Decoherence: an Explanation of Quantum Measurement

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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. Iwill 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 |\psi t\rangle at time t; and tensor product |\psi \rangle1 \otimes\phi | \rangle2 to denote picks out a preferred basis of states that are not (sig- the entangled state of ``object 1 in a state \psi 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 |\psi \rangle1 |\phi \rangle 2 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., \scrS ), and sets of states by subscripted Latin letters would be terribly inconvenient if your measurement re- in curly braces (e.g., \{s | i\rangle \}). 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 | \psi1 \rangle and | \psi2 \rangle are valid we usually measure a system by interacting with a small states, then so is (up to normalization) \alpha |\psi 1\rangle + \beta | \psi2 \rangle 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 description 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 \scrS which has possible pend on how we measure that system. states \psi i, we can suppose that we have an ensemble of such systems, in which the probability of one such system being in state \psi i is P (\psii ). If we make a measurement of II. ENVIRONMENT, SYSTEM, OBSERVERS a physical observable \phi which can take on values \phi j, then the probability of measuring it to be \phi j 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- \sum vironment, which will generally be denoted \scrE , the system P\mathrm{C}(\phi j) = P (\phi j| \psii )P (\psii ). (1) we are studying, which will generally be denoted \scrS or \scrA, i and the observer, or the measuring device, which will often be denoted \scrO . 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 \scrS which has possible states spanned by an or- coherence. The measurement device interacts with the thonormal basis of (classically) observable states \{ |\psi i\rangle. \} 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 \sum |\psi \rangle \equiv \alpha |\psi \rangle. (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 | \phi j\rangle, then the Hilbert space \scrE have little effect on each other; that the probability of measuring a system to be in a state |\phi j\rangle 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- P\mathrm{Q}\mathrm{M}(\phi j) \equiv\phi |\langle j| \psi \rangle | = \langle \phi j|\psi \rangle \langle\psi |\phi j\rangle alent to the ensemble average of that measurement over \sum 2 \sum \ast the states of the environment. This will be made more = |\alpha i \langle\phi j| \psii \rangle | + \alphai \alphak \langle\phi j|\psi i\rangle\psi \langle k|\phi j\rangle. i i,k rigorous in section IV B. i\not =k (3)

Then Equation 3 becomes Then the probability of measuring the system \scrS to be in a state | \phi \rangle \in\scrA is \sum \sum \ast j P\mathrm{Q}\mathrm{M}(\phi j) = P (\phi j| \psii )P (\psii ) + \alphai \alphak \langle\phi j|\psi i\rangle \langle\psik |\phi j\rangle i i,k \sum i\not =k P\scrE (\phi j) \equiv \langle\phi j| \otimes \langle\psi\scrE ,k| \psi| \rangle \langle\psi | \psi| \scrE ,k\rangle \otimes|\phi j\rangle \bigm| \bigm| k \bigm| \bigm| \sum 2 \Biggl\langle \bigm| \bigm| \Biggr\rangle = |\alpha \langle \phi |\psi \rangle | (7) \bigm| \sum \ast \bigm| m j \scrA,i m = P\mathrm{C}(\phi j) + \phi j\bigm| \alphai \alpha |\psi i\rangle \langle\psi k|\bigm| \phi j . (4) \bigm| k \bigm| m \bigm| i,k \bigm| \sum \ast \bigl\langle \bigm| \bigr\rangle \prime \bigm| i\not =k \bigm| + \alpham \alpha m \langle\phi j| \psi\scrA ,im \rangle \psi\scrA ,im\prime \bigm| \phi j \delta km,km\prime m,m\prime Note that for every term appearing in the sum over i m\not =m\prime and k on the first line, the conjugate of that term also appears, and so the sum is purely real. 2 If we again define P (\phi j|\psi \scrA,i ) \equiv |\alpham \langle\phi j|\psi \scrA,i \rangle | , then Thus, the quantum mechanical probability of measur- the first term is just the classical probability of measuring ing a particular outcome is, in some sense, the classical \phi , and so Equation 7 becomes probability of measuring that outcome, plus the expec- j tation value of the sum of the cross-terms, the interfer- \sum \ast \bigl\langle \bigm| \bigr\rangle P (\phi ) = P (\phi ) + \alpha \alpha \prime \langle\phi | \psi \rangle \psi \phi ence terms, when