Linear position measurements with minimum error-disturbance in each minimum uncertainty state

TL;DR

This paper constructs linear position measurements that achieve minimum error-disturbance bounds in minimum uncertainty states, providing explicit solutions and analyzing their properties within the framework of the Branciard-Ozawa error-disturbance relation.

Contribution

It introduces a necessary and sufficient condition for linear position measurements to attain the minimum error-disturbance bound in minimum uncertainty states and provides explicit measurement models.

Findings

Explicit linear measurements achieving the lower bound in minimum uncertainty states

Derived probability distributions and post-measurement states

Established conditions for optimal measurement performance

Abstract

In quantum theory, measuring process is an important physical process; it is a quantum description of the interaction between the system of interest and the measuring device. Error and disturbance are used to quantitatively check the performance of the measurement, and are defined by using measuring process. Uncertainty relations are a general term for relations that provide constraints on them, and actively studied. However, the true error-disturbance bound for position measurements is not known yet. Here we concretely construct linear position measurements with minimum error-disturbance in each minimum uncertainty state. We focus on an error-disturbance relation (EDR), called the Branciard-Ozawa EDR, for position measurements. It is based on a quantum root-mean-square (q-rms) error and a q-rms disturbance. We show the theorem that gives a necessary and sufficient condition for a linear position measurement to achieve its lower bound in a minimum uncertainty state, and explicitly give exactly solvable linear position measurements achieving its lower bound in the state. We then give probability distributions and states after the measurement when using them. It is expected to construct measurements with minimum error-disturbance in a broader class of states in the future, which will lead to a new understanding of quantum limits, including uncertainty relations.