REVIEWS OF MODERN PHYSICS, VOLUME 75, JULY 2003 Decoherence, einselection, and the quantum origins of the classical Wojciech Hubert Zurek Theory Division, LANL, Mail Stop B210, Los Alamos, New Mexico 87545 (Published 22 May 2003) The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal ‘‘Schro¨ dinger-cat states.’’ The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective ‘‘collapse of the wave packet.’’ The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand ‘‘classical reality’’ as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be ‘‘found out’’ without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their ‘‘fitness’’ in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born’s rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection. CONTENTS A. Models of einselection 730 1. Decoherence of a single qubit 730 2. The classical domain and quantum halo 731 I. Introduction 716 3. Einselection and controlled shifts 732 A. The problem: Hilbert space is big 716 B. Einselection as the selective loss of information 733 1. Copenhagen interpretation 716 1. Conditional state, entropy, and purity 733 2. Many-worlds interpretation 717 2. Mutual information and discord 733 B. Decoherence and einselection 717 C. Decoherence, entanglement, dephasing, and C. The nature of the resolution and the role of noise 734 envariance 718 D. Predictability sieve and einselection 735 D. Existential interpretation and quantum V. Einselection in Phase Space 736 Darwinism 719 A. Quantum Brownian motion 737 II. Quantum Measurements 720 A. Quantum conditional dynamics 720 B. Decoherence in quantum Brownian motion 740 1. Decoherence time scale 740 1. Controlled NOT and bit-by-bit measurement 720 2. Measurements and controlled shifts 721 2. Phase-space view of decoherence 741 3. Amplification 722 C. Predictability sieve in phase space 742 B. Information transfer in measurements 723 D. Classical limit in phase space 744 1. Reduced density matrices and correlations 723 1. Mathematical approach (ប!0) 744 2. Action per bit 723 2. Physical approach: The macroscopic limit 744 C. ‘‘Collapse’’ analog in a classical measurement 724 3. Ignorance inspires confidence in classicality 745 III. Chaos and Loss of Correspondence 725 E. Decoherence, chaos, and the second law 745 A. Loss of quantum-classical correspondence 725 1. Restoration of correspondence 745 B. Moyal bracket and Liouville flow 727 2. Entropy production 746 C. Symptoms of correspondence loss 728 3. Quantum predictability horizon 749 1. Expectation values 728 VI. Einselection and Measurements 749 2. Structure saturation 729 A. Objective existence of einselected states 749 IV. Environment-Induced Superselection 729 B. Measurements and memories 750 0034-6861/2003/75(3)/715(61)/$35.00 715 ©2003 The American Physical Society 716 Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classical C. Axioms of quantum measurement theory 751 with a focus on decoherence, einselection, and the emer- 1. Observables are Hermitian—axiom (iii a) 751 gence of classicality, and also attempts to offer a preview 2. Eigenvalues as outcomes—axiom (iii b) 751 of the future of this exciting and fundamental area. 3. Immediate repeatability, axiom (iv) 752 4. Probabilities, einselection, and records 752 D. Probabilities from envariance 753 A. The problem: Hilbert space is big 1. Envariance 754 2. Born’s rule from envariance 755 The interpretation problem stems from the vastness of 3. Relative frequencies from envariance 757 Hilbert space, which, by the principle of superposition, 4. Other approaches to probabilities 757 admits arbitrary linear combinations of any states as a VII. Environment as a Witness 758 possible quantum state. This law, thoroughly tested in A. Quantum Darwinism 759 the microscopic domain, bears consequences that defy 1. Consensus and algorithmic simplicity 759 classical intuition: It appears to imply that the familiar 2. Action distance 760 3. Redundancy and mutual information 760 classical states should be an exceedingly rare exception. 4. Redundancy ratio rate 762 And, naively, one may guess that the superposition prin- B. Observers and the existential interpretation 762 ciple should always apply literally: Everything is ulti- C. Events, records, and histories 763 mately made out of quantum ‘‘stuff.’’ Therefore there is VIII. Decoherence in the Laboratory 764 no a priori reason for macroscopic objects to have defi- A. Decoherence due to entangling interactions 765 nite position or momentum. As Einstein noted1 localiza- B. Simulating decoherence with classical noise 766 tion with respect to macrocoordinates is not just inde- 1. Decoherence, noise, and quantum chaos 766 pendent of, but incompatible with, quantum theory. 2. Analog of decoherence in a classical system 767 How, then, can one establish a correspondence between C. Taming decoherence 767 the quantum and the familiar classical reality? 1. Pointer states and noiseless subsystems 767 2. Environment engineering 768 1. Copenhagen interpretation 3. Error correction and resilient quantum computing 768 Bohr’s solution was to draw a border between the IX. Concluding Remarks 769 quantum and the classical and to keep certain objects— X. Acknowledgments 771 especially measuring devices and observers—on the References 771 classical side (Bohr, 1928, 1949). The principle of super- position was suspended ‘‘by decree’’ in the classical do- main. The exact location of this border was difficult to I. INTRODUCTION pinpoint, but measurements ‘‘brought to a close’’ quan- tum events. Indeed, in Bohr’s view the classical domain The interpretation of quantum theory has been an is- was more fundamental. Its laws were self-contained sue ever since its inception. Its tone was set by the dis- (they could be confirmed from within) and established ¨ cussions of Schrodinger (1926, 1935a, 1935b), Heisen- the framework necessary to define the quantum. berg (1927), and Bohr (1928, 1949; see also Jammer, The first breach in the quantum-classical border ap- 1974; Wheeler and Zurek, 1983). Perhaps the most inci- peared early: In the famous Bohr-Einstein double-slit sive critique of the (then new) theory was that of Ein- debate, quantum Heisenberg uncertainty was invoked stein, who, searching for inconsistencies, distilled the es- by Bohr at the macroscopic level to preserve wave- sence of the conceptual difficulties of quantum particle duality. Indeed, since the ultimate components mechanics through ingenious gedanken experiments. We of classical objects are quantum, Bohr emphasized that owe to him and Bohr clarification of the significance of the boundary must be moveable, so that even the human quantum indeterminacy in the course of the Solvay Con- nervous system could be regarded as quantum, provided gress debates (see Bohr, 1949) and elucidation of the that suitable classical devices to detect its quantum fea- nature of quantum entanglement (Bohr, 1935; Einstein, tures were available. In the words of Wheeler (1978, ¨ Podolsky, and Rosen, 1935; Schrodinger, 1935a, 1935b). 1983), who has elucidated Bohr’s position and decisively The issues they identified then are still a part of the contributed to the revival of interest in these matters, subject. ‘‘No [quantum] phenomenon is a phenomenon until it is Within the