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Environment–Induced Decoherence and the Transition from Quantum to Classical

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Environment–Induced Decoherence and the Transition from Quantum to Classical COURSE Lectures given by both authors at the 72nd Les Houches Summer School on ”Coherent Matter Waves”, July–August 1999 ENVIRONMENT{INDUCED DECOHERENCE AND THE TRANSITION FROM QUANTUM TO CLASSICAL JUAN PABLO PAZ1 WOJCIECH HUBERT ZUREK2 1Departamento de F´ısica J.J. Giambiagi, FCEN, UBA, Pabell´on 1, Ciudad Universitaria 1428 Buenos Aires, Argentina 2Theoretical Astrophysics, MS B288 Los Alamos National Laboratory, Los Alamos, NM87545, USA Contents 1 Introduction and overview 3 2 Quantum Measurements 7 2.1Bit-by-bitmeasurementandquantumentanglement........ 9 2.2 Interactions and the information transfer in quantum measurements 12 2.3Monitoringbytheenvironmentanddecoherence.......... 14 2.4One–bitenvironmentforabit-by-bitmeasurement......... 16 2.5Decoherenceofasingle(qu)bit.................... 17 2.6Decoherence,einselection,andcontrolledshifts........... 22 3 Dynamics of quantum open systems: master equations 25 3.1Masterequation:Perturbativeevaluation.............. 26 3.2 Example 1: Perturbative master equation in Quantum Brownian Motion.................................. 28 3.3 Example 2: Perturbative master equation for a two–level system coupledtoabosonicheatbath.................... 32 3.4 Example 3: Perturbative master equation for a particle interacting withaquantumfield.......................... 34 3.5ExactmasterequationforQuantumBrownianMotion....... 35 4 Einselection in quantum Brownian motion 41 4.1Decoherenceofasuperpositionoftwocoherentstates....... 41 4.2 Predictability sieve and preferred states for QBM . ....... 45 4.3 Energy eigenstates can also be selected by the environment! . 47 5 Deconstructing decoherence: landscape beyond the standard models 48 5.1Saturationofthedecoherencerateatlargedistances........ 49 5.2Decoherenceatzerotemperature................... 50 5.3 Preexisting correlations between the system and the environment. 52 6 Decoherence and chaos 55 6.1 Quantum predictability horizon: How the correspondence is lost. 56 6.2 Exponential instability vs. decoherence . ........... 58 6.3 The arrow of time: a price of classicality? . ........... 60 6.4Decoherence,einselection,andtheentropyproduction........ 64 7 How to fight against decoherence: Quantum error correcting codes. 65 7.1 How to protect a classical bit . .................. 66 7.2Howtoprotectaquantumbit..................... 66 7.3Stabilizerquantumerror–correctingcodes.............. 72 8 Discussion 75 9 Acknowledgment 77 ENVIRONMENT{INDUCED DECOHERENCE AND THE TRANSITION FROM QUANTUM TO CLASSICAL Juan Pablo Paz1, Wojciech Hubert Zurek2 Abstract We study dynamics of quantum open systems, paying special at- tention to those aspects of their evolution which are relevant to the transition from quantum to classical. We begin with a discussion of the conditional dynamics of simple systems. The resulting mod- els are straightforward but suffice to illustrate basic physical ideas behind quantum measurements and decoherence. To discuss deco- herence and environment-induced superselection (einselection)ina more general setting, we sketch perturbative as well as exact deriva- tions of several master equations valid for various systems. Using these equations we study einselection employing the general strategy of the predictability sieve. Assumptions that are usually made in the discussion of decoherence are critically reexamined along with the “standard lore” to which they lead. Restoration of quantum-classical correspondence in systems that are classically chaotic is discussed. The dynamical second law —it is shown— can be traced to the same phenomena that allow for the restoration of the correspondence prin- ciple in decohering chaotic systems (where it is otherwise lost on a very short time-scale). Quantum error correction is discussed as an example of an anti-decoherence strategy. Implications of decoherence and einselection for the interpretation of quantum theory are briefly pointed out. 1 Introduction and overview The quantum origin of the classical world was so difficult to imagine for the forefathers of quantum theory that they were often willing to either postu- late its independent existence (Bohr), or even to give up quantum theory 1Departamento de F´ısica J.J. Giambiagi, FCEN, UBA, Pabell´on 1, Ciudad Universi- taria, 1428 Buenos Aires, Argentina 2Theoretical Astrophysics, MS B288 Los Alamos National Laboratory, Los Alamos, NM87545, USA c EDP Sciences, Springer-Verlag 1999 4 The title will be set by the publisher. and look for something with more fundamental classical underpinnings (de Broglie, and, to a lesser extent, also Einstein). The source of the problem is the quantum principle of superposition, which, in effect, exponentially