Uncertainty relation and simultaneous measurements in quantum theory

Shiro Ishikawa

Ratan Dasgupta

Relativity of simultaneity is shown to limit the accuracy in simultaneous measurement of position and momentum of a moving body by an inertial observer. The inaccuracy or uncertainty product derived from special relativity theory is found to be similar to the much known quantum mechanical uncertainty relation. However, whether the above two uncertainty relations indicate the same physics is debatable and any conclusion is not attempted here. A modification of this relativistic uncertainty formula under gravitational field is achieved using general relativity principles, and a small gravitational correction term has to be introduced. This gravitational term being small remains negligible in ordinary conditions but becomes appreciable at small time scale.

Paul Busch

Kazuya Okamura

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.

John Maddox

Wonmin Son