Environment as a witness: Selective proliferation of information and emergence of objectivity in a quantum universe

We study the role of the information deposited in the environment of an open quantum system in the course of the decoherence process. Redundant spreading of information—the fact that some observables of the system can be independently read off from many distinct fragments of the environment—is investigated as the key to effective objectivity, the essential ingredient of classical reality. This focus on the environment as a communication channel through which observers learn about physical systems underscores the importance of quantum Darwinism—selective proliferation of information about “the fittest states” chosen by the dynamics of decoherence at the expense of their superpositions—as redundancy imposes the existence of preferred observables. We demonstrate that the only observables that can leave multiple imprints in the environment are the familiar pointer observables singled out by environment-induced superselection (einselection) for their predictability. Many independent observers monitoring the environment will therefore agree on properties of the system as they can only learn about preferred observables. In this operational sense, the selective spreading of information leads to appearance of an objective classical reality from within the quantum substrate.

Figure 1

The role and the structure of the environment in decoherence and in quantum Darwinism. (a) Decoherence treats the environment as a monolithic entity inaccessible to observers. The only role of $\mathcal{E}$ is to be a sink of information about $\mathcal{S}$. This is enough to understand einselection—the emergence of the preferred pointer observable in the system, with decoherence-free pointer eigenspaces. In our Universe (b) environments are typically not monolithic. Rather, they consist of subsystems, e.g., atoms or photons. From the point of view of decoherence, there is no difference between the paradigms illustrated by (a) and (b): The same environment will lead to the same evolution of the system (e.g., in quantum Brownian motion the same environment can be modeled either as a monolithic field or as a collection of harmonic oscillators). Quantum Darwinism (c) recognizes that observers acquire their information about $\mathcal{S}$ without interacting with it directly—from the imprints left by $\mathcal{S}$ on the fragments of $\mathcal{E}$. Such fragments are generally much smaller than $\mathcal{E}$. The central result of this paper is to show that the selected information about the system that is inscribed redundantly—in many copies—on the environment, and can be therefore found out independently from different fragments of $\mathcal{E}$ by many observers, concerns the einselected pointer observable.

Figure 2

Venn diagram for classical information. $H\left(\mathbf{A}\right)$ is the entropy associated with a measurement of $\mathbf{A}$: it is the information needed to completely determine its outcome. $H\left(\mathbf{X}\right)$ is the same quantity for the observable $\mathbf{X}$. $H(\mathbf{A},\mathbf{X})$ is the joint entropy of $\mathbf{A}$ and $\mathbf{X}$. $H(\mathbf{A}\mid \mathbf{X})$ is the average uncertainty about $\mathbf{A}$ remaining after a measurement of $\mathbf{X}$. The information learned about $\mathbf{A}$ by measuring $\mathbf{X}$ is thus $I(\mathbf{A}:\mathbf{X})=H\left(\mathbf{A}\right)-H(\mathbf{A}\mid \mathbf{X})$. The equivalent definition $I(\mathbf{A}:\mathbf{X})=H\left(\mathbf{A}\right)+H\left(\mathbf{X}\right)-H(\mathbf{A},\mathbf{X})$ can also be understood from the diagram. These two definitions of the mutual information are equivalent [26] when probabilities can be consistently assigned to the outcomes of all the possible measurements of $\mathbf{A}$ and $\mathbf{X}$ both separately and jointly: this is ensured when $\mathbf{A}$ and $\mathbf{X}$ commute. They need not coincide otherwise.

Figure 3

Suppose $\mathbf{B}$ and $\mathbf{C}$ are redundantly imprinted in the environment, but are associated, respectively, with decompositions (a) and (b), represented by different arrangements of the subsystems of $\mathcal{E}$ into fragments. Then, from Theorem 1 one cannot yet conclude that there exists a maximally refined observable $\mathbf{A}$ from which both $\mathbf{B}$ and $\mathbf{C}$ can be inferred. This is because the two decompositions do not match. However, when the redundancy of $\mathbf{B}$ and $\mathbf{C}$ satisfies $R_0\left(\mathbf{B}\right)R_0\left(\mathbf{C}\right)>N$, then Theorem 2 proves that the maximally refined observables obtained for each decomposition of $\mathcal{E}$ into fragments must commute (i.e., they are compatible).