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

Suppose that A1,…,AN 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(Aj,Ak)}, using the commutators [Aj,Ak]; 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,Aj] 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.