Fidelity of quantum blobs

Abstract

Quantum blobs are the smallest units of phase space that are compatible with the Robertson-Schrödinger indeterminacy relation and invariant under general symplectic transformations [1]. In this article, the distinguishability between pairs of quantum blobs, as measured by quantum fidelity, is defined on complex phase space. Fidelity is physically interpreted as the probability that the pair are mistaken for each other upon a measurement. The mathematical representation is based on the concept of symplectic capacity in symplectic topology. The fidelity is the absolute square of the complex-valued overlap between the symplectic capacities of the pair of quantum blobs. The symplectic capacity for a given quantum blob, onto any conjugate plane of degrees of freedom, is postulated to be bounded from below by the Gromov width h/2. This generalizes the Liouville theorem in classical mechanics, which states that the volume of a region of phase space is invariant under the Hamiltonian flow of the system, by constraining the shape of the flow. It is shown that for closed Hamiltonian systems, the Schrödinger equation is the mathematical representation for the conservation of fidelity.