Weak vs. approximate values in quantum state determination

TL;DR

This paper extends the concept of weak values in quantum measurements to positive operator measures, analyzes their operational meaning, and compares state reconstruction methods, highlighting limitations of weak measurements.

Contribution

It introduces a generalized definition of weak values for arbitrary positive operator measures and evaluates their effectiveness in quantum state reconstruction.

Findings

Weak values can be defined for arbitrary positive operator measures.

Weak measurement-based reconstruction cannot fully determine unknown quantum states.

Phase space measurement methods are more general than weak measurement approaches.

Abstract

We generalize the concept of a weak value of a quantum observable to cover arbitrary real positive operator measures. We show that the definition is operationally meaningful in the sense that it can be understood within the quantum theory of sequential measurements. We then present a detailed analysis of the recent experiment of Lundeen et al. concerning the reconstruction of the state of a photon using weak measurements. We compare their method with the reconstruction method through informationally complete phase space measurements and show that it lacks the generality of the phase space method. In particular, a completely unknown state can never be reconstructed using the method of weak measurements.