  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.
