Weak Measurements with Arbitrary Probe States

Abstract

The exact conditions on valid pointer states for weak measurements are derived. It is demonstrated that weak measurements can be performed with any pointer state with vanishing probability current density. This condition is found both for weak measurements of noncommuting observables and for c-number observables. In addition, the interaction between pointer and object must be sufficiently weak. There is no restriction on the purity of the pointer state. For example, a thermal pointer state is fully valid.PACS numbers: 03.65.Ta In "orthodox" quantum mechanics, the result of a "measurement" is always one of the eigenvalues of the observable. In the last pages of his textbook on quantum mechanics [1], von Neumann provided a model of a measurement where the object under study was interacting with a measurement pointer. By assuming that the initial uncertainty of the pointer was small, von Neumann demonstrated that the pointer would display one of the eigenvalues of the object observable.Aharonov, Albert and Vaidman (AAV) considered the same experimental arrangement [2]. However, they made the opposite assumption, namely that the initial uncertainty of the pointer was large. They demonstrated that despite this, the pointer would on average show the correct expectation value of an observableĉ, although it could not distinguish separate eigenvalues. Their most interesting discovery, though, was that if a projective measurement of a second observabled was made on the object after the interaction, the average meter reading conditioned on the result of the measurement on the object would be the real part of the quantityThe authors introduced the name "weak value" for this quantity. They observed that the values of c w might lie outside the range of eigenvalues of the observableĉ. It has been contested whether the experimental arrangement of AAV qualifies as a "measurement", and whether it has any meaning to ascribe to c w a significance as a "value" of the observableĉ [3][4][5]. Despite initial scepticism, weak values have found applications in a variety of systems. A classical optical analog [6] of the original experiment proposed by AAV has been realized experimentally [7]. The polarization state of a classical radiation field can be treated as analogous to a spin-1 2 system, and the weak value of polarization may exceed the eigenvalue range [8,9]. It has also been found that weak values have applications within classical optical communication [10,11]. These examples demonstrate * Electronic address: lars.m.johansen@hibu.no that weak values have applications beyond quantum mechanics. In fact, it is not unfamiliar that the quantum formalism can be applied even to classical systems where two observables cannot be jointly measured with arbitrary accuracy. This is known e.g. in signal processing, where the Wigner distribution for time and frequency has gained popularity [12].It was recently found that "weak values" have a deeper significance when considered from the viewpoint of standard Bayesian estimation theory [13]....

Nonclassicality in weak measurements

“…Weak measurements may be performed in exactly the same way as standard von Neumann measurements, but with a weakened interaction [3]. In a weak measurement, the pointer will on average register the expectation value of the observable that is measured.…”

Quantum averages of weak values

“…(4.3). These expressions can be further simplified if the initial apparatus state has a real φ(q) [45,46] and vanishing expectation value of p, in which case the posterior expectation p ≺ and the correlation {∆p, ∆A} ≺ vanish. Under such conditions, the first two central moments of (4.5) are indistinguishable from those obtained from classically averaging weak values with a variability defined through the posterior distribution (4.3).…”

Complete characterization of post-selected quantum statistics using weak measurement tomography

“…These developments in the field of weak measurement may be especially significant in the context of new experimental possibilities pioneered by quantum information related research [9,10,11]. However, there seems to be a certain mismatch between the conventional approach to weak measurements, which is based on the notion of measurement interaction dynamics mediated by operator observables [12,13,14,15,16], and the more general approach to measurements based on the measurement operators widely used in quantum information [17]. In particular, the conventional analysis seems to overemphasize the exotic and surprising aspects of the weak measurements, while the operator based approach shows more clearly how weak measurements fit into the general framework of quantum physics [18].…”