276 Quantum Mechanics: Weak Measurements it is argued that statistical fluctuations around the Ghirardi G, Pearle P, and Rimini A (1990) Markov processes in canonical ensemble can give rise to the behavior of Hilbert space and continuous spontaneous localization of wave-function collapse, of the kind discussed here, systems of identical particles. Physical Review A 42: 78–89. Holland PR (1993) The Quantum Theory of Motion: An Account both energy-driven and CSL-type mass-density-driven of the de Broglie–Bohm Causal Interpretation of Quantum collapse so that, with the latter, comes the Born Mechanics. Cambridge: Cambridge University Press. probability interpretation of the algebra. The Hamil- Nelson E (1966) Derivation of the Schro¨ dinger equation from tonian needed for this theory to work is not provided Newtonian mechanics. Physical Review 150: 1079–1085. but, as the argument progresses, its necessary features Pearle P (1979) Toward explaining why events occur. Interna- tional Journal of Theoretical Physics 18: 489–518. are delimited. Pearle P (1989) Combining stochastic dynamical state-vector reduction with spontaneous localization. Physical Review A See also: Quantum Mechanics: Foundations. 39: 2277–2289. Pearle P (1999) Collapse models. In: Petruccione F and Breuer HP (eds.) Open Systems and Measurement in Relativistic Quan- tum Theory, pp. 195–234. Heidelberg: Springer. (ArXiv: Further Reading quant-ph/9901077). Valentini A (1991) Signal-locality, uncertainty, and the subquan- Adler SL (2004) Quantum Theory as an Emergent Phenomenon. tum H-theorem. I and II. Physics Letters A 156: 5–11 and Cambridge: Cambridge University Press. 158: 1–8. Bassi A and Ghirardi GC (2003) Dynamical reduction models. Valentini A (2002a) Signal-locality in hidden-variables theories. Physics Reports 379: 257–426 (ArXiv: quant-ph/0302164). Physics Letters A 297: 273–278. Bell JS (1987) Speakable and Unspeakable in Quantum Valentini A (2002b) Subquantum information and computation. Mechanics. Cambridge: Cambridge University Press. Pramana – Journal of Physics 59: 269–277 (ArXiv: quant-ph/ Bohm D (1952) A suggested interpretation of the quantum theory 0203049). in terms of ‘hidden’ variables. I and II. Physical Review Valentini A (2004a) Black holes, information loss, and hidden 85: 166–179 and 180–193. variables, ArXiv: hep-th/0407032. Bohm D and Bub J (1966) A proposed solution of the Valentini A (2004b) Universal signature of non-quantum systems. measurement problem in quantum mechanics by a hidden Physics Letters A 332: 187–193 (ArXiv: quant-ph/0309107). variable theory. Reviews of Modern Physics 38: 453–469. Valentini A and Westman H (2005) Dynamical origin of quantum Du¨ rr D, Goldstein S, and Zanghı` N (2003) Quantum equilibrium probabilities. Proceedings of the Royal Society of London and the role of operators as observables in quantum theory, Series A 461: 253–272. ArXiv: quant-ph/0308038. Wallstrom TC (1994) Inequivalence between the Schro¨ dinger Ghirardi G, Rimini A, and Weber T (1986) Unified dynamics for equation and the Madelung hydrodynamic equations. Physical microscopic and macroscopic systems. Physical Review D Review A 49: 1613–1617. 34: 470–491.
