  So-called quantum limits and their achievement are important themes in
physics. Heisenberg's uncertainty relations are the most famous of them but are
not universally valid and violated in general. In recent years, the
reformulation of uncertainty relations is actively studied, and several
universally valid uncertainty relations are derived. On the other hand, several
measuring models, in particular, spin-1/2 measurements, are constructed and
quantitatively examined. However, there are not so many studies on simultaneous
measurements of position and momentum despite their importance. Here we show
that an error-trade-off relation (ETR), called the Branciard-Ozawa ETR, for
simultaneous measurements of position and momentum gives the achievable bound
in minimum uncertainty states. We construct linear simultaneous measurements of
position and momentum that achieve the bound of the Branciard-Ozawa ETR in each
minimum uncertainty state. To check their performance, we then calculate
probability distributions and families of posterior states, sets of states
after the measurements, when using them. The results of the paper show the
possibility of developing the theory of simultaneous measurements of
incompatible observables. In the future, it will be widely applied to quantum
information processing.
