The real and imaginary parts of a weak value appearing as back-actions via a post-selection

TL;DR

This paper explores how both the real and imaginary parts of a weak value can act as back-actions in post-selected measurements, and demonstrates how they can be inferred from probabilities without additional systems.

Contribution

It introduces a case where the real part of a weak value also functions as a back-action, expanding the understanding of weak measurement effects.

Findings

Both real and imaginary parts can be inferred from measurement probabilities.

The real part can act as a back-action similar to the imaginary part.

Inference of weak value components does not require an auxiliary pointer system.

Abstract

In a weak measurement the real and imaginary parts of a weak value participate in the shifts of the complementary variables of a pointer. While the real part represents the value of an observable in the limit of zero measurement strength, the imaginary one is regarded as the back-action due to the measurement with a post-selection, which has an influence on the post-selection probability. In this paper I give a case in which a real part could also appear as such a back-action in a post-selection probability on an equal footing with an imaginary one. It is also shown that both of the real and imaginary parts can be inferred by observing the probability in practice, which has an advantage that an additional system of a pointer is not needed.