Quantum Mechanics: Weak Measurements L Dio´si, Research Institute for Particle and Nuclear to extend the notion of mean for normalized bilinear Physics, Budapest, Hungary expressions (Aharonov et al. 1988):
ª 2006 Elsevier Ltd. All rights reserved. hf jA^jii A ¼: ½2 w hf jii However unusual is this structure, standard quan- Introduction tum theory provides a plausible statistical interpre- tation for it, too. The two pure states jii, jf i play the In quantum theory, the mean value of a certain roles of the prepared initial and the postselected observable A^ in a (pure) quantum state jii is defined final states, respectively. The statistical interpreta- by the quadratic form: tion relies upon the concept of weak measurement. In a single weak measurement, the notorious ^ ^ hAii ¼: hijAjii½1 decoherence is chosen asymptotically small. In physical terms, the coupling between the measured Here A^ is Hermitian operator on the Hilbert space state and the meter is assumed asymptotically weak. H of states. We use Dirac formalism. The above The novel mean value [2] is called the (complex) mean is interpreted statistically. No other forms had weak value. been known to possess a statistical interpretation in The concept of quantum weak measurement standard quantum theory. One can, nonetheless, try (Aharonov et al. 1988) provides particular Quantum Mechanics: Weak Measurements 277
conclusions on postselected ensembles. Weak mea- respectively. Here G is the central Gaussian surements have been instrumental in the interpreta- distribution of variance . Note that, as expected, tion of time-continuous quantum measurements on eqn [5] implies eqn [4]. Nonzero means that the single states as well. Yet, weak measurement itself measurement is nonideal, yet the expectation value can properly be illuminated in the context of E[a] remains calculable reliably if the statistics N is classical statistics. Classical weak measurement as suitably large. well as postselection and time-continuous measure- Suppose the spread of A in state is finite: ment are straightforward concepts leading to con- 2 2 2 clusions that are natural in classical statistics. In A ¼: hA i hAi < 1½7 quantum context, the case is radically different and Weak measurement will be defined in the asympto- certain paradoxical conclusions follow from weak tic limit (eqns [8] and [9]) where both the stochastic measurements. Therefore, we first introduce the error of the measurement and the measurement classical notion of weak measurement on postse- statistics go to infinity. It is crucial that their rate is lected ensembles and, alternatively, in time-contin- kept constant: uous measurement on a single state. Certain idioms from statistical physics will be borrowed and certain ; N !1 ½8 not genuinely quantum notions from quantum 2 theory will be anticipated. The quantum counterpart 2 ¼: ¼ const: ½9 of weak measurement, postselection, and continuous N measurement will be presented afterwards. The Obviously for asymptotically large , the precision apparent redundancy of the parallel presentations of individual measurements becomes extremely is of reason: the reader can separate what is weak. This incapacity is fully compensated by the common in classical and quantum weak measure- asymptotically large statistics N. In the weak ments from what is genuinely quantum. measurement limit (eqns [8] and [9]), the probability distribution pw of the arithmetic mean a of the N independent outcomes converges to a Gaussian distribution: Classical Weak Measurement Given a normalized probability density (X) over pwða Þ!G a hAi ½10 the phase space {X}, which we call the state, the The Gaussian is centered at the mean hAi , and the mean value of a real function A(X) is defined as Z variance of the Gaussian is given by the constant rate [9]. Consequently, the mean [3] is reliably hAi ¼: dXA ½3 calculable on a statistics N growing like 2. With an eye on quantum theory, we consider two Let the outcome of an (unbiased) measurement of A situations – postselection and time-continuous be denoted by a. Its stochastic expectation value measurement – of weak measurement in classical E[a] coincides with the mean [3]: statistics.
E½a ¼hAi ½4 Performing a large number N of independent Postselection measurements of A on the elements of the ensemble For the preselected state , we introduce postselec- of identically prepared states, the arithmetic mean a tion via the real function (X), where 0 1. of the outcomes yields a reliable estimate of E[a] The postselected mean value of a certain real and, this way, of the theoretical mean hAi . function A(X) is defined by Suppose, for concreteness, the measurement outcome a is subject to a Gaussian stochastic h Ai hAi ¼: ½11 error of standard dispersion >0. The probability h i distribution of a and the update of the state where h i is the rate of postselection. Postselection corresponding to the Bayesian inference are means that after having obtained the outcome a described as regarding the measurement of A, we measure the pðaÞ¼hiG ða AÞ ½5 function , too, in ideal measurement with random outcome upon which we base the following 1 random decision. With probability , we include ! G ða AÞ ½6 pðaÞ the current a into the statistics and we discard it 278 Quantum Mechanics: Weak Measurements with probability 1 . Then the coincidence of E[a] Equations [17] and [18] are the special case of the and hAi ,asineqn [4], remains valid: Kushner–Stratonovich equations of time-continuous Bayesian inference conditioned on the continuous E½a ¼ hAi ½12 measurement of A yielding the time-dependent Therefore, a large ensemble of postselected states outcome value at. Formal time derivatives of both sides of eqn [17] yield the heuristic equation allows one to estimate the postselected mean hAi . Classical postselection allows introducing the at ¼hAi þ g t ½19 effective postselected state: t Accordingly, the current measurement outcome is ¼: ½13 always equal to the current mean plus a term h i proportional to standard white noise t. This Then the postselected mean [11] of A in state can, plausible feature of the model survives in the by eqn [14], be expressed as the common mean of A quantum context as well. As for the other equation in the effective postselected state : [18], it describes the gradual concentration of the distribution t in such a way that the variance A hAi ¼hAi ½14 t tends to zero while A tends to a random h i t As we shall see later, quantum postselection is asymptotic value. The details of the convergence more subtle and cannot be reduced to common depend on the character of the continuously mea- sured function A(X). Consider a stepwise A(X): statistics, that is, to that without postselection. The X quantum counterpart of postselected mean does not AðXÞ¼ a P ðXÞ½20 exist unless we combine postselection and weak measurement. The real values a are step heights all differing from Time-Continuous Measurement each other. The indicator functions P take values 0 or 1 and form a complete set of pairwise disjoint For time-continuous measurement, one abandons the functions on the phase space: ensemble of identical states. One supposes that a single X time-dependent state t is undergoing an infinite P 1 ½21 sequence of measurements (eqns [5] and [6])ofA employed at times t = t, t = 2 t, t = 3 t, ... . The rate P P ¼ P ½22 =:1= t goes to infinity together with the mean squared error 2. Their rate is kept constant: In a single ideal measurement of A, the outcome a is one of the a ’s singled out at random. The ; !1 ½15 probability distribution of the measurement out- 2 come and the corresponding Bayesian update of the g2 ¼: ¼ const: ½16 state are given by In the weak measurement limit (eqns [15] and [16]), p P 23 ¼h i 0 ½ the infinite frequent weak measurements of A constitute the model of time-continuous measure- 1 ! P ¼: ½24 ment. Even the weak measurements will signifi- 0 p 0 cantly influence the original state , due to the 0 respectively. Equations [17] and [18] of time- accumulated effect of the infinitely many Bayesian continuous measurement are a connatural time- updates [6]. The resulting theory of time-continuous continuous resolution of the ‘‘sudden’’ ideal measurement is described by coupled Gaussian measurement (eqns [23] and [24]) in a sense that processes [17] and [18] for the primitive function they reproduce it in the limit t !1. The states of the time-dependent measurement outcome t are trivial stationary states of the eqn [18]. It can be and, respectively, for the time-dependent Bayesian shown that they are indeed approached with conditional state : t probability p for t !1. d A dt g dW 17 t ¼h i t þ t ½ Quantum Weak Measurement d g 1 A A dW 18 t ¼ h i t t t ½ In quantum theory, states in a given complex Here dWt is the Itoˆ differential of the Wiener Hilbert space H are represented by non-negative process. density operators ˆ, normalized by tr ˆ = 1. Like the Quantum Mechanics: Weak Measurements 279 classical states , the quantum state ˆ is interpreted Hilbert space L2 of a hypothetic meter. Suppose statistically, referring to an ensemble of states with R 2 ( 1, 1) is the position of the ‘‘pointer.’’ Let its ^ the same ˆ. Given a Hermitian operator A, called initial state ˆM be a pure central Gaussian state of observable, its theoretical mean value in state ˆ is width ; then the density operator ˆM in Dirac defined by position basis takes the form Z Z ^ ^ hAi ^ ¼ trðA ^Þ½25 0 1=2 1=2 0 0 ^M ¼ dR dR G ðRÞG ðR ÞjRihR j½30 Let the outcome of an (unbiased) quantum measure- ment of A^ be denoted by a. Its stochastic expectation We are looking for a certain dynamical interaction value E[a] coincides with the mean [25]: to transmit the ‘‘value’’ of the observable A^ onto the pointer position R^. To model the interaction, we ^ E½a ¼hAi ^ ½26 define the unitary transformation [31] to act on the tensor space H L2: Performing a large number N of independent ^ measurements of A on the elements of the ensemble U^ ¼ expðiA^ K^Þ½31 of identically prepared states, the arithmetic mean a ^ of the outcomes yields a reliable estimate of E[a] Here K is the canonical momentum operator ^ conjugated to R^: and, this way, of the theoretical mean hAi ˆ . If the measurement outcome a contains a Gaussian sto- ^ chastic error of standard dispersion , then the expðiaKÞjRi¼jR þ ai½32 probability distribution of a and the update, called The unitary operator U^ transforms the initial collapse in quantum theory, of the state are uncorrelated quantum state into the desired corre- described by eqns [27] and [28], respectively. (We lated composite state: adopt the notational convenience of physics litera- ^ ^ ^ y ture to omit the unit operator ^I from trivial ¼: U ^ ^MU ½33 expressions like a^I.) DE Equations [30]–[33] yield the expression [34] for the ^ state ˆ : pðaÞ¼ G ða AÞ ½27 ^ Z Z ^ 0 1=2 ^ 1=2 ¼ dR dR G ðR AÞ ^G 1 1=2 ^ 1=2 ^ ^ ! G ða AÞ ^G ða AÞ½28 pðaÞ ðR0 A^Þ jRihR0j½34
Nonzero means that the measurement is nonideal, Let us write the pointer’s coordinate operator R^ into but the expectation value E[a] remains calculable the standard form [35] in Dirac position basis: reliably if N is suitably large. Z Weak quantum measurement, like its classical R^ ¼ dajaihaj½35 counterpart, requires finite spread of the observable A^ on state ˆ: The notation anticipates that, when pointer R^ is 2 ^ ^ 2 ^ 2 ^A ¼: hA i ^ hAi ^ < 1½29 measured ideally, the outcome a plays the role of the nonideally measured value of the observable A^. Weak quantum measurement, too, will be defined in Indeed, let us consider the ideal von Neumann the asymptotic limit [8] introduced for classical weak measurement of the pointer position on the corre- measurement. Single quantum measurements can no lated composite state ˆ . The probability of the ^ more distinguish between the eigenvalues of A. Yet, outcome a and the collapse of the composite state the expectation value E[a]oftheoutcomea remains are given by the following standard equations: 2 calculable on a statistics N growing like . hi Both in quantum theory and classical statistics, pðaÞ¼tr ð^I jaihajÞ ^ ½36 the emergence of nonideal measurements from ideal hi ones is guaranteed by general theorems. For com- 1 ^ ! ð^I jaihajÞ ^ð^I jaihajÞ ½37 pleteness of this article, we prove the emergence of pðaÞ the nonideal quantum measurement (eqns [27] and [28]) from the standard von Neumann theory of respectively. We insert eqn [34] into eqns [36] and ideal quantum measurements (von Neumann 1955). [37]. Furthermore, we take the trace over L2 of both The source of the statistical error of dispersion sides of eqn [37]. In such a way, as expected, eqns ^ is associated with the state ˆM in the complex [36] and [37] of ideal measurement of R yield the 280 Quantum Mechanics: Weak Measurements earlier postulated eqns [27] and [28] of nonideal The interpretation of postselection itself reduces to a measurement of A^. simple procedure. One performs the von Neumann ideal measurement of the Hermitian projector jf ihf j, then includes the case if the outcome is 1 and Quantum Postselection discards it if the outcome is 0. The rate of 2 A quantum postselection is defined by a Hermitian postselection is jhf jiij . We note that a certain operator satisfying 0^ ˆ ^I. The corresponding statistical interpretation of Im Aw, too, exists postselected mean value of a certain observable A^ is although it relies upon the details of the ‘‘meter.’’ defined by We outline a heuristic proof of the central equation [40]. One considers the nonideal measure- ^ ^ h Ai^ ment (eqns [27] and [28])ofA^ followed by the ideal hA^i ¼: Re ½38 ^ ^ ^ measurement of ˆ . Then the joint distribution of the h i ^ corresponding outcomes is given by eqn [42]. The ˆ The denominator h i ˆ is the rate of quantum probability distribution of the postselected outcomes postselection. Quantum postselection means that a is defined by eqn [43], and takes the concrete form ^ after the measurement of A, we measure the [44]. The constant N assures normalization: observable ˆ in ideal quantum measurement and ^ 1=2 ^ 1=2 ^ we make a statistical decision on the basis of the pð ;aÞ¼tr ð ÞG ða AÞ ^G ða AÞ ½42 outcome . With probability , we include the case Z in question into the statistics while we discard it 1 with probability 1 . By analogy with the classical pðaÞ¼: pð ; aÞ d ½43 N case [12], one may ask whether the stochastic expectation value E[a] of the postselected measure- DE 1 1=2 ^ ^ 1=2 ^ ment outcome does coincide with pðaÞ¼: G ða AÞ G ða AÞ ½44 N ^ ? ^ ^ E½a ¼ ^ hAi ^ ½39 Suppose, for simplicity, that A is bounded. When !1, eqn [44] yields the first two moments of Contrary to the classical case, the quantum equation the outcome a: [39] does not hold. The quantum counterparts of ^ classical equations [12]–[14] do not exist at all. E½a ! ^ hAi ½45 ^ ^ Nonetheless, the quantum postselected mean ˆ hAi ˆ possesses statistical interpretation although E½a2 2 ½46 restricted to the context of weak quantum measure- Hence, by virtue of the central limit theorem, the ments. In the weak measurement limit (eqns [8] and probability distribution [40] follows for the average [9]), a postselected analog of classical equation [10] a of postselected outcomes in the weak measurement holds for the arithmetic mean a of postselected weak limit (eqns [8] and [9]). quantum measurements: ^ pwða Þ!G a ^ hAi ^ ½40 Quantum Weak-Value Anomaly The Gaussian is centered at the postselected mean Unlike in classical postselection, effective postse- ^ lected quantum states cannot be introduced. We can ˆ hAi ˆ , and the variance of the Gaussian is given by the constant rate [9]. Consequently, the mean [38] ask whether eqn [47] defines a correct postselected becomes calculable on a statistics N growing like 2. quantum state: Since the statistical interpretation of the postse- ^ ^ lected quantum mean [38] is only possible for weak ^? ¼: Herm ½47 ^ ^ ^ h i ^ measurements, therefore ˆ hAi ˆ is called the (real) ^ weak value of A. Consider the special case when This pseudo-state satisfies the quantum counterpart both the state ˆ = jiihij and the postselected operator of the classical equation [14]: =ˆ jf ihf j are pure states. Then the weak value ^ ? ˆ hAi ˆ takes, in usual notations, a particular form ^ ^ ^ hAi ^ ¼ tr A ^ ^ ½48 [41] yielding the real part of the complex weak value A [1]: ? w In general, however, the operator ˆ ˆ is not a density operator since it may be indefinite. Therefore, eqn ^ ^ hf jAjii [47] does not define a quantum state. Equation [48] f hAi ¼: Re ½41 i hf jii does not guarantee that the quantum weak value Quantum Mechanics: Weak Measurements 281
^ ˆ hAi ˆ lies within the range of the eigenvalues of the from eqn [9] after replacing N by the size of the observable A^. postselected statistics which is approximately N=4:
Let us see a simple example for such anomalous 2 weak values in the two-dimensional Hilbert space. ¼ 400=N ½55 Consider the pure initial state given by eqn [49] and Accordingly, if N = 3600 independent quantum the postselected pure state by eqn [50], where measurements of precision = 10 are performed 2 [0, ] is a certain angular parameter. regarding the observable A^, then the arithmetic mean a of the 900 postselected outcomes a will be 1 ei =2 jii¼pffiffiffi ½49 2 0.33. This exceeds significantly the largest 2 e i =2 eigenvalue of the measured observable A^. Quantum postselection appears to bias the otherwise unbiased 1 e i =2 nonideal weak measurements. jf i¼pffiffiffi ½50 2 ei =2 Quantum Time-Continuous Measurement 2 The probability of successful postselection is cos . The mathematical construction of time-continuous If 6¼ =2, then the postselected pseudo-state quantum measurement is similar to the classical one. follows from eqn [47]: We consider the weak measurement limit (eqns [15] and [16]) of an infinite sequence of nonideal 1 1 cos 1 ^? ¼ ½51 quantum measurements of the observable A^ at ^ 2 cos 1 1 t = t,2 t, ..., on the time-dependent state ˆt. The resulting theory of time-continuous quantum mea- This matrix is indefinite unless = 0, its two surement is incorporated in the coupled stochastic eigenvalues are 1 cos 1 . The smaller the post- equations [56] and [57] for the primitive function selection rate cos2 , the larger is the violation of the t of the time-dependent outcome and the conditional positivity of the pseudo-density operator. Let the time-dependent state ˆ , respectively (Dio´si 1988): weakly measured observable take the form t ^ d t ¼hAi dt þ gdWt ½56 01 ^t A^ ¼ ½52 10 d ^ ¼ 1 g 2½A^; ½A^; ^ dt t 8 t Its eigenvalues are 1. We express its weak value 1 ^ ^ þ g Herm A hAi ^t dWt ½57 from eqns [41], [49], and [50] or, equivalently, from ^t eqns [48] and [51]: Equation [56] and its classical counterpart [17] are perfectly similar. There is a remarkable difference 1 hA^i ¼ ½53 between eqn [57] and its classical counterpart [18]. f i cos In the latter, the stochastic average of the state is This weak value of A^ lies outside the range of the constant: E[d t] = 0, expressing the fact that classi- eigenvalues of A^. The anomaly can be arbitrarily cal measurements do not alter the original ensemble large if the rate cos2 of postselection decreases. if we ‘‘ignore’’ the outcomes of the measurements. Striking consequences follow from this anomaly On the contrary, quantum measurements introduce if we turn to the statistical interpretation. For irreversible changes to the original ensemble, a ^ phenomenon called decoherence in the physics concreteness, suppose = 2 =3 so that f hAii = 2. On average, 75% of the statistics N will be lost literature. Equation [57] implies the closed linear in postselection. We learnt from eqn [40] that first-order differential equation [58] for the stochas- tic average of the quantum state ˆt under time- the arithmetic mean a of the postselected outcomes ^ of independent weak measurements converges continuous measurement of the observable A: stochastically to the weak value upto the Gaussian dE½ ^ t ¼ 1g 2½A^; ½A^; E½ ^ ½58 fluctuation , as expressed symbolically by dt 8 t a ¼ 2 ½54 This is the basic irreversible equation to model the gradual loss of quantum coherence (decoherence) Let us approximate the asymptotically large error under time-continuous measurement. In fact, the of our weak measurements by = 10 which is very equation models decoherence under the influ- already well beyond the scale of the eigenvalues 1 ence of a large class of interactions, for example, of the observable A^. The Gaussian error derives with thermal reservoirs or complex environments. In 282 Quantum Mechanics: Weak Measurements two-dimensional Hilbert space, for instance, we can resolution of the ‘‘sudden’’ ideal quantum measure- consider the initial pure state hij =: [ cos , sin ] and ment (eqns [65] and [66]) in a sense that they the time-continuous measurement of the diagonal reproduce it in the limit t !1. The states ˆ are observable [59] on it. The solution of eqn [58] is stationary states of eqn [57]. It can be shown that given by eqn [60]: they are indeed approached with probability p for t !1 (Gisin 1984). 10 A^ ¼ ½59 0 1 Related Contexts "# cos2 e t=4g2 cos sin In addition to the two particular examples as E½ ^t ¼ 2 ½60 in postselection and in time-continuous measure- e t=4g cos sin sin2 ment, respectively, presented above, the weak The off-diagonal elements of this density matrix measurement limit itself has further variants. go to zero, that is, the coherent superposition A most natural example is the usual thermodynamic represented by the initial pure state becomes an limit in standard statistical physics. Then weak incoherent mixture represented by the diagonal measurements concern a certain additive micro- scopic observable (e.g., the spin) of each constituent density matrix ^1. Apart from the phenomenon of decoherence, the and the weak value represents the corresponding stochastic equations show remarkable similarity additive macroscopic parameter (e.g., the magneti- with the classical equations of time-continuous zation) in the infinite volume limit. This example measurement. The heuristic form of eqn [56] is indicates that weak values have natural interpreta- eqn [61] of invariable interpretation with respect tion despite the apparent artificial conditions of to the classical equation [19]: their definition. It is important that the weak value, with or without postselection, plays the physical role ^ similar to that of the common mean hA^i . If, at ¼hAi ^ þ g t ½61 ˆ t between their pre- and postselection, the states ˆ Equation [57] describes what is called the time- become weakly coupled with the state of another continuous collapse of the quantum state under quantum system via the observable A^, their average ^ ^ ^ time-continuous quantum measurement of A. For influence will be as if A took the weak value ˆ hAi ˆ . concreteness, we assume discrete spectrum for A^ and Weak measurements also open a specific loophole to consider the spectral expansion circumvent quantum limitations related to the X irreversible disturbances that quantum measure- A^ ¼ a P^ ½62 ments cause to the measured state. Noncommuting observables become simultaneously measurable in the weak limit: simultaneous weak values of non- The real values a are nondegenerate eigenvalues. commuting observables will exist. The Hermitian projectors P^ form a complete Literally, weak measurement had been coined orthogonal set: X in 1988 for quantum measurements with (pre- and) P^ ^I ½63 postselection, and became the tool of a certain time- symmetric statistical interpretation of quantum states. Foundational applications target the paradoxical ^ ^ ^ P P ¼ P ½64 problem of pre- and retrodiction in quantum theory. In a broad sense, however, the very principle of weak In a single ideal measurement of A^, the outcome a is measurement encapsulates the trade between asymp- one of the a ’s singled out at random. The totically weak precision and asymptotically large probability distribution of the measurement out- statistics. Its relevance in different fields has not yet come and the corresponding collapse of the state are been fully explored and a growing number of founda- given by tional, theoretical, and experimental applications are being considered in the literature – predominantly in p ¼hP^ i ½65 ^0 the context of quantum physics. Since specialized 1 monographs or textbooks on quantum weak measure- ^ ! P^ ^ P^ ¼: ^ ½66 0 p 0 ment are not yet available, the reader is mostly referred to research articles, like the recent one by Aharonov respectively. Equations [56] and [57] of continuous and Botero (2005),coveringmanytopicsofpostse- measurements are an obvious time-continuous lected quantum weak values. Quantum n-Body Problem 283
