Quantum decoherence

Abstract

Quantum decoherence plays a pivotal role in the dynamical description of the quantum-to-classical transition and is the main impediment to the realization of devices for quantum information processing. This paper gives an overview of the theory and experimental observation of the decoherence mechanism. We introduce the essential concepts and the mathematical formalism of decoherence, focusing on the picture of the decoherence process as a continuous monitoring of a quantum system by its environment. We review several classes of decoherence models and discuss the description of the decoherence dynamics in terms of master equations. We survey methods for avoiding and mitigating decoherence and give an overview of several experiments that have studied decoherence processes. We also comment on the role decoherence may play in interpretations of quantum mechanics and in addressing foundational questions.

Introduction

Hilbert space is a vast and seemingly egalitarian place. If $\left|{\psi }_1\right.〉$ and $\left|{\psi }_2\right.〉$ represent two possible physical states of a quantum system, then quantum mechanics postulates that an arbitrary superposition $\alpha \left|{\psi }_1\right.〉+\beta \left|{\psi }_2\right.〉$ constitutes another possible physical state. The question, then, is why most such states, especially for mesoscopic and macroscopic systems, are found to be very difficult to prepare and observe, often prohibitively so. For example, it turns out to be extremely challenging to prepare a macroscopic quantum system in a spatial superposition of two macroscopically separated, narrow wave packets, with each individual wave packet approximately representing the kind of spatial localization familiar from the classical world of our experience. Even if one succeeded in generating such a superposition and confirming its existence – for example, by measuring fringes arising from interference between the wave-packet components – one would find that it becomes very rapidly unobservable. Thus, we arrive at the dynamical problem of the quantum-to-classical transition: Why are certain “nonclassical” quantum states so fragile and easily degraded? The question is of immense importance not only from a fundamental point of view, but also because quantum information processing and quantum technologies crucially depend on our ability to generate, maintain, and manipulate such nonclassical superposition states.

The key insight in addressing the problem of the quantum-to-classical transition was first spelled out almost fifty years ago by Zeh [1], and it gave birth to the theory of quantum decoherence, sometimes also called dynamical decoherence or environment-induced decoherence [1], [2], [3], [4], [5], [6], [7], [8], [9]. The insight is that realistic quantum systems are never completely isolated from their environment, and that when a quantum system interacts with its environment, it will in general become rapidly and strongly entangled with a large number of environmental degrees of freedom. This entanglement dramatically influences what we can locally observe upon measuring the system, even when from a classical point of view the influence of the environment on the system (in terms of dissipation, perturbations, noise, etc.) is negligibly small. In particular, quantum interference effects with respect to certain physical quantities (most notably, “classical” quantities such as position) become effectively suppressed, making them prohibitively difficult to observe in most cases of practical interest.

This, in a nutshell, is the process of decoherence [1], [2], [3], [4], [5], [6], [7], [8], [9]. Stated in general and interpretation-neutral terms, decoherence describes how entangling interactions with the environment influence the statistics of future measurements on the system. Formally, decoherence can be viewed as a dynamical filter on the space of quantum states, singling out those states that, for a given system, can be stably prepared and maintained, while effectively excluding most other states, in particular, nonclassical superposition states of the kind epitomized by Schrödinger’s cat [10]. In this way, decoherence lies at the heart of the quantum-to-classical transition. It ensures consistency between quantum and classical predictions for systems observed to behave classically. It provides a quantitative, dynamical account of the boundary between quantum and classical physics. In any concrete experimental situation, decoherence theory specifies the physical requirements, both qualitatively and quantitatively, for pushing the quantum–classical boundary toward the quantum realm. Decoherence is a genuinely quantum-mechanical effect, to be carefully distinguished from classical dissipation and stochastic fluctuations.

One of the most surprising aspects of the decoherence process is its extreme efficiency, especially for mesoscopic and macroscopic quantum systems. Furthermore, due to the many uncontrollable degrees of freedom of the environment, the dynamically created entanglement between system and environment is usually irreversible for all practical purposes; indeed, this effective irreversibility is a hallmark of decoherence. Increasingly realistic models of decoherence processes have been developed, progressing from toy models to complex models tailored to specific experiments (see Section 4). Advances in experimental techniques have made it possible to observe the gradual action of decoherence in experiments such as cavity QED [11], matter-wave interferometry [12], superconducting systems [13], and ion traps [14], [15] (see Section 6).