we have picked a basis of classically \scrE j \mathrm{C} j m m j \scrA ,im \scrA ,im\prime \bigm| j m,m\prime observable states; which terms are considered ``interfer- m\not =m\prime ence terms"" depends on which states we consider tobe km=km\prime classically observable. (8) We again see that the quantum mechanical probability is the classical probability, plus an interference term. B. Ensemble Averages Note, again, that the interference term is dependent on our choice of basis for \scrE . (The overall value, however, is Suppose now that the states of our system \scrS live in the basis independent. This is due to a general property of tensor product of the Hilbert spaces \scrA and \scrE , which have the operation that we are doing, better known as a par- orthonormal bases \{\psi | \scrA,i \rangle \} and \{ |\psi\scrE ,k\rangle \} of (classically) tial trace.) This time, the interference term dependent observable states of \scrA and \scrE , respectively. If we have on how many terms of |\phi j\rangle project on to the same vector \prime the ability to measure states of \scrA, but are incapable of in \scrE (i.e., for how many m \not = m we have km = km\prime ). distinguishing states of \scrE , so that the states | \phi j\rangle that we If, by and large, when considering only the terms in \psi are measuring are elements of \scrA, then we can define the with non-vanishing coefficients, there is a one-to-one cor- environment-agnostic overlap of a state |\psi \rangle with a vector respondence between basis elements | \psi\scrA ,i\rangle of \scrA and basis |\psi \scrA,i \rangle \in\scrA to be elements |\psi \scrE ,k\rangle of \scrE , then we see that the probability of \sum any particular outcome is the same as it is classically, P\scrE (\psi\scrA ,i) \equiv \langle \psi\scrA i | \otimes \langle\psi\scrE ,k| \psi| \rangle \langle\psi | \psi| \scrE ,k\rangle \otimes|\psi \scrAi \rangle and we cannot distinguish this quantum system from the k corresponding ensemble of classical ones. \sum 2 This treatment, and the result that the relevant quan- = | \psi\langle \scrAi | \otimes \langle\psi \scrE ,k| \psi| \rangle | (5) k tities to determining how far a system is from being classical are the terms that couple different basis elements of 1 which is a sum over an orthonormal basis of \scrE . This the environment, will be important for section VI C. is the norm squared of the overlap of the wave-function with | \psi\scrA ,i\rangle. If we assume that all basis states |\psi \scrE ,k\rangle of \scrE are equally likely, then we are simply adding up the joint V. MEASUREMENT probabilities P (\psi\scrA ,i and \psi \scrE ,k). If we express | \psi \rangle in the tensor basis, then we can say A. Premeasurement \sum | \psi \rangle = \alpha m| \psi\scrA ,im \rangle \otimes|\psi \scrE ,km \rangle (6) m Consider a quantum system \scrS initially in a state \sum |\psi \rangle = ai|s i\rangle (9) 1 I have tacitly assumed in this presentation that the environment i is equally likely to be found in any classically observable pure state, and that this is a good representation of the environment. and a measuring apparatus \scrA initially in a state |A 0\rangle. The more accurate way to model this is to construct a density Let \{ |si\rangle \} and \{A | i\rangle \} be orthonormal bases for the sys- matrix of combined \scrA \otimes\scrE system, and then take the partial trace tem and the apparatus, respectively; the | s \rangle are the pos- over the environment. See, for example, [6, \S 2.4] and [7, \S 2.1.8], i for a good introduction to density matrices. Equation 5 is the sible states of the system that we can measure with our appropriate matrix element of the partial trace over \scrE of the apparatus, and each |A i\rangle is the state that we observe the density matrix | \psi \rangle \langle\psi | , denoted T r\scrE (| \psi \rangle \langle\psi | ). In this language, apparatus to have when it measures the system to be Equation 5 becomes P (\psi ) = \bigl\langle \psi \bigm| T r (| \psi \rangle\psi \langle | )\bigm|\psi \bigr\rangle . \scrE \scrA ,i \scrA ,i\bigm| \scrE \bigm| \scrA ,i in the corresponding state |s i\rangle. Letting |A 0\rangle denote the Decoherence: An Explanation of Quantum Measurement 4 initial state of the apparatus, a premeasurement on the It is tempting to think that the flow of information in system is a unitary time evolution from (10) to (11): the quantum controlled not