Typical behaviour of genuine multimode entanglement of pure Gaussian states

TL;DR

This paper investigates the typical genuine multimode entanglement in pure Gaussian states, deriving analytical expressions for the average entanglement and analyzing how it concentrates as the number of modes increases.

Contribution

It provides a closed-form expression for the Haar averaged GGM of multimode Gaussian states and analyzes the distribution and concentration of entanglement with increasing modes.

Findings

Haar averaged GGM increases with the number of modes.

Distribution of GGM shifts towards higher entanglement as modes increase.

Deviation from average GGM is exponentially suppressed for large mode numbers.

Abstract

Trends of genuine entanglement in Haar uniformly generated multimode pure Gaussian states with fixed average energy per mode are explored. A distance-based metric known as the generalized geometric measure (GGM) is used to quantify genuine entanglement. The GGM of a state is defined as its minimum distance from the set of all non-genuinely entangled states. To begin with, we derive an expression for the Haar averaged value of any function defined on the set of energy-constrