**1. Introduction**

The emergence of objective, classical information from quantum systems is due to amplification: Many pieces of the environment—e.g., many photons—each interact with a quantum system and acquire an imprint of certain states, the pointer states. This is the process by which select information becomes redundant and accessible to many different observers. The framework, where the environment decoheres systems and acts as a communication channel for the resulting information, is known as quantum Darwinism [1–20]. It is the pointer states that survive the interaction with the environment and create "copies" of themselves from which observers can infer the pointer state of the system. This process has been seen experimentally in both natural [21] and engineered [22,23] settings, and both theory and practical calculations are steadily progressing [24–38].

Within this framework, one primary question concerns the information available within an environment fragment as its size increases. This allows one to quantify redundancy: If small fragments F of the environment E all contain the same information about the system S, then that information is available to many observers. Given a global state, *ρ*SE , the accessible information

$$I\_{\mathsf{acc}}(\Pi\_{\mathcal{S}}) = \max\_{\Pi\_{\mathcal{F}}} I(\Pi\_{\mathcal{S}} : \Pi\_{\mathcal{F}}) \tag{1}$$

**Citation:** Zwolak, M. Amplification, Inference, and the Manifestation of Objective Classical Information. *Entropy* **2022**, *24*, 781. https:// doi.org/10.3390/e24060781

Academic Editor: Ronnie Kosloff

Received: 13 March 2022 Accepted: 23 May 2022 Published: 1 June 2022

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can quantify the amount of information an observer learns about ΠS (a positive operatorvalued measure, a POVM, on S) by making a measurement ΠF on only F. The quantity *<sup>I</sup>*(<sup>Π</sup>S : ΠF ) is the classical mutual information computed from the joint probability distribution from outcomes of ΠS and ΠF . The POVM ΠS has elements *πs* that generate an ensemble "*ps*, *ρ*F|*s* # of outcomes *s* with probability *ps* = trSE*πsρ*SE and conditional states *ρ*F|*s* = trSE/F *<sup>π</sup>sρ*SE/*ps* = trSE/F √*<sup>π</sup>sρ*SE√*<sup>π</sup>s*/*ps* on F (i.e., assuming the POVM acts on only S and an auxiliary system but F is not directly affected). Allowing ΠS to be arbitrary, the accessible information, Equation (1), depicts a situation where some auxiliary system A, perhaps a special observer or another part of the environment, has access directly only to S, makes a measurement ΠS, and holds a record of the outcome *s*, leaving a joint state (after tracing out the now irrelevant S)

$$\sum\_{s} p\_{s} |s\rangle\_{\mathcal{A}} \langle s| \otimes \rho\_{\mathcal{F}|s} \,. \tag{2}$$

An observer O then wants to predict the outcome *s* by making measurements only on F, e.g., correlations are generated between A and O but indirectly from separate measurements on S and F, for which Equation (1) quantifies this capability. One could then maximize the accessible information over all ΠS to see what quantity the observer can learn most about. This allows one to quantify the structure of correlations between S and F induced by, e.g., a decohering interaction between them.

Within the context of physical processes that give rise to quantum Darwinism, ΠS is not arbitrary, however. For redundant information to be present, there must be at least two records of some information, which, when decoherence is the main interaction, will be the pointer information. Hence, there must be an F that almost, to a degree we want to quantify, makes a measurement of the pointer states. At the same time, the remaining part of the environment, E/F, has already made an effective measurement for all intents and purposes, to a degree that we can retroactively validate. This entails that the correlations are effectively of the form of Equation (2) but with A = E/F or S and ΠS = Πˆ S (the pointer observable),

$$\sum\_{\mathfrak{s}} p\_{\mathfrak{s}} |\mathfrak{s}\rangle\langle\mathfrak{s}| \otimes \rho\_{\mathfrak{r}}\mathfrak{r}|\_{\mathfrak{s}'} \tag{3}$$

where *s*ˆ labels the pointer states (see Refs. [39,40] for a discussion of pointer states). This form is a consequence of "branching" [3] and appears in the good decoherence limit of purely decohering models, which will be extensively discussed below. Here, it is sufficient to note that the state, Equation (3), is the most relevant to quantum Darwinism. It makes little difference if one treats the A as E/F or as just the fully decohered, or directly measured, S, even when F is extremely large in absolute terms. Only for "global" questions, where F is some sizable fraction of the environment, does it matter. Since the environment is huge for most problems of everyday interest, such as photon scattering, F can be very large—even asymptotically large—without concern for this. However, Equation (3) does drop exponentially small corrections in the size of E/F and one can not formally take the asymptotic limit of F without first doing so in E. The degree to which asymptotic approximations work thus relies on the balance sheet—how well records are kept in the environment components compared to E's absolute size. Ref. [14] has dealt with retaining corrections to Equation (3). Hereon, I treat the auxiliary system A as if it were S.
