Remarks on the Sequential Products

TL;DR

This paper demonstrates that various sequential products proposed in previous quantum measurement literature are essentially equivalent to a standard form involving the square root of the first operator, simplifying their understanding.

Contribution

It shows that all previously proposed sequential products are unitarily equivalent to the standard form A∘B=A^{1/2}BA^{1/2}, unifying different approaches.

Findings

All proposed sequential products are unitarily equivalent to the standard form.

The standard form simplifies the analysis of sequential quantum measurements.

Clarifies the structure of sequential products in quantum theory.

Abstract

In this paper, we show that those sequential products which were proposed by Liu and Shen and Wu in [J. Phys. A: Math. Theor. {\bf 42}, 185206 (2009), J. Phys. A: Math. Theor. {\bf 42}, 345203 (2009)] are just unitary equivalent to the sequential product $A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}$.