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Quantum Mechanics: Weak Measurements 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 ÁA 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 Á2 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 pwðaÞ!GÁ 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. E½a¼hAi 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 Å(X), where 0 Å 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 hÅAi ÅhAi ¼: ½11 error of standard dispersion >0. The probability hÅi distribution of a and the update of the state where hÅi 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 pðaÞ¼hiGða À AÞ ½5 function Å, too, in ideal measurement with random outcome upon which we base the following 1 random decision. With probability , we include ! Gða À AÞ ½6 pðaÞ the current a into the statistics and we discard it 278 Quantum Mechanics: Weak Measurements with probability 1 À . Then the coincidence of E[a] Equations [17] and [18] are the special case of the and ÅhAi,asineqn [4], remains valid: Kushner–Stratonovich equations of time-continuous Bayesian inference conditioned on the continuous E½a¼ hAi ½12 Å 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 ÅhAi. Classical postselection allows introducing the at ¼hAi þ gt ½19 effective postselected state: t Accordingly, the current measurement outcome is Å Å ¼: ½13 always equal to the current mean plus a term hÅi 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 Å: [18], it describes the gradual concentration of the distribution t in such a way that the variance Á A hAi ¼hAi ½14 t Å Å tends to zero while A tends to a random h it 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 aPðXÞ½20 exist unless we combine postselection and weak measurement.
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