system is also unambiguous. However, this is not the case. If we prepare the target bit \Biggl( \Biggr) \sum in the state |0 \ranglet or | 1\ranglet , then premeasurement transfers |\Psi 0\rangle \equiv|\psi \rangle \otimes|A0\rangle = ai| si\rangle \otimes |A0\rangle (10) information from the control to the target. Consider, i however, the | \pm \rangle basis defined by \downarrow \sum 1 |\Psi \rangle \equiv a |s \rangle \otimes|A \rangle . (11) | \pm \rangle\equiv \surd (| 0\rangle \pm|1\rangle) . (15) t i i i 2 i Informally, a premeasurement is an interaction between Application of the transformation rule of Equation 13 to the system and the apparatus which does not change the Equation 15, after straightforward calculation, gives us state of the system when the apparatus is ignored.2 the new transformation rule This step of premeasurement is not sufficient to mea- | \pm \rangle \otimes |+\rangle - \rightarrow |\pm\rangle \otimes |+\rangle sure \scrS , as we shall see in the next section, but it expresses c t c t | \pm \rangle \otimes |-\rangle - \rightarrow |\mp\rangle \otimes |-\rangle . (16) the reason that we call \scrA a measuring device: \scrA attains c t c t a specific state corresponding to each possible state of If we prepare the ``target"" (the ``measuring device"") in the system, without disturbing the system. the state |+ \rangle t or | - \ranglet, then the target bit is unaffected by the premeasurement, and we have precisely the re- verse of the situation we had before; knowing the state B. Toy Model: c- not of the target bit before and after premeasurement gives no information about the control bit, but knowing the Two-state systems are useful toy models for some state of the control bit before and after gives complete quantum processes. The simplest toy model for mea- information about the target bit. The same result oc- surement is when the system we are trying to measure curs if we prepare the control bit in | +\rangle c or | - \ranglec. Zurek is a two state system, often called the ``target bit"", and claims in [1] that this basis ambiguity is present in all our measuring apparatus is also a two state system, often such target-control systems. called the ``control bit"". When we refer to this objects as ``bits"", the two states are labeled 0and1. Consider, as in [1], the logical operation ``controlled C. Three Descriptions of Measurement not"", which flips a target bit when a control bit is1,and leaves the target bit alone when the control bit is 0: Following Zurek in [1], I describe three different quan- 0 a - \rightarrow 0 a ; 1 a - \rightarrow 1 \nega (12) tum mechanical descriptions of the same measurement. c t c t c t c t Consider a quantum system \scrS with orthonormal states where \nega t is logical negation, sending 0 to 1 and 1 to 0. \{s | i\rangle \}, and three potential observers (measuring devices) The quantum analog of this classical operation evolves \scrI , \scrD , and \scrO , all of which can measure \scrS . Say that the by the rule possible (classical) states of the measuring devices are \{I | i\rangle, \} \{D | i\rangle, \} and \{ |Oi\rangle \}, respectively; say that ``\scrI is in (\alpha |0 \ranglec + \beta | 1\ranglec )| a\ranglet \rightarrow \alpha |0 \ranglec |a \rangle t + \beta |1 \rangle c| \neg a\ranglet (13) state | Ii\rangle"" corresponds to \scrI measuring \scrS to be in state |s i\rangle (for that i), and similarly for \scrD and \scrO . where the logical negation | a\neg \rangle t \equiv \neg|a\ranglet is given by the rule 1. The Insider \neg( \gamma |0 \rangle + \delta |1 \rangle ) \equiv \gamma |1 \rangle + \delta | 0\rangle (14) The simplest case of quantum measurement corre- Letting | A0\rangle0 \equiv| \ranglet and |A 1\rangle1 \equiv| \ranglet turns c-not in to a toy model for premeasurement. sponds to when the (classical) state of the system is al- In the classical controlled not, information unambigu- ready known with certainty; this is usually because it has ously flows from the control to the target; knowing the just been measured. An insider is an observer already state of the target bit before and after premeasurement perfectly correlated with the state of the system; the in- is sufficient to determine the control bit. Knowing the sider can predict the outcome of his next measurement state of the control bit before and after the premeasure- with certainty. ment, on the other hand, tells you nothing; the control More formally, an insider, who already knows the state bit is unchanged by the target bit. of \scrS to be si, has this information recorded somewhere, say in measuring device \scrI which is now in state Ii. A sub- sequent measurement of the system, using a measuring device \scrI \prime , looks like 2 More precisely, a premeasurement fixes the projection of the state \prime \prime onto \scrS . |I 0\rangle \otimes|Ii\rangle \otimes|si\rangleI -\rightarrow| i\rangle \otimes|Ii\rangle \otimes|si\rangle. (17) Decoherence: An Explanation of Quantum Measurement 5

