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Decoherence: an Explanation of Quantum Measurement

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Decoherence: an Explanation of Quantum Measurement Decoherence: An Explanation of Quantum Measurement Jason Gross\ast Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139-4307 (Dated: May 5, 2012) The description of the world given by quantum mechanics is at odds with our classical experience. Most of this conflict resides in the concept of \measurement". After explaining the originsof the controversy, and introducing and explaining some of the relevant mathematics behind density matrices and partial trace, I will introduce decoherence as a way of describing classical measurements from an entirely quantum perspective. I will discuss basis ambiguity and the problem of information flow in a system + observer model, and explain how introducing the environment removes this ambiguity via environmentally-induced superselection. I will use a controlled not gate as a toy model of measurement; including a one-bit environment will provide an example of how interaction can pick out a preferred basis, and included an N-bit environment will give a toy model of decoherence. I. INTRODUCTION function over each independent observable. Moreover, the law of superposition, which states that any linear For any physicist who learns classical mechanics before combination of valid quantum states is also valid, gives quantum mechanics, the theory of quantum mechanics at rise to states that do not seem to correspond to any clas- first appears to be a strange and counter-intuitive wayto sical system anyone has experienced. The most famous describe the world. The correspondence between classical example of such a state is that of Schr\"odinger'shypo- experience and quantum theory is not obvious, and has thetical cat, which is in a superposition of being alive been at the core of debates about the interpretation of and being dead. quantum mechanics since its inception. As Zurek notes in [1], one might na\"{\i}vely assume that In this paper, I will begin by describing the measure- the law of superposition should always be taken literally, ment and interpretation problems of quantum mechanics. that fundamentally, reality consists of quantum wave- I will talk about interference, one of the essential charac- functions. From this point of view, there is no a priori teristics of quantum systems. After describing a criterion reason to expect systems to have the well-defined local- for determining how close to classical a quantum system ized behavior that we observe (such as definite position is, I will introduce decoherence, the thermodynamically and momenta). Einstein noted in a 1954 letter to Born irreversible transfer of information from a system to the that such localization is incompatible with quantum me- environment; it provides a description of how to recover chanics, because time evolution tends to smear localized the illusion of classical mechanics from quantum mechan- states. [1, 3] Additionally, such superpositions often in- ics. I will present a toy model of decoherence|a one-bit volve interference effects, where probabilities do not add system with a one-bit measuring device, coupled to an N- classically. bit environment|and calculate decoherence time. I will close with remarks about how close decoherence gets us to solving the interpretation and measurement problems, B. The Role of The Environment and what's left to be explained. Most systems in the world are open in the thermody- namic sense; they interact with the environment, which A. The Problem: Classicality can be modeled as a \bath" with which a system can exchange information and energy. Decoherence is a de- The so-called \measurement problem" is the problem scription of the thermodynamically irreversible exchange of determining the correspondence between quantum me- of information with the environment. If we are largely chanical descriptions of a system and our classical expe- incapable of measuring the environment, then decoher- rience with measurements. One of the essential problems ence quickly renders most quantum systems indistin- in finding such a correspondence is that there are far guishable from classical ensembles of their possible (clas- more permissible quantum mechanical states of a system sical) states. We will see how quickly this happens by than there are classical states. Classically, to specify the developing a toy model for decoherence and calculating state of a system, you give the values of some small set the decoherence time in section VI C. of numbers, such as position, velocity, mass, charge, etc. The environment is constantly interacting with most Quantum mechanically, you must pick out an element systems, especially large ones, and most measuring de- of a vast Hilbert space, i.e., you must specify a complex vices are rather large. The strength of that interaction usually dominates the strength of the interaction between the system being measured and the device being used to measure it. This results in what is called environmentally \ast [email protected] induced super-selection, or einselection for short. [1] The Decoherence: An Explanation of Quantum Measurement 2 interaction between the observer and the environment j ti at time t; and tensor product j i1 \otimes j\phi i2 to denote picks out a preferred basis of states that are not (sig- the entangled state of \object 1 in a state and object nificantly) perturbed by further interactions; these are 2 in a state \phi ". Occasionally, the tensor product sym- the pointer states that remain correlated with the sys- bol will be dropped and j i1j\phi i2 will be used to save tem despite further interactions with the environment. space. Quantum systems are denoted by script letters Other states don't make for good measuring devices; it (e.g., S), and sets of states by subscripted Latin letters would be terribly inconvenient if your measurement re- in curly braces (e.g., fjsiig). Unless otherwise specified, sults were scrambled every time a bit of light shined on the states in these sets are assumed to be orthonormal. them, or every time a bit of air passed by. We will see effect this more formally in section IV B and section VIB. Though it is beyond the scope of this paper, Zurek IV. THE HALLMARK OF THE QUANTUM: develops a complementary description of this process in INTERFERENCE AND RELATIVE PHASES [1, 4], which he calls quantum Darwinism, in which the classically observable, einselected states are the ones that The primary difference between quantum mechanical are best at propagating information about themselves states and classical states is the quantum mechanical law through the environment. Very roughly, the idea is that of superposition. It states that if j 1i and j 2i are valid we usually measure a system by interacting with a small states, then so is (up to normalization) \alpha j 1i + \beta j 2i for subset of the environment (e.g., by measuring some small arbitrary complex \alpha and \beta . This is most easily seen in subset of all the photons that bounce off the system in relative phases between terms and in interference exper- question). So the relative frequencies of states that we iments. [5] observe are the relative ``fitnesses"" of those states inen- coding their information in the environment. This de- scription has the benefit that it does not necessarily re- A. Interference Terms quire us to specify the measuring device as fundamental; the intrinsic characteristics of a system should not de- Classically, if there is a system S which has possible pend on how we measure that system. states i, we can suppose that we have an ensemble of such systems, in which the probability of one such system being in state i is P ( i). If we make a measurement of II. ENVIRONMENT, SYSTEM, OBSERVERS a physical observable \phi which can take on values \phij , then the probability of measuring it to be \phij is the sum of the The formalism of decoherence relies on partitioning the products of conditional probabilities: universe into three disjoint, interacting systems: the en- X vironment, which will generally be denoted E, the system PC(\phij ) = P (\phij j i)P ( i): (1) we are studying, which will generally be denoted S or A, i and the observer, or the measuring device, which will of- ten be denoted O. The environment and the system are Quantum mechanically, we may suppose that there is continuously interacting, and this forms the basis of de- a system S which has possible states spanned by an or- coherence. The measurement device interacts with the thonormal basis of (classically) observable states fj iig. system for some short period of time, after which it is Consider an ensemble of identically prepared systems, considered to have measured the system. We assume each in state that the observers are largely incapable of measuring the X j i \equiv \alpha j i: (2) environment. As we will see in section IV B and in our i i toy model in section VI C, after a short time, parts of the i wave-function that project on to orthogonal elements of If we measure the observable with eigenvectors j\phij i, then the Hilbert space E have little effect on each other; that the probability of measuring a system to be in a state j\phij i is, there is very little interference between different states is of the environment. Thus, from the point of view of the 2 observer, the outcome of any measurement will be equiv- PQM(\phij ) \equiv jh\phij j ij = h\phij j ih j\phij i alent to the ensemble average of that measurement over X 2 X \ast the states of the environment. This will be made more = j\alphai h\phij j iij + \alphai \alphak h\phij j iih kj\phij i: i i;k rigorous in section IV B. i6=k (3) III.
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