Uncertainty principle with quantum Fisher information

Abstract

In this paper we prove a nontrivial lower bound for the determinant of the covariance matrix of quantum mechanical observables, which was conjectured by Gibilisco, Isola and Imparato. The lower bound is given in terms of the commutator of the state and the observables and their scalar product, which is generated by an arbitrary symmetric operator monotone function.Comment: 8 pages, LaTeX; added reference

Quantum Covariance, Quantum Fisher Information, and the Uncertainty Relations

“…If it should generate a Riemannian metric, then it should depend on D 0 smoothly [1]. 2 From coarse-graining to Fisher information and covariance Heuristically, coarse-graining implies loss of information, therefore Fisher information should be monotone under coarse-graining. This was proved in [3] in probability theory and a similar approach was proposed in [18] for the quantum case.…”

Mathematical Uncertainty Relations and their Generalization for Multiple Incompatible Observables

“…Our results have been derived from generalizing the classical Cauchy-Schwarz inequality. Alternative stronger uncertainty relations for multi observables, their associative lower bounds and minimal states have been investigated recently [7][8][9][10]. These…”

“…As we have seen in previous section, the Cauchy-Schwarz inequality (3) is the mathematical foundation of the Heisenberg uncertainty relation (7). In this section, we first propose a novel generalized Cauchy-Schwarz inequality for multiple vectors, and subsequently, using this inequality we formulate a generalized uncertainty relation for multiple incompatible observables.…”

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

“…In the case N = 1, a reasonable candidate for a lower bound is an expression involving some commutation relation between A and ρ. An inequality of this kind, valid for any N , has been recently proved in [1,11] (see also [5,9,10,12,[15][16][17][18][19][20]30]). To describe this result we need the theory of operator monotone functions.…”

“…In the present paper we prove that, with due modifications, inequality (1.3) is true on an arbitrary von Neumann algebra. Despite the general setting, the proof we present here appears simpler than the existing ones (see [1,11]). Intermediate results have been previously proved by the authors [6,7].…”