This is the familiar classical case of measurement; if VI. DECOHERENCE you know the initial state of a classical system, and you know the (deterministic) laws according to which the sys- A. Environmentally Induced Superselection tem evolves, then you can predict with certainty what the outcome of a future measurement will be. B. Toy Model: c-not + Environment As Zurek notes in [1], every classical observer could aim, in principle, to be the ultimate insider; classically, if Consider again the controlled not not gate of sec- you ignore details about information storage limits, it is tion V B, but instead of just a control/system \scrS and a possible, in principle, to know the initial conditions of ev- target/apparatus \scrA, consider also the environment \scrE . ery particle in the universe, and thus be able to calculate, As before, the first step is premeasurement by the ap- in principle, the outcome of every measurement ahead of paratus of the system, in which the system, apparatus, time. In quantum mechanics, this is not possible, even and environment evolve as in principle.

|0 \ranglec \otimes |A0\rangle \otimes|\varepsilon \ast \rangle0 -\rightarrow| \ranglec \otimes |A0\rangle \otimes|\varepsilon \ast \rangle |1 \rangle \otimes |A \rangle \otimes|\varepsilon \rangle1 -\rightarrow| \rangle \otimes |A \rangle \otimes|\varepsilon \rangle. (20) 2. The Discoverer c 0 \ast c 1 \ast where |\varepsilon \ast \rangle is a generic state of the environment, and the Consider the standard description of quantum mea- other states are as they were defined above. In general, surement: Observer \scrD measures the system \scrS , and we have the transformation rule records a particular outcome s . If \scrD did not ahead of i (a|0 \rangle + b|1 \rangle )|A \rangle\varepsilon | \rangle -\rightarrow (a|0 \rangle | A \rangle + b| 1\rangle | A \rangle) |\varepsilon \rangle time what the outcome of the measurement would be c c 0 \ast c 0 c 1 \ast (21) (i.e., if \scrD describes the state of the system, before measurement, as a superposition of the states |s i\rangle), then this Let us suppose that the pointer states of the apparatus observer is called a discoverer. If the discoverer also ob- are einselected. That is, suppose that the states of the serves an insider \scrI , in addition to the system \scrS , then apparatus corresponding to our possible measurements, we can describe this observation mathematically by the | A0\rangle and | A1\rangle, are eigenstates of the interaction between (non-unitary) time evolution the measurement device \scrA and the environment \scrE . The environment then preforms a premeasurement on the ap- \sum |D 0\rangle \otimes ai| Ii\rangle \otimes|si\rangleD -\rightarrow| i\rangle \otimes|Ii\rangle \otimes|si\rangle. (18) paratus and system, resulting in the transformation i (a| 0\ranglec |A 0\rangle + b| 1\ranglec |A 1\rangle) | \varepsilon \ast \rangle This discoverer describes his observation with what \downarrow (22) is commonly called ``collapse of the wave-function""; the a| 0\rangle |A \rangle\varepsilon | \rangle + b|1 \rangle |A \rangle\varepsilon | \rangle discoverer might describe his measurement by saying: c 0 0 c 1 1 ``When I measure \scrS , which is currently in a superposi- where |\epsilon 0\rangle and | \epsilon 1\rangle are the states of the environment that tion state, I see it collapse into one of the classical states have premeasured the system. 3 2 |s i\rangle with respective probabilities |a i| . No matter which As Zurek notes in [1], the basis ambiguity of sec- state I measure \scrS to be in, I will percieve the insider \scrI tion V B is gone; in a tensor product of three or more to be in agreement with my measurement, and to have spaces, the form of the interaction Hamiltonian and time known this outcome ahead of time."" evolution pick out an unambiguously preferred basis for us.

