# Separability gap and large deviation entanglement criterion

## TL;DR

This paper investigates the separability gap in quantum Hamiltonians, providing bounds for its size in random models and proposing an entanglement criterion based on deviations in expectation values.

## Contribution

It introduces explicit bounds for the separability gap in random Hamiltonians and develops an entanglement criterion based on large deviation analysis.

## Key findings

- Bounds for the separability gap depend on the number of subsystems.
- The entanglement criterion applies to generic multipartite systems.
- Large deviations in expectation values indicate almost certain entanglement.

## Abstract

For a given Hamiltonian $H$ on a multipartite quantum system, one is interested in finding the energy $E_0$ of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value $\lambda_{\rm min}^{\otimes}$ of $H$ among all product states. For several concrete model Hamiltonians, we investigate the difference $\lambda_{\rm min}^{\otimes}-E_0$, called separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable among a given state is sufficiently distant from the average value, the state is almost surely entangled.