Matrix geometric means and uncertainty relation ∗

The research of geometric means of matrices is attracting more and more attention from scholars since it has very important applications in mathematics and physics, especially in the von Neuman entropy involving the quantum physics and statistical thermal physical. There are many achievements on the entropy, among which the entropy uncertainty relation, the generalization of the Heisenberg uncertainty relation, has been applied in mathematics, physics and signal processing community. In this paper, we consider the entropy uncertainty relation in the geometric means type. By using the definition of the geometric means for the trace class operator, and associating with the Rényi entropy uncertainty relation and the Shannon entropy uncertainty relation, we obtain the Rényi entropy uncertainty relation for the density and the entropy uncertainty relation for the trace class operator.