# A stable quantum Darmois-Skitovich theorem

## TL;DR

This paper extends the Darmois-Skitovich theorem to quantum Gaussian states, demonstrating its stability under small errors and providing explicit stability constants based on physical parameters.

## Contribution

It introduces a non-commutative quantum version of the Darmois-Skitovich theorem and proves its stability with explicit stability constants.

## Key findings

- Quantum Darmois-Skitovich theorem established
- Theorem is stable under small deviations
- Explicit stability constants derived

## Abstract

The Darmois-Skitovich theorem is a simple characterization of the normal distribution in terms of the independence of linear forms. We present here a non-commutative version of this theorem in the context of Gaussian bosonic states and show that this theorem is stable under small errors in its underlying conditions. An explicit estimate of the stability constants which depend on the physical parameters of the problem is given.