Quantum measurements and equilibration: the emergence of objective outcomes via entropy maximisation

TL;DR

This work introduces the Measurement-Equilibration Hypothesis, positing that quantum measurements emerge from entropy-maximizing equilibration of a system and its environment under unitary dynamics, rather than from a fundamental postulate. By analyzing equilibration on average, it shows that exact encoding of measurement outcomes into multiple observers via Spectrum Broadcast Structure is impossible for generic closed-system dynamics, though approximate objectivity can be achieved by coherently aggregating many microscopic observers into macro-observers. The results contrast with the standard measurement model, which requires temporally controlled interactions to generate system–environment correlations; under spontaneous equilibration, such correlations do not arise unless a more complex evolution is considered. The paper also outlines practical routes to approximate projective measurements through coarse-graining and discusses extensions, including pre-measurement stages and relaxations of independence assumptions, pointing toward a thermodynamically consistent framework for quantum measurements and the quantum–classical transition.

Abstract

The measurement postulate of quantum theory stands in conflict with the laws of thermodynamics and has evoked debate regarding what actually constitutes a measurement. With the help of modern quantum statistical mechanics, we take the first step in formalising the hypothesis that quantum measurements are driven by the natural tendency of closed systems to maximize entropy, a notion that we call the Measurement-Equilibration Hypothesis. In this paradigm, we investigate how classical measurement outcomes can emerge within a purely unitary framework, and find that: (i) the interactions used in standard measurement models fail to spontaneously encode information classically and (ii) while ideal projective measurements are impossible, one can (for a given form of Hamiltonian) approximate them exponentially well as more physical systems are collected together into an ``observer'' system. We thus lay the groundwork for self-contained models of quantum measurement, proposing improvements to our simple scheme.

Figures (3)

• Figure 1: Illustration of the equilibration (on average) of an observable $A$ over time. The difference between the instantaneous expectation value of an observable $A$ at some time $t$ and its equilibrium value, decreases over time, despite the system continuing to evolve unitarily. The inset illustrates finite fluctuations around the equilibrium, which generally depend on the effective dimension of the system, in accordance with Eq. \ref{['Eq:reimannbound']}.
• Figure 2: Measurement as an equilibration process. The system of interest $S$ and some observer systems $\lbrace\mathcal{E}_{q}\rbrace_{q}$ begin in an uncorrelated, out-of-equilibrium initial state. During the measurement, the collective system undergoes closed-system equilibration, correlating them in the measurement basis $\lbrace\left|\right.\!{i}\!\left.\right\rangle\rbrace_{i}$, and encoding the corresponding probabilities $\lbrace p_i=\left\langle\right.\!{i}\!\left.\right|\rho_{\mathrm{S} ,0}\left|\right.\!{i}\!\left.\right\rangle\rbrace_{i}$ in the observer systems in a redundant and objective (i.e. classical) manner.
• Figure 3: A schematic representation of the conditional Hamiltonian given in Eq. \ref{['eq:InteractionHamiltonianAprr']}. It depicts the interaction of system $\mathcal{H}_\mathrm{S}$ (in yellow) with individual parts of the environment $\mathcal{H}_k$ (in red). The coupling strength between the system and each environment is indicated by the dashed lines and the labels $c_k$. The light-red frames exemplify ways of coarse-graining the environments into macro-observers $N_k$.