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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.