Typical weak and superweak values

Abstract

Weak values, resulting from the action of an operator on a preselected state when measured after postselection by a different state, can lie outside the spectrum of eigenvalues of the operator: they can be 'superweak'. This phenomenon can be quantified by averaging over an ensemble of the two states, and calculating the probability distribution of the weak values. If there are many eigenvalues, distributed within a finite range, this distribution takes a simple universal generalized lorentzian form, and the 'superweak probablility', of weak values outside the spectrum, can be as large as 1-1/ √ 2 = 0.293.. .. By contrast, the familiar expectation values always lie within the spectral range, and their distribution, although approximately gaussian for many eigenvalues, is not universal.

Weak Measurements: Typical Weak and Superweak Values

“…The above indicates the universality of the superweak probability and as shown in Berry and Shukla 7 by the examples of interesting eigenvalue distributions, e.g. uniform, semicircle, bimodal, etc.…”

“…the superweak probability of weak values lying outside the spectrum can be as large as 0.293 (almost 30% chance of a weak value being superweak). By contrast, the familiar expectation values always lie within the spectral range, and their distribution, although approximately Gaussian for many eigenvalues, is not universal 7 . As indicated by eq.…”

“…It is also important to know the influence, if any, of the type of randomness and whether superweak values are common or rare? Seeking the answers in Berry and Shukla 7,8 , the probability distribution P(A wr ) of A wr  ReA w was calculated for random pre-and post-states in the eigenbasis of A (i.e. |i = i n |a n  and | f  =  n f n |a n  with i n and f n randomly distributed); note this corresponds to considering an ensemble of states |i and | f  which should not be confused with pre-and postselected ensembles of systems, each one of which is at the same state |i and | f  respectively.…”

Limits to superweak amplification of beam shifts

“…With appropriate choice of initial and final polarizations, the beam shift close to Brewster or null-reflection angles can be very large. In the quantum analogy, this is the "superweak" regime [13], where the value of the beam shift in the postselected component takes on a larger value than the shift of total intensity for either s or p incident polarization, albeit with a greatly reduced intensity. For general incident and analyzer polarizations, the net component shift is arbitrary (it has a spatial and angular shift both with longitudinal and transverse components), and the apparent shift of the total beam is an appropriately weighted sum of orthogonal analyzer polarizations [8].…”

Spatially resolved characterization of electron localization and delocalization in molecules: Extending the Kohn‐Resta approach

“…denotes anticommutator; and being electron density of a system and its electronic current density, respectively. This splitting is in the spirit of a concept of the so‐called quantum‐mechanical weak values and, more particularly, “weak momentum.” Integration of Equation over all space leads to the total electronic momentum while the integral of Equation is a zero. Hence, it would be reasonable to associate with electronic momentum density which reflects averaged directional flow of electronic fluid.…”