# Bounds for fidelity of semiclassical Lagrangian states in K{\"a}hler quantization

## TL;DR

This paper establishes bounds for the fidelity between semiclassical Lagrangian states in Kähler quantization, providing estimates for their similarity in the semiclassical limit and exploring specific examples on the sphere.

## Contribution

It introduces bounds for fidelity between states associated with intersecting Lagrangian submanifolds and analyzes particular cases on the sphere, advancing understanding in Kähler quantization.

## Key findings

- Derived lower and upper bounds for fidelity in the semiclassical limit.
- Obtained improved upper bounds for specific spherical examples.
- Proposed a conjecture for the general fidelity case.

## Abstract

We define mixed states associated with submanifolds with probability densities in quantizable closed K{\"a}hler manifolds. Then, we address the problem of comparing two such states via their fidelity. Firstly, we estimate the sub-fidelity and super-fidelity of two such states, giving lower and upper bounds for their fidelity, when the underlying submanifolds are two Lagrangian submanifolds intersecting transversally at a finite number of points, in the semiclassical limit. Secondly, we investigate a family of examples on the sphere, for which we manage to obtain a better upper bound for the fidelity. We conclude by stating a conjecture regarding the fidelity in the general case.