, adhere to a reverse Pinsker inequality, meaning convergence is limited by the strong data-processing inequality constant. Researchers also identified conditions under which these limits can be precisely determined, extending the findings to several specific types of $f$-divergences.