The superposition states necessary for quantum information processing are typically also those most susceptible to decoherence. Thus, decoherence is a major barrier to the implementation of devices for quantum information processing such as quantum computers. Qubit systems must be engineered to minimize environmental interactions detrimental to the preparation and longevity of the desired superposition states. At the same time, these systems must remain sufficiently open to allow for their control. Strategies for combatting the adverse effects of decoherence include decoherence avoidance, such as the encoding of information in decoherence-free subspaces (see Section 5.1), and quantum error correction [16], which can undo the decoherence-induced degradation of the superposition state (see Section 5.3). Such strategies will be an integral part of quantum computers. Not only is decoherence relevant to quantum information, but also vice versa. An information-centric view of quantum mechanics proves helpful in conveying the essence of the decoherence process and is also used in recent explorations of the role of the environment as an information channel (see Sections 2.2 Environmental monitoring and information transfer, 2.5 Proliferation of information and quantum Darwinism).

Decoherence is a technical result concerning the dynamics and measurement statistics of open quantum systems. From this view, decoherence merely addresses a consistency problem, by explaining how and when the quantum probability distributions approach the classically expected distributions. Since decoherence follows directly from an application of the quantum formalism to interacting quantum systems, it is not tied to any particular interpretation of quantum mechanics, and it neither supplies such an interpretation nor amounts to a theory that could make predictions beyond those of standard quantum mechanics. However, the bearing decoherence has on the problem of the relation between quantum and classical has been frequently invoked to assess or support various interpretations of quantum mechanics, and the implications of decoherence for the so-called quantum measurement problem have been analyzed extensively (see Section 7). Indeed, historically decoherence theory arose in the context of Zeh’s independent formulation of an Everett-style interpretation [1]; see Ref. [17] for an analysis of the connections between the roots of decoherence and matters of interpretation.

It is a curious “historical accident” (Joos’s term [18, p. 13]) that the implications of environmental entanglement were appreciated only relatively late. While one can find – for example, in Heisenberg’s writings (see Section 7.3 and Ref. [19]) – a few early anticipatory remarks about the role of environmental interactions in the quantum-mechanical description of physical systems, it was not until the 1970s that the ubiquity and implications of environmental entanglement were realized by Zeh [1], [20]. In the 1980s, the formalism of decoherence was further developed, chiefly by Zurek [2], [3], and the first concrete decoherence models and numerical estimates of decoherence rates were worked out by Joos and Zeh [21] and Zurek [22] (see also Refs [23], [24], [25]). Zurek’s 1991 Physics Today article [26] was an important factor in introducing a broader audience of physicists to decoherence theory. Such dissemination and maturing of decoherence theory came at a perfect time, as the 1990s also saw the blossoming of quantum information [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], as well as experimental advances in the creation of superpositions of mesoscopically and macroscopically distinct states [38], [39], [40], [41]. The quantum states relevant to quantum information processing and Schrödinger-cat-type experiments required the insights of decoherence theory, and conversely the new experiments served as a fertile ground for testing the predictions of decoherence theory. Accordingly, these developments led to a rapid rise in interest and research activity in the field of decoherence. Today, decoherence has become a central topic of modern quantum mechanics and is studied intensely both theoretically and experimentally.

Existing reviews of decoherence include the papers by Zurek [5], Paz and Zurek [4], and Hornberger [42]. Two books dedicated to decoherence are presently available: a volume by Joos et al. [8] (a collection of chapters written by different authors), and a monograph by this author [9], which offers, among other material, a detailed treatment of the topics surveyed in this paper. Textbooks on open quantum systems, such as Ref. [43], also contain a substantial amount of material on decoherence, especially in the context of quantum master equations.

This article is organized as follows. Section 2 introduces the theory, formalism, and fundamental concepts of decoherence. Section 3 discusses the description of decoherence dynamics in terms of master equations. Section 4 reviews several classes of important decoherence models. Section 5 describes methods for avoiding and mitigating the influence of decoherence. Section 6 gives an overview of several experiments that have demonstrated the gradual, controlled action of decoherence. Section 7 comments on the implications of decoherence for foundational issues in quantum mechanics and for the different interpretations of quantum mechanics. Section 8 offers concluding remarks.