Uncertainty principle and quantum Fisher information

Abstract

Heisenberg and Schrödinger uncertainty principles give lower bounds for the product of variances Var ͑A͒Var ͑B͒ if the observables A , B are not compatible, namely, if the commutator ͓A , B͔ is not zero. In this paper, we prove an uncertainty principle in Schrödinger form where the bound for the product of variances Var ͑A͒Var ͑B͒ depends on the area spanned by the commutators i͓ , A͔ and i͓ , B͔ with respect to an arbitrary quantum version of the Fisher information.

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

“…In addition, now we can see a new trend of the information geometry which is called Quantum information theory [12,21,23,37,42]. This theory is based on information geometry and quantum mechanics.…”

“…For example, Nagaoka [37] and Petz [42] studied the information geometry of quantum probability. On the other hand, Gibilisco and Isola in [21] gave a mathematical foundation that extended information geometry to the function space. In particular, they showed that a family of inequalities, which relates to the uncertainty principle, has a geometric interpretation in terms of quantum Fisher information.…”

Quantum Covariance, Quantum Fisher Information, and the Uncertainty Relations

“…In the case N = 2 the inequality was proved by Luo, Q. Zhang and Z. Zhang [16,17,15], by Kosaki [11] and by Yanagi, Furuichi and Kuriyama [27] for some special functions f. The general case is due to Gibilisco, Imparato and Isola [7,4]. Gibilisco and Isola emphasized the geometric aspects of the inequality (10) and conjectured it for general quantum Fisher information [4].…”

“…This is actually the same relation as (4). Therefore, condition (7) (8) This is considered to be the general de nition.…”

“…Gibilisco and Isola emphasized the geometric aspects of the inequality (10) and conjectured it for general quantum Fisher information [4].…”