How weak values emerge in joint measurements on cloned quantum systems

TL;DR

This paper explores how weak measurement statistics can be derived from joint measurements on cloned quantum systems, revealing new insights into quantum measurement processes and their relation to cloning and swap operations.

Contribution

It introduces a novel connection between weak measurement statistics and joint measurements on cloned systems, utilizing an idealized cloning process and swap operations.

Findings

Joint probabilities on clones relate to real parts of weak measurement probabilities.

Imaginary parts are obtained via partial swap operations.

Controlled-swap operations unify the measurement processes.

Abstract

A statistical analysis of optimal universal cloning shows that it is possible to identify an ideal (but non-positive) copying process that faithfully maps all properties of the original Hilbert space onto two separate quantum systems. The joint probabilities for non-commuting measurements on separate clones then correspond to the real parts of the complex joint probabilities observed in weak measurements on a single system, where the measurements on the two clones replace the corresponding sequence of weak measurement and post-selection. The imaginary parts of weak measurement statics can be obtained by replacing the cloning process with a partial swap operation. A controlled-swap operation combines both processes, making the complete weak measurement statistics accessible as a well-defined contribution to the joint probabilities of fully resolved projective measurements on the two output systems.