We have found that the usual measuring procedure for preselected and postselected ensembles of quantum systems gives unusual results. Under some natural conditions of weakness of the measurement, its result consistently defines a new kind of value for a quantum variable, which we call the weak value. A description of the measurement of the weak value of a component of a spin for an ensemble of preselected and postselected spin-2 particles is presented.PACS numbers: 03.65.Bz This paper will describe an experiment which measures a spin component of a spin--, ' particle and yields a result which is far from the range of "allowed" values. We shall start with a brief description of the standard measuring procedure. Considering measurements on an ensemble of preselected and postselected systems, we shall define a new concept: a weak value of a quantum variable. And, finally, we shall describe the measurement of the weak value on the example of a spin--, ' particle.In quantum theory, the result of a measurement of a variable A which has discrete eigenvalues a; must necessarily be one of those values. The Hamiltonian of the standard measurement procedure ' is H = -g(t)qA, where g(t) is a normalized function with a compact support near the time of measurement, and q is a canonical variable of the measuring device with a conjugate momentum tr. The initial state of the measuring device in the ideal case has to be such that tr is well defined.After the interaction (I) we can ascertain the value of A from the final value of tr: A Btr.As a reasonable approximation for a real situation, we may take the initial state of the measuring device as a Gaussian in the q (and consequently also in the tr) representation. For this case, the Harniltonian (1) leads to the transformation t I H dte a2I4(-tea) ' g-(2) where g; a; i A =a;) is the initial state of our system. If the spread of the tt distribution hatt is much smaller than the differences between the a;, then, after the interaction, we shall be left with the mixture of Gaussians located around a; correlated with different eigenstates of A. A measurement of tt will then indicate the value of A.In the opposite limit, where htr is much bigger than all a;, the final probability distribution will be again close to a Gaussian with the spread hatt. The center of the Gaussian will be at the mean value of A: (A) =g; i a; i a;.One measurement like this will give no information because htr»(A); but we can make this same measurement on each member of an ensemble of W particles prepared in the same state, and that will reduce the relevant uncertainty by the factor I/JN, while the mean value of the average will remain (A). By enlarging the number X of particles in the ensemble, we can make the measurement of (A) with any desired precision. The outcome of the measurement is the average of the obtained values tr of the measuring devices. As we explained earlier, it will yield, for a sufficiently large ensemble, the value (A). We now raise the question: Can we change the above outcome by taking in...

Aharonov and Vaidman Reply:In our recent Letter we defined a new concept: a weak value of a quantum variable. We showed that a standard measuring procedure with weakened coupling, performed on an ensemble of both preselected and postselected systems, yields the weak value. The intuitive picture can be seen from our general approach 2 in which we consider two wave functions for a single system at a given time: the usual one evolving toward the future, and another evolving backward in time toward the past. Weak enough measurements do not disturb the above two wave functions and thus, the outcomes of such measurements should reflect properties of both states. The weakness of the interaction, therefore, is the essential requirement for the above measuring process. We claim that for any measuring procedure of a physical variable the coupling can be made weak enough such that the effective value of the variable for a preselected and postselected ensemble will be its weak value.Leggett 3 argues that our result has "little relevance to the theory of measurements as conventionally understood" because it relies on a "very specific choice" of the interaction: our (and his) Eq. (1). This equation represents the measuring interaction of the von Neumann formalism, the conventional theory of measurements, and we used it only for the proof; the result itself does not rest on the specific form of the interaction. The only requirement is the weakness of the interaction.Another point of Leggett is that our result is valid only up to first order in X. Our requirement of weakness [Eq. (4)] ensures that all contributions beyond the first order can be neglected; therefore, the first order is all that we need. Our "weakness" requirement [Eq. (4)] is, however, too strong. We have since refined it, 2 and it turns out that there are situations (as the one in Ref. 4) in which our result is valid for A» lA, i.e., for higher orders in X.

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