A Dynamical Uncertainty Principle in von Neumann Algebras by Operator Monotone Functions

Suppose that A ₁,…,A _N are observables (selfadjoint matrices) and ρ is a state (density matrix). In this case the standard uncertainty principle, proved by Robertson, gives a bound for the quantum generalized variance, namely for det {Cov  _ρ (A _j ,A _k )}, using the commutators [A _j ,A _k ]; this bound is trivial when N is odd. Recently a different inequality of Robertson-type has been proved by the authors with the help of the theory of operator monotone functions. In this case the bound makes use of the commutators [ρ,A _j ] and is non-trivial for any N. In the present paper we generalize this new result to the von Neumann algebra case. Nevertheless the proof appears to simplify all the existing ones.