Quantifying Irreversibility via Bayesian Subjectivity for Classical & Quantum Linear Maps

TL;DR

The paper proposes Bayesian subjectivity as an information-theoretic irreversibility measure for classical and quantum maps, defining it as the prior-dependence of Bayesian retrodiction via the classical hat{\varphi}_{\gamma} and quantum hat{\mathcal{F}}_{\gamma} maps. It shows that subjectivity vanishes for bijective/unitary dynamics and peaks for erasure-like processes, with analytical results and extensive numerics across bits, qubits, and qutrits supporting the link to quasistaticity, entropy production, and the data-processing paradigm. The work also uncovers subtle reversibility features through absorbing and pseudo-absorbing maps, including ridges in trit systems that reflect embedded deterministic transitions. Overall, Bayesian subjectivity serves as a principled, priorsensitive irreversibility metric that integrates geometric, thermodynamic, and information-theoretic perspectives, with clear avenues for analytic DPI proofs and higher-dimensional generalizations.

Abstract

In both classical and quantum physics, irreversible processes are described by maps that contract the space of states. The change in volume has often been taken as a natural quantifier of the amount of irreversibility. In Bayesian inference, loss of information results in the retrodiction for the initial state becoming increasingly influenced by the choice of reference prior. In this paper, we import this latter perspective into physics, by quantifying the irreversibility of any process with its Bayesian subjectivity -- that is, the sensitivity of its retrodiction to one's prior. From this perspective, we review analytical and numerical results that highlight both intuitive and subtle insights that this measure sheds on irreversible processes.

Figures (8)

• Figure 1: An illustration of the information geometric action of various, generic maps on their respective state spaces. Namely, for generic maps acting on qubits, trits and bits. $\bar{\mathcal{F}}$ and $\bar{\varphi}$ are the respective channels iterated over an arbitrarily long time \ref{['eq:ite']}.
• Figure 2: An illustration of various edge cases of reversibility and irreversibility, for the quantum and classical maps, and their respective Bayesian retrodictions.
• Figure 3: A cartoon illustration of the key interest in this work: quantifying the irreversibility of a map by a formal measure of the dependence of its Bayesian inversion on reference priors. That is, understanding physical irreversibility of processes through the subjectivity involved in doing Bayesian inference for their past inputs.
• Figure 4: Colour density plots for bit, qubit and trit channels of Bayesian subjectivity $\mathrm{I}^\mathsf{s}$ against absolute determinant and the fixed centroid distance.
• Figure 5: Colour density plots for bit, qubit and trit channels of average change in divergence $\mathrm{I}^\mathsf{d}$ against absolute determinant and the fixed centroid distance.

Theorems & Definitions (31)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• Theorem 1
• Theorem 2
• Theorem 3