expands the set of available states to all of the conceivable superpositions. Thus, coherent superpositions of dead and alive cats have —in the light of the quantum theory— the same right to exist as either of the two classical alternatives. Within the Hilbert space describing a given system “classically legal” states are exceptional. The set of all states in the Hilbert space is enormous as compared with the size of the set of states where one finds classical systems. Yet, it is a fact of life that classical objects are only found in a very small subset of all possible (and in principle, allowed) states. So, one has to explain the origin of this apparent “super-selection” rule that prevents the existence of most states in the Hilbert space of some physical systems. Decoherence and its principal consequence —environment-induced super-selection or einselection— account for this experimental fact of life. Decoherence is caused by the interaction between the system and its environment. Under a variety of conditions, which are particularly easy to satisfy for macroscopic objects, it leads to the einselection of a small subset of quasi-classical states from within the enormous Hilbert space. The classicality is then an emergent property, induced in the system by its interaction with the environment. Arbitrary superpositions are dismissed, and a preferred set of “pointer states” emerges. These preferred states are the candidate classical states. They correspond to the definite readings of the apparatus pointer in quantum measurements, as well as to the points in the phase space of a classical dynamical system. The role of the process of decoherence in inducing classicality has become clear only relatively recently —within the past two decades. The key idea is relatively simple: An environment of a quantum system can, in effect, mon- itor its states through continuous interaction. The imprint of the system left on the environment will contain information about selected states of the system. The states that leave the imprint without getting perturbed in the process are the preferred states. Thus, the key property of quasi-classical pointer states is their insensitivity to monitoring by —and consequently their resistance to the entanglement caused by— interaction with the envi- ronment: states that entangle least are most stable. They are also, almost by definition, the only states that remain an accurate description of the the system alone: All other states evolve into joint system-environment states, preserving their purity (and, consequently, the information the observer has about them) only when both the system and the environment are included in a larger “super system”. The fact that the interaction between quantum systems produces en- tanglement was well known almost since the beginning of quantum theory. Indeed, because the ideas of decoherence and einselection rely on quantum Decoherence 5 theory, and on quantum theory alone, it may be useful to ask why it took so long to arrive at a natural explanation of the quantum origins of clas- sicality. There are several possible explanations for this delay. We shall return to them later in the paper. But, for the moment, it is useful to note that the ability of environment-induced decoherence to result in the same set of preferred states, essentially independently of the initial state of the system and the environment, is crucial. This was not appreciated until relatively recently [1, 2]. It is precisely this stability of the set of preferred states that allows them to be regarded as good candidates for the quan- tum counterparts of classical reality. Indeed, only still more recent research on the predictability sieve has allowed for more fundamental and general understanding of the emergent classicality (see [3, 4] and also [5]). The prejudice that seems to have delayed serious study of the role of the “openness” of a quantum system in the emergence of classicality is it- self rooted in the classical way of thinking about the Universe. Within the context of classical physics, all fundamental questions were always settled in the context of closed systems. The standard strategy to ensure isolation involved enlarging a system —i. e., by including the immediate environ- ment. The expectation was that in this manner one can always reduce any open system to a larger closed system. This strategy does indeed work in classical physics, where the enlargement can help in satisfying conservation laws for quantities such as energy or momentum. It fails in the quantum case under discussion, because now it is the information (about the state of the system) that must be prevented from spreading. Information is much harder to contain when the system in question becomes larger. Thus, in the end, the only truly isolated macroscopic system is the Universe as a whole. And we, the observers, are certainly
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