Equilibration of objective observables in a dynamical model of quantum measurements

Abstract

The challenge of understanding quantum measurement persists as a fundamental issue in modern physics. Particularly, the abrupt and energy-non-conserving collapse of the wave function appears to contradict classical thermodynamic laws. The contradiction can be resolved by considering measurement itself to be an entropy-increasing process, driven by the second law of thermodynamics. One such resolution explains the apparently irreversible emergence of objective outcomes in an isolated, unitarily-evolving quantum system via the theory of closed-system equilibration. Working within this framework, we construct the set of \textit{`objectifying observables'} that best encode the measurement statistics of a system in an objective manner, and establish a measurement error bound to quantify the probability an observer will obtain an incorrect measurement outcome. Using this error bound, we show that the objectifying observables readily equilibrate on average under the set of Hamiltonians which preserve the outcome statistics on the measured system. Using a random matrix model for this set, we numerically determine the measurement error bound, finding that the error only approaches zero with increasing environment size when the environment is coarse-grained into so-called observer systems. This indicates the necessity of coarse graining an environment for the emergence of objective, classical measurement outcomes.

Figures (5)

• Figure 1: Conceptual illustration. We model a measurement as an isolated system-environment interaction. The system being measured, $S$, interacts unitarily with an environment (blue background). The environment is composed of many sub-environments (orange spheres), and observer systems are collections of sub-environments (white envelopes). We imagine a scenario in which each observer has access to an observer system.
• Figure 2: Schematic illustrating two system-environment configurations. In both cases, we model a system $S$ being measured via a unitary interaction with its surrounding environment. The environment is composed of many sub-environments (orange spheres). The observer systems (indicated by shaded boxes) are collections of one or more sub-environments. Objectifying observables, corresponding to $\lbrace\Pi_{k}^{i}\rbrace_{i}$ are associated with each observer system. The black lines connecting the system to the sub-environments indicate interactions via the broadcasting Hamiltonian (Eq. \ref{['ham']}). In (a), we consider observer systems of single high-dimensional qudit sub-environments, where each qudit has dimension $d$ (here $d=16$). In (b) we consider observer systems of $n$ qubit sub-environments ($n=4$ in this illustration and so the total dimension of each observer system is $2^4 = 16$).
• Figure 3: Numerical simulation of $\mathcal{E}_{\text{eq}}$, defined in Eq. \ref{['eq:D_term']}, when sampled over the GUE. Each sample in histograms (a) and (b) considers an initially pure and uncorrelated system-environment state, evolving according to conditional Hamiltonians drawn from the GUE. There are 10,000 samples for each considered observer system dimension $d_{k}$. (a) A single qudit observer system of dimension $d_{k}$ (the scenario in Fig. \ref{['fig:schematic']}a). (b) A coarse-grained observer system of $n$ qubits with total dimension $d_{k} = 2^n$ (the scenario in Fig. \ref{['fig:schematic']}b). In (c) we plot the average over the GUE of each histogram in single qudit and many-qubit observer system cases as a function of $d_{k}$. In (d) we plot the variance over the GUE.
• Figure 4: Numerical simulation of $\mathcal{E}_{\text{obj}}$, defined in Eq. \ref{['eq:F_term']}, via sampling over the GUE. (a) shows histograms for a single qudit observer system (the scenario in Fig. \ref{['fig:schematic']}a). Each sample considers an initially pure and uncorrelated system-environment, evolving according to conditional Hamiltonians drawn from the GUE. There are 10,000 samples for each considered dimension $d_{k}$. In (b) we show $\left\langle \mathcal{E}_{\text{obj}} \right \rangle_{\text{GUE}}$ for a coarse-grained observer system consisting of $n$ qubits (the scenario in Fig. \ref{['fig:schematic']}b). We present the average fidelity on a log scale. Here we show the necessity of coarse graining in order to approach zero fidelity in the large observer system limit.
• Figure 5: (a) shows $\left\langle \mathcal{E}_{\text{eq}} \right \rangle_{\text{GUE}}$ for a single-qudit observer system with dimension $d_{k}$ (the scenario in Fig. \ref{['fig:schematic']}a), that is initially thermal at temperature $T=1/ \beta$ and energy gap $\omega$. (b) shows $\left\langle \mathcal{E}_{\text{obj}} \right \rangle_{\text{GUE}}$ for the same thermal initial states as in (a). Each data point in (a) and (b) is an average of 10,000 samples over the GUE.