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

Abstract

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.