Connection-state approach to pre- and post-selected quantum measurements

TL;DR

This paper introduces connection matrices as a novel tool to describe post-selected quantum ensembles, extending density matrices to include non-Hermitian cases, and explains their role in understanding weak values and quantum complementarity.

Contribution

The paper presents connection matrices as a new framework for analyzing post-selected quantum measurements, including a method for their experimental determination and a new quantum detector tomography approach.

Findings

Connection matrices extend density matrices to non-Hermitian cases.

Unusual weak values are explained by quantum complementarity.

Connection matrices can be experimentally reconstructed.

Abstract

We discuss the concept of connection states (or connection matrices) that describe posterior ensembles, post-selected according to the outcomes of a quantum measurement. Connection matrices allow one to obtain results of any weak and some non-weak pre- and post-selected measurements, in the same manner as density matrices allow one to predict the results of conventional quantum measurements. Connection matrices are direct extensions of the density matrices and are generally non-Hermitian, which we show to be a direct consequence of quantum complementarity. This implies that the ultimate reason for unusual weak values is quantum complementarity. We show that connection matrices can be determined experimentally. We also show that retrodictive states are a special case of connection states. We propose a new method of tomography of quantum detectors.