3. The Outsider C. Toy Model: One Qubit + N Spins An outsider is an observer who describes the process of \scrI and \scrD measuring \scrS , but taking no measurements of 1. The Model \scrS (nor of \scrI nor of \scrD ) himself. Such an observer \scrO would say that the insider starts off entangled with the state of Following Zurek, Cucchietti, and Paz in [1, 8], suppose the system, and, in measuring the system, the discoverer that instead of a single bit, the environment consists of becomes entangled with the state of the system, but no collapse occurs. The mathematical description of this process (excluding the state of \scrO for simplicity) is the 3 I am abusing notation somewhat by not specifying | \varepsilon\ast \rangle ; to make time evolution this a unitary transformation, either the coefficients have to \sum \sum change slightly, other terms have to be introduced, or the envi- |D 0\rangle ai|I i\rangles | i\rangle -\rightarrow ai|D i\rangleI | i\rangles | i\rangle. (19) ronment must be prepared in a special state. When we explicitly i i calculate decoherence time in the next section, we will see that this representation of the transformation is only approximate, This is a unitary process, and well-described by what we and that in general other terms with exponentially suppressed called premeasurement above. coefficients are introduced. Decoherence: An Explanation of Quantum Measurement 6

N bits, or spins, initially in states \alphak |0 \ranglee + \beta k| 1\ranglee , each states of the environment are equally likely, the ensemble coupled to the apparatus by a coupling coefficient gk. In average over the environment is the simplest case, we can say that the self-Hamiltonian \sum \ast of both the system and the apparatus are zero. I con- \langle\varepsilon i|2 \Re (ab | 0\rangle a\langle 1|a \otimes |\scrE0(t)\rangle1 \langle\scrE (t)| ) | \varepsilon i\rangle sider the case where the system is first premeasured by i \ast the apparatus, and then undergoes decoherence. In ana- = 2\Re (ab | 0\ranglea \langle1 |a \langle1 \scrE (t)| 0\scrE (t)\rangle) (29) lyzing decoherence, I drop the state of the system from the wave-function, for convenience, because it does not Thus we see that the time dependence of the interfer- change. The interaction Hamiltonian is then ence terms is given by

N \sigma z = |0 \rangle \langle0| - |1\rangle \langle1| . (24) \prod \bigl( 2 \bigr) r(t) = cos(2gkt/\hbar) + i(1 - 2| \beta k| ) sin(2gkt/\hbar ) . If we make the simplifying assumption that the en- k=1 vironment begins in a pure state, then the Schr\"odinger (31) equation gives the time evolution transformation rule We are interested primarily in how quickly this drops off to zero, if at all. N \bigotimes Suppose that the \beta k and the gk are randomly sam- | \Psi\rangle (0) \equiv (a|0 \rangle + b|1 \rangle ) g \bigl( \alpha | 0\rangle + \beta | 1\rangle \bigr) a a k k ek k ek pled from some distribution. If we assume that we are k=1 sampling from and ensemble of systems where | \beta k| are \downarrow uniformly distributed between 0 and 1, then we can calculate the ensemble average of r(t) by integrating over |\Psi (t)\rangle = a| 0\rangle a| \scrE 0(t)\rangle + b|1 \ranglea | 1\scrE (t)\rangle (25) each | \beta k| from 0 to 1. If we suppose further, for simplicity, that the gk are distributed normally with mean where 2 \mug and variance \sigma g , then we can calculate the ensemble N average over g by integrating over a normal distribution \bigotimes \Bigl( \Bigr) | \scrE (t)\rangle \equiv \alpha eigkt/\hbar | 0\rangle + \beta e - igkt/\hbar |1 \rangle for each gk: 0 k ek k ek k=1 \biggl( \biggl( \biggr) \biggr)N 2 1 \equiv |\scrE ( - t)\rangle (26) \langler (t)\rangle = e - 2(\sigma g t/\hbar ) cos (2t\mu / ) + i sin (2t\mu / ) . 