Aharonov and Vaidman reply

Abstract

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.

What Is a Quantum-Mechanical “Weak Value” the Value of?

“…Of the previous criticisms of weak values that seems most similar to mine, I should mention [29,30] (to which Aharonov and Vaidman replied [31]) and [32]. Of the more recent ones, I emphasize [33] and, even more [34], in which Kastner explicitly points out that a weak value "should be thought of as an amplitude and not an expectation value at all."…”

Analysis of quantum-state disturbance in a protective measurement of a spin-1/2 particle

“…In a protective measurement [1][2][3][4][5], the quantum system interacts weakly with a measuring apparatus via an interaction Hamiltonian given bŷ…”

“…This paradigmatic and experimentally relevant model was first studied in Refs. [1,4], but those studies merely demonstrated one basic feature of protective measurement, namely, how information about the expectation value becomes encoded in the shift of the apparatus pointer. The studies only considered in the limit of an infinitesimally weak inhomogeneous magnetic field applied for an infinite amount of time (and infinitely rapidly turned on and off), without attending to the crucial issue of state disturbance.…”

“…Besides being an important and unusual instance of a quantum measurement, protective measurement offers the possibility of successively carrying out measurements of expectation values of different observables while the system is likely to remain in its initial quantum state. This state may then be reconstructed from the measured expectation values with in principle arbitrarily high fidelity [1][2][3][4]. In this way, protective measurement may provide a route toward state tomography of single quantum systems.…”

“…Protective measurement [1][2][3][4][5][6] is a scheme for measuring expectation values of a quantum system in a way that makes this state disturbance arbitrarily small. For such a measurement to obtain, the system is coupled weakly to the apparatus, and the initial state of the system is required to be in a nondegenerate eigenstate of its Hamiltonian.…”