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arXiv:0911.4307 · Physical Review A

Redundant imprinting of information in nonideal environments: Objective reality via a noisy channel

Michael P. Zwolak, H. T. Quan, Wojciech Hubert Zurek

Published Jun 10, 2010• 71 citations
arXivHTMLPDF

Abstract

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, $\mathcal{E}$, 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, $\mathcal{S}$, and an environment fragment, $\mathcal{F}$, is given solely by $\mathcal{F}$'s entropy increase. This demonstrates that the environment's capacity for recording the state of $\mathcal{S}$ is directly related to its ability to increase its entropy. Environments that remain nearly invariant under the interaction with $\mathcal{S}$, 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 $\mathcal{F}$'s entropy change is quantified by the quantum discord, i.e., the excess mutual information between $\mathcal{S}$ and $\mathcal{F}$ is information regarding the initial coherence between pointer states of $\mathcal{S}$. 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 $\mathcal{S}$. 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.

DOI
10.1103/physreva.81.062110

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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\mathcal{E}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\mathcal{S}S, and an environment fragment, F\mathcal{F}F, is given solely by F\mathcal{F}F's entropy increase. This demonstrates that the environment's capacity for recording the state of S\mathcal{S}S is directly related to its ability to increase its entropy. Environments that remain nearly invariant under the interaction with S\mathcal{S}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\mathcal{F}F's entropy change is quantified by the quantum discord, i.e., the excess mutual information between S\mathcal{S}S and F\mathcal{F}F is information regarding the initial coherence between pointer states of S\mathcal{S}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\mathcal{S}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.
Quantum Darwinism recognizes that we-the observers-acquire our information about the "systems of interest" indirectly from their imprints on the environment. Here, we show that information about a system can be acquired from a mixed-state, or hazy, environment, but the storage capacity of an environment fragment is suppressed by its initial entropy. In the case of good decoherence, the mutual information between the system and the fragment is given solely by the fragment's entropy increase. For fairly mixed environments, this means a reduction by a factor 1-h, where h is the haziness of the environment, i.e., the initial entropy of an environment qubit. Thus, even such hazy environments eventually reveal the state of the system, although now the intercepted environment fragment must be larger by approximately (1-h)(-1) to gain the same information about the system.
Quantum Darwinism recognizes the role of the environment as a communication channel: Decoherence can selectively amplify information about the pointer states of a system of interest (preventing access to complementary information about their superpositions) and can make records of this information accessible to many observers. This redundancy explains the emergence of objective, classical reality in our quantum Universe. Here, we demonstrate that the amplification of information in realistic spin environments can be quantified by the quantum Chernoff information, which characterizes the distinguishability of partial records in individual environment subsystems. We show that, except for a set of initial states of measure zero, the environment always acquires redundant information. Moreover, the Chernoff information captures the rich behavior of amplification in both finite and infinite spin environments, from quadratic growth of the redundancy to oscillatory behavior. These results will considerably simplify experimental testing of quantum Darwinism, e.g., using nitrogen vacancies in diamond.
The framework of Quantum Darwinism strives at characterizing the quantum-to-classical transition by introducing the concept of redundancy of information-as measured by Mutual Information-that a set of observers would acquire on the state of a physical system of interest. Further development on this concept, in the form of Strong Quantum Darwinism and Spectrum Broadcast Structures, has recently led to a more fine-grained identification of the nature of such information, which should not involve any quantum correlations between observing and observed systems, while the assessment of information proliferation from individual systems has attracted most of the attention so far, the way such mechanism takes place in more complex states is open to exploration. To this end, we shall consider a two-qubit state, sharing initial quantum correlations in the form of Quantum Discord, and different dephasing-like interactions between them and an observing environment. We will focus on the amount of information regarding the subsystem not involved in the interaction that is proliferated to the environment. We shall refer to this as mediated redundancy. We will show that, in some cases, the channel capacity of the subsystems, given these interactions, can exceed that of the fragments.
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.
Quantum Darwinism recognizes that decoherence imprints redundant records of preferred quasi-classical pointer states on the environment. These redundant records are then accessed by observers. We show how redundancy enables and even implies consensus between observers who use fragments of that decohering environment to acquire information about systems of interest. We quantify consensus using information-theoretic measures that employ mutual information to assess the correlation between the records available to observers from distinct -- hence, independently accessible -- fragments of the environment. We prove that when these fragments have enough information about a system, observers that access them will attribute the same pointer state to that system. Thus, those who know enough about the system agree about what they know. We then test proposed measures of consensus in a solvable model of decoherence as well as in numerical simulations of a many-body system. These results provide detailed understanding of how our classical everyday world arises from within the fundamentally quantum Universe we inhabit.