1 g \hbar 3 g \hbar \bigotimes (32) and where k denotes a tensor product indexed by k. Thus we see that we expect the interference terms to drop 2 2- N(\sigma g t/ ) 2. Calculating Decoherence Time off as e \hbar , and the characteristic\surd decoherence time scale of the system is \hbar/ ( 2N\sigma g). Zurek, Cucchietti, and Paz treat this model more gen- We can ask the following question: as a function of erally in [8], making fewer assumptions about the form time, how far from classical should a discoverer associ- of the interaction Hamiltonian. ated with the apparatus expect his measurements to be? Recall from section IV that what makes the quantum law of superposition different from the classical (probabilis- VII. CONCLUDING REMARKS; tic) law of superposition is the interference terms, or cross WHAT'S LEFT? terms. Thus asking this question is thus approximately equivalent to asking for the expectation value of the sum This paper provides only a brief introduction to deco- of the interference terms, that is, the expectation over herence. For more complicated systems, density matri- the environment of ces and reduced density matrices are powerful tools that ab\ast |0 \rangle | \scrE (t)\rangle \langle1| \langle \scrE (t)| + a\ast b|1 \rangle | \scrE (t)\rangle0 \langle | \langle \scrE (t)| (27) unify the calculations I presented. (See [6, \S2.4] and [7, a 0 a 1 a 1 a 0 \S 2.1.8] for more details.) Zurek develops an alternative which can be shortened to description of decoherence in [1, 4] based on information transfer and the ability of a state to propagate its infor- \ast mation through the environment. 2\Re (ab | 0\rangle a\langle1 |a \otimes |\scrE0(t)\rangle1 \langle\scrE (t)|) . (28) Decoherence provides a mathematical formalism for Pick an orthonormal basis of the environment \{ |\varepsilon i\rangle \}. talking about the apparent ``collapse"" of the wave- Then, as seen in section IV B, if we assume that all basis function. With the aid of decoherence, we can recover the Decoherence: An Explanation of Quantum Measurement 7 illusion of the classical; we can describe why observers of the point of view of an outsider (see section V C 3), there open quantum systems, both of which are constantly in- seems to be no similar description for probability in de- teracting with (and being measured by) the environment, scribing a discoverer making a measurement. There is observe systems to behave classically, on average. no ambiguity about the outcome of the measurement, as Everett and Gleason independently proved in [9, 10] described by the outsider. So not only does probability that the norm squared measure is the only possible prob- seem to be subjective, in the sense that its definition re- ability measure on a Hilbert space of dimension \geq 3. quires you to be a discoverer, but the probability space Zurek derived Born's Rule in [1, 4, 11] from the stand- itself, the set of possible outcomes that you label with point of decoherence, using the idea that if there are mul- likelihoods, seems to be subjective, in that it does not tiple possible states, and we are completely ignorant of admit any remotely familiar classical description, so long which state the system is in, then there must be an equal as you are an outsider. While Zurek et al. give an op- probability of measuring the system to be in any of those erational definition of probability from decoherence, it given states. does not seem to me be philosophically satisfying; de- As stated by Yudkowsky in [12], however, decoherence coherence is almost, but not quite, enough, to solve the does not completely solve the measurement problem, at measurement and interpretation problems. least in so far as I can tell. When we talk about probabilities classically, we talk about the probability of an event happening. We specify the set of possible outcomes ACKNOWLEDGMENTS of a particular process, and after the process completes, we will have observed one of those outcomes. One way The author gratefully acknowledges Fakhri Zahedy of characterizing the probability of a particular outcome ([email protected]), his peer reviewer, Francesco D'Eramo is by the frequency of that outcome in the limit of an ([email protected]), his writing assistant, and Professor infinite sequence of identically prepared processes. From Jesse Thaler ([email protected]), his 8.06 Professor.

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