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So-called quantum limits and their achievement are important themes inphysics. Heisenberg's uncertainty relations are the most famous of them but arenot universally valid and violated in general. In recent years, thereformulation of uncertainty relations is actively studied, and severaluniversally valid uncertainty relations are derived. On the other hand, severalmeasuring models, in particular, spin-1/2 measurements, are constructed andquantitatively examined. However, there are not so many studies on simultaneousmeasurements of position and momentum despite their importance. Here we showthat an error-trade-off relation (ETR), called the Branciard-Ozawa ETR, forsimultaneous measurements of position and momentum gives the achievable boundin minimum uncertainty states. We construct linear simultaneous measurements ofposition and momentum that achieve the bound of the Branciard-Ozawa ETR in eachminimum uncertainty state. To check their performance, we then calculateprobability distributions and families of posterior states, sets of statesafter the measurements, when using them. The results of the paper show thepossibility of developing the theory of simultaneous measurements ofincompatible observables. In the future, it will be widely applied to quantuminformation processing.

Linear simultaneous measurements of position and momentum with minimum error-trade-off in each minimum uncertainty state