Measuring processes and the Heisenberg picture

TL;DR

This paper develops a framework for quantum measurement theory in the Heisenberg picture, establishing a correspondence between measurement correlations and processes, and analyzing the extendability and realizability of instruments.

Contribution

It introduces the concept of measurement correlations in the Heisenberg picture and proves a unitary dilation theorem linking these correlations to measuring processes.

Findings

Established a one-to-one correspondence between measurement correlations and measuring processes.

Proved that all CP instruments can be extended into measurement correlations.

Analyzed the approximate realizability of CP instruments within error limits.

Abstract

In this paper, we attempt to establish quantum measurement theory in the Heisenberg picture. First, we review foundations of quantum measurement theory, that is usually based on the Schr\"{o}dinger picture. The concept of instrument is introduced there. Next, we define the concept of system of measurement correlations and that of measuring process. The former is the exact counterpart of instrument in the (generalized) Heisenberg picture. In quantum mechanical systems, we then show a one-to-one correspondence between systems of measurement correlations and measuring processes up to complete equivalence. This is nothing but a unitary dilation theorem of systems of measurement correlations. Furthermore, from the viewpoint of the statistical approach to quantum measurement theory, we focus on the extendability of instruments to systems of measurement correlations. It is shown that all completely positive (CP) instruments are extended into systems of measurement correlations. Lastly, we study the approximate realizability of CP instruments by measuring processes within arbitrarily given error limits.