Objective properties from subjective quantum states: Environment as a witness

TL;DR

This work addresses how objective classical properties emerge from quantum substrates by treating the environment as an active witness that broadcasts information about the system (quantum Darwinism). It formalizes objectivity via an operational criterion based on mutual information $I(\sigma: \mathfrak e)$ and redundancy $R_\delta(\sigma)$, showing that a complete and redundantly imprinted observable can be read by many observers without perturbing the system. A central result proves the existence of a unique maximally refined observable $\pi$ such that for $m_\delta(\pi) \le m \ll N$, $\hat{I}_m(\sigma)=I(\sigma: \pi)$ for all $\sigma$ in the relevant set, thereby selecting pointer states as the objective observables. Using a concrete spin–environment model, the authors illustrate that only pointer observables acquire robust, redundant information across environmental fragments, while other observables are constrained by their correlations with $\pi$. Overall, the paper links decoherence, einselection, and Darwinian proliferation of information to explain the emergence of a single, objective classical reality from quantum dynamics.

Abstract

We study the emergence of objective properties in open quantum systems. In our analysis, the environment is promoted from a passive role of reservoir selectively destroying quantum coherence, to an active role of amplifier selectively proliferating information about the system. We show that only preferred pointer states of the system can leave a redundant and therefore easily detectable imprint on the environment. Observers who--as it is almost always the case--discover the state of the system indirectly (by probing a fraction of its environment) will find out only about the corresponding pointer observable. Many observers can act in this fashion independently and without perturbing the system: they will agree about the state of the system. In this operational sense, preferred pointer states exist objectively.

Figures (1)

• Figure 1: Quantum Darwinism can be illustrated using a model introduced in Zur82a. The system ${\cal S}$, a spin-$\frac{1}{2}$ particle, interacts with $N=50$ two-dimensional subsystems of the environment through $\hat{H}^{{\cal S}{\cal E}} = \sum_{k=1}^N g_k \sigma_z^{{\cal S}} \otimes\sigma_y^{{\cal E}_k}$ for a time $t$. The initial state of $\mathcal{S}\otimes{\cal E}$ is $\frac{1}{\sqrt 2}(| 0 \rangle + | 1 \rangle) \otimes | 0 \rangle ^{{\cal E}_1}\otimes\ldots\otimes| 0 \rangle^{{\cal E}_N}$. All the plotted quantities are function of the system's observable $\sigma(\mu) = \cos(\mu)\sigma_z + \sin(\mu)\sigma_x$, where $\mu$ is the angle between its eigenstates and the pointer states of ${\cal S}$---here the eigenstates of $\sigma_z^{\cal S}$. a) Information acquired by the optimal measurement $\mathfrak e$ on the whole environment, $\hat{I}_N(\sigma)$, as a function of the inferred observable $\sigma(\mu)$ and the action $a_k = g_k t = a$ for all $k$. A large amount of information is accessible in the whole environment for any observables $\sigma(\mu)$ except when the interaction action $a_k$ is very small. Thus, complete imprinting of an observable of ${\cal S}$ in ${\cal E}$ is not sufficient to claim objectivity. b) Redundancy of the information about the system as a function of the inferred observable $\sigma(\mu)$ and the action $a_k = g_k t = a$. It is measured by $R_{\delta=0.1}(\sigma)$, which counts the number of times 90% of the total information can be "read off" independently by measuring distinct fragments of the environment. For all values of the action $a_k = g_k t =a$, redundant imprinting is sharply peaked around the pointer observable. Redundancy is a very selective criterion. The number of copies of relevant information is high only for the observables $\sigma(\mu)$ falling inside the theoretical bound (see text) indicated by the dashed line. c) Information about $\sigma(\mu)$ extracted by an observer restricted to local random measurements on $m$ environmental subsystems (e.g. $\mathfrak e = \mathfrak e^{{\cal E}_1}\otimes\ldots \otimes\mathfrak e^{{\cal E}_m}$ where each $\mathfrak e^{{\cal E}_k}$ is chosen at random). The interaction action $a_k = g_k t$ is randomly chosen in $[0,\pi/4]$ for each $k$. Because of redundancy, pointer states---and only pointer states---can be found out through this far-from-optimal measurement strategy. Information about any other observable $\sigma(\mu)$ is restricted by our theorem to be equal to the information brought about it by the pointer observable $\sigma_z^{\cal S}$, Eq. (\ref{['bound_th']}).