Abstract
Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value: a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in our theory of classical mechanics with an epistemic restriction. By using this epistemically restricted theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimized and the limits of weak measurement.
Highlights
One of the most distinctive features of quantum mechanics is the necessary disturbance to the quantum state associated with any measurement that acquires information about the state
Within epistemically restricted Liouville (ERL) theory, as in quantum mechanics, we find that the weak value appears operationally as shifts in the mean position and momentum distributions of the measurement device upon postselection
A weak measurement can be obtained by using an initial state of the measurement device with μP = 0 and ΔP → 0, which would imply ΔQ → ∞ from (5). (While these two limits lead to identical measurement statistics within quantum theory, we explore each of them separately, as they will correspond to different processes in the context of the epistemically-restricted theory of classical mechanics explored .) In both of these limits, the disturbance caused by measurement, as well as the amount of information gained, are both very small
Summary
One of the most distinctive features of quantum mechanics is the necessary disturbance to the quantum state associated with any measurement that acquires information about the state. Much of the difficulty in interpreting the weak value may be because it seeks to analyse the measurement outcomes of two noncommuting observables on a given state of a particle, which is known to be problematic in quantum theory due to the lack of an ontology for measurement outcomes associated with observables It is worthwhile, to consider whether the weak value can arise in a theory that does possess a clear ontology. We analyse weak values using a theory of classical mechanics (thereby possessing a clear ontology) supplemented with a restriction on the observers knowledge This theory is the epistemically restricted Liouville (ERL) mechanics of [28], and it is known to reproduce many of the features of quantum measurement. Consistent with the results of [24], our model is noncontextual: the ERL mechanics provides an explicit noncontextual ontological model for all procedures described here
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