Amplification was regarded, since the early days of quantum theory, as a mysterious ingredient that endows quantum microstates with macroscopic consequences, key to the "collapse of the wave packet," and a way to avoid embarrassing problems exemplified by Schrödinger's cat. Such a bridge between the quantum microworld and the classical world of our experience was postulated ad hoc in the Copenhagen interpretation. Quantum Darwinism views amplification as replication, in many copies, of the information about quantum states. We show that such amplification is a natural consequence of a broad class of models of decoherence, including the photon environment we use to obtain most of our information. This leads to objective reality via the presence of robust and widely accessible records of selected quantum states. The resulting redundancy (the number of copies deposited in the environment) follows from the quantum Chernoff information that quantifies the information transmitted by a typical elementary subsystem of the environment.
Quantum Darwinism provides an information-theoretic framework for the emergence of the objective, classical world from the quantum substrate. The key to this emergence is the proliferation of redundant information throughout the environment where observers can then intercept it. We study this process for a purely decohering interaction when the environment,
E, is in a nonideal (e.g., mixed) initial state. In the case of good decoherence, that is, after the pointer states have been unambiguously selected, the mutual information between the system,
S, and an environment fragment,
F, is given solely by
F's entropy increase. This demonstrates that the environment's capacity for recording the state of
S is directly related to its ability to increase its entropy. Environments that remain nearly invariant under the interaction with
S, either because they have a large initial entropy or a misaligned initial state, therefore have a diminished ability to acquire information. To elucidate the concept of good decoherence, we show that, when decoherence is not complete, the deviation of the mutual information from
F's entropy change is quantified by the quantum discord, i.e., the excess mutual information between
S and
F is information regarding the initial coherence between pointer states of
S. In addition to illustrating these results with a single-qubit system interacting with a multiqubit environment, we find scaling relations for the redundancy of information acquired by the environment that display a universal behavior independent of the initial state of
S. Our results demonstrate that Quantum Darwinism is robust with respect to nonideal initial states of the environment: the environment almost always acquires redundant information about the system but its rate of acquisition can be reduced.
The objective, classical world emerges from the underlying quantum substrate via the proliferation of redundant copies of selected information into the environment, which acts as a communication channel, transmitting that information to observers. These copies are independently accessible, allowing many observers to reach consensus about the state of a quantum system via its imprints in the environment. Quantum Darwinism recognizes that the redundancy of information is thus central to the emergence of objective reality in the quantum world. However, in addition to the ``quantum system of interest,'' there are many other systems ``of no interest'' in the Universe that can imprint information on the common environment. There is therefore a danger that the information of interest will be diluted with irrelevant bits, suppressing the redundancy responsible for objectivity. We show that mixing of the relevant (the ``wheat'') and irrelevant (the ``chaff'') bits of information makes little quantitative difference to the redundancy of the information of interest. Thus, we demonstrate that it does not matter whether one separates the wheat (relevant information) from the (irrelevant) chaff: The large redundancy of the relevant information survives dilution, providing evidence of the objective, effectively classical world.
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.
Our everyday reality is characterized by objective information-information that is selected and amplified by the environment that interacts with quantum systems. Many observers can accurately infer that information indirectly by making measurements on fragments of the environment. The correlations between the system, S, and a fragment, F, of the environment, E, is often quantified by the quantum mutual information, or the Holevo quantity, which bounds the classical information about S transmittable by a quantum channel F. The latter is a quantum mutual information but of a classical-quantum state where measurement has selected outcomes on S. The measurement generically reflects the influence of the remaining environment, E/F, but can also reflect hypothetical questions to deduce the structure of SF correlations. Recently, Touil et al. examined a different Holevo quantity, one from a quantum-classical state (a quantum S to a measured F). As shown here, this quantity upper bounds any accessible classical information about S in F and can yield a tighter bound than the typical Holevo quantity. When good decoherence is present-when the remaining environment, E/F, has effectively measured the pointer states of S-this accessibility bound is the accessible information. For the specific model of Touil et al., the accessible information is related to the error probability for optimal detection and, thus, has the same behavior as the quantum Chernoff bound. The latter reflects amplification and provides a universal approach, as well as a single-shot framework, to quantify records of the missing, classical information about S.
Quantum Darwinism demonstrates not only that preferred states are selected for their stability but also that information about them is broadcast by the same environment that causes decoherence and einselection. That environment acts both as a censor and as an advertising agent that disseminates information about pointer states while suppressing complementary information. Chapter 8 explores the implications and limitations of quantum Darwinism using models inspired by the structure of the Universe we inhabit. We perceive our Universe using light and other means of information transmission. We explore models that have a well-defined relation with our everyday reality, and where one can also selectively relax some of the idealized assumptions and investigate the consequences. Light is the communication channel through which we obtain most of our information. Fortunately, it is an ideal channel in the sense of quantum Darwinism, and simple but realistic cases are exactly solvable. The solution presented herein demonstrates the inevitability of the consensus between observers who rely on scattered photons: The emergence of classical objective reality (classical because pointer states are einselected, and objective because redundancy imposes consensus) is inevitable. This is how the classical world we perceive emerges from within the quantum Universe we inhabit.