The AAV effect is the well-known phenomenon where a weak measurement followed by post-selection leads to a pointer shift proportional to the weak value of the measured observable. The effect is usually derived by considering a perturbative expansion of the time-evolution operator corresponding to the measurement. We show that the AAV effect can be instead derived given three simple conditions on the pointer states. The most important condition is that the pointer states be approximately indistinguishable. The other two conditions relate the observables used to characterize the pointer with the measured system observable. This intuitive approach does not require any model for the measurement process, but is completely consistent with the usual approach based on the von Neumann Hamiltonian. Moreover, this approach predicts and explains a new phenomenon, called weak echo, where the usual weak value behaviour can re-emerge at high interaction strengths.
The AAV effect is the well-known phenomenon where a weak measurement followed by post-selection leads to a pointer shift proportional to the weak value of the measured observable. The effect is usually derived by considering a perturbative expansion of the time-evolution operator corresponding to the measurement. We show that the AAV effect can be instead derived given three simple conditions on the pointer states. The most important condition is that the pointer states be approximately indistinguishable. The other two conditions relate the observables used to characterize the pointer with the measured system observable. This intuitive approach does not require any model for the measurement process, but is completely consistent with the usual approach based on the von Neumann Hamiltonian. Moreover, this approach predicts and explains a new phenomenon, called weak echo, where the usual weak value behaviour can re-emerge at high interaction strengths.
In the weak measurement (WM) scenario involving weak interaction and postselection by projective measurement, the empirical significance of weak values is manifested in terms of shifts in the measurement pointer's mean position and mean momentum. In this context, a general quantitative treatment is presented in this paper by taking into account the hitherto unexplored effect of correlations among the pointer degrees of freedom which pertain to an arbitrary multidimensional preselected pointer state. This leads to an extension of the earlier results, showing that, for complex weak values, the correlations among different pointer degrees of freedom can crucially affect the way the imaginary parts of the weak values are related to the observed shifts of the mean pointer position and momentum. The particular relevance of this analysis is discussed in the case of sequential weak interactions followed by a projective measurement enabling postselection (called sequential WM) which, in the special case, reduces to the usual WM scheme (involving a single weak interaction and postseletion) modified by the effect of pointer-state correlations.
A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an ``anomalous weak value'', is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures.
Weak values are usually associated with weak measurements of an observable on a pre- and post-selected ensemble. We show that more generally, weak values are proportional to the correlation between two pointers in a successive measurement. We show that this generalized concept of weak measurements displays a symmetry under reversal of measurement order. We show that the conditions for order symmetry are the same as in classical mechanics. We also find that the imaginary part of the weak value has a counterpart in classical mechanics. This scheme suggests new experimental possibilities.
Imaginary weak values are a peculiarity of standard quantum mechanics subject to post-selection and weak measurements. We examine their possible role and physical meaning in teleological or post-selected quantum cosmology, in the context of the Hubble diagram. We only consider the linear Hubble diagram, controlled by the Hubble parameter and the redshift effect. We set up a variety of von Neumann models entangling the geometry with different pointers, either upstream, via the redshift itself, or downstream, via the instruments used to measure the arriving frequency of calibrated lines. We find that the AAV (Aharonov, Albert and Vaidman) formula for weak values remains valid in these models, even though the assumptions are wildly different. Hence the imaginary part of the weak value of the Hubble parameter translates into a shift in the conjugate variable of the pointer, and we evaluate the very diverse expressions this takes for the various meters considered. Furthermore, in situations where meters are coupled, the effect may only appear in the first meter explicitly sensitive to the geometry and, unlike real weak values, need not propagate down the chain. The imaginary weak value is therefore distinctly in the ``eye of the beholder'', as a result of quantum measurement theory. This could distinguish teleological quantum explanations of cosmic acceleration from dark-energy alternatives that reproduce the same real Hubble diagram.
The variance of an arbitrary pointer observable is considered for the general case that a complex weak value is measured using a complex valued pointer state. For the typical cases where the pointer observable is either its position or momentum, the associated expressions for the pointer's variance after the measurement contain a term proportional to the product of the weak value's imaginary part with the rate of change of the third central moment of position relative to the initial pointer state just prior to the time of the measurement interaction when position is the observable---or with the initial pointer state's third central moment of momentum when momentum is the observable. These terms provide a means for controlling pointer position and momentum variance and identify control conditions which, when satisfied, can yield variances that are smaller after the measurement than they were before the measurement. Measurement sensitivities which are useful for estimating weak-value measurement accuracies are also briefly discussed.
A weak measurement on a system is made by coupling a pointer weakly to the system and then measuring the position of the pointer. If the initial wave function for the pointer is real, the mean displacement of the pointer is proportional to the so-called weak value of the observable being measured. This gives an intuitively direct way of understanding weak measurement. However, if the initial pointer wave function takes complex values, the relationship between pointer displacement and weak value is not quite so simple, as pointed out recently by Jozsa [R. Jozsa, Phys. Rev. A 76, 044103 (2007)]. This is even more striking in the case of sequential weak measurements [G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A 76, 062105 (2007)]. These are carried out by coupling several pointers at different stages of the evolution of the system, and the relationship between the products of the measured pointer positions and the sequential weak values can become extremely complicated for an arbitrary initial pointer wave function. Surprisingly, all this complication vanishes when one calculates the cumulants of pointer positions. These are directly proportional to the cumulants of sequential weak values. This suggests that cumulants have a fundamental physical significance for weak measurement.