# Non-Orthogonal Bases for Quantum Metrology

## TL;DR

This paper introduces a method using non-orthogonal bases and Gramian matrices to simplify quantum Fisher information calculations, especially for states like noisy Schrödinger cat states, avoiding complex diagonalization.

## Contribution

The authors develop a new approach leveraging Gramian matrices to evaluate quantum Fisher information without orthogonalization, applicable to non-orthonormal bases in quantum metrology.

## Key findings

- Simplified analytical evaluation of quantum Fisher information.
- Application to noisy Schrödinger cat states.
- New estimation results for quantum states with superpositions.

## Abstract

Many quantum statistical models are most conveniently formulated in terms of non-orthonormal bases. This is the case, for example, when mixtures and superpositions of coherent states are involved. In these instances, we show that the analytical evaluation of the quantum Fisher information may be greatly simplified by bypassing both the diagonalization of the density matrix and the orthogonalization of the basis. The key ingredient in our method is the Gramian matrix (i.e. the matrix of scalar products between basis elements), which may be interpreted as a metric tensor for index contraction. As an application, we derive novel analytical results for several estimation problems involving noisy Schroedinger cat states.