Nomenclature Aharonov Y and Botero A (2005) Quantum averages of weak values. Physical Review A (in print). a measurement outcome Aharonov Y, Popescu S, Rohrlich D, and Vaidman L (1993) a arithmetic mean of measurement Measurements, errors, and negative kinetic energy. Physical outcomes Review A 48: 4084–4090. A^ Hermitian operator, quantum observable Aharonov Y and Vaidman L (1990) Properties of a quantum A(X) real phase-space function system during the time interval between two measurements. Physical Review A 41: 11–20. E[ ...] stochastic expectation value ^ Belavkin VP (1989) A new equation for a continuous hf jAjii matrix element nondemolition measurement. Physics Letters A 140: hf jii inner product 355–358. H Hilbert space Dio´si L (1988) Continuous quantum measurement and Itoˆ- 2 L space of Lebesgue square-integrable formalism. Physics Letters A 129: 419–432. complex functions Gisin N (1984) Quantum measurements and stochastic processes. p probability distribution Physical Review Letters 52: 1657–1660. tr trace Giulini D, Joos E, Kiefer C, Kupsch J, Stamatescu IO et al. (1996) U^ unitary operator Decoherence and the Appearance of a Classical World in Quantum Theory. Berlin: Springer. Wt Wiener process Gough J and Sobolev A (2004) Stochastic Schro¨ dinger equations white noise process t as limit of discrete filtering. Open Systems and Information h...i postselected mean value Dynamics 11: 1–21. ˆ density operator Kraus K (1983) States, Effects, and Operations: Fundamental (X) phase-space distribution Notions of Quantum Theory. Berlin: Springer. direct product von Neumann J (1955) Mathematical Foundations of Quantum y operator adjoint Mechanics. Princeton: Princeton University Press. j ...i state vector Nielsen MA and Chuang IL (2000) Quantum Computation and h...j adjoint state vector Quantum Information. Cambridge: Cambridge University Press. h...i mean value Stratonovich R (1968) Conditional Markov Processes and Their Application to the Theory of Optimal Control. New York: Further Reading Elsevier. Wiseman HM (2002) Weak values, quantum trajectories, and the Aharonov Y, Albert DZ, and Vaidman L (1988) How the result of cavity-QED experiment on wave-particle correlation. Physical measurement of a component of the spin of a spin-1/2 particle Review A 65: 32111–32116. can turn out to be 100. Physical Review Letters 60: 1351–1354.
Quantum n-Body Problem
R G Littlejohn, University of California at Berkeley, ‘‘classical n-body problem,’’ in which V is usually Berkeley, CA, USA assumed to consist of the sum of the pairwise ª 2006 Elsevier Ltd. All rights reserved. gravitational interactions of the particles. In this article, we shall only assume that V (hence H)is invariant under translations, proper rotations, par- Introduction ity, and permutations of identical particles. The Hamiltonian H is also invariant under time reversal. This article concerns the nonrelativistic quantum This Hamiltonian describes the dynamics of isolated mechanics of isolated systems of n particles inter- atoms, molecules, and nuclei, with varying degrees acting by means of a scalar potential, what we shall of approximation, including the case of molecules in call the ‘‘quantum n-body problem.’’ Such systems the Born–Oppenheimer approximation, in which V are described by the kinetic-plus-potential is the Born–Oppenheimer potential. We shall ignore Hamiltonian, the spin of the particles, and treat the wave function Xn 2 jP j as a scalar. We assume that is an eigenfunction H ¼ T þ V ¼ þ VðR1; ...; RnÞ½1 of H, H =E . In practice, the value of n typically 2m ¼1 ranges from 2 to several hundred. Often the cases where R , P , = 1, ..., n are the positions and n = 3andn = 4 are of special interest. In this article, momenta of the n particles in three-dimensional we shall assume that n 3, since n = 2 is the trivial space, m are the masses, and V is the potential case of central-force motion. The quantum n-body energy. This Hamiltonian also occurs in the problem is not to be confused with the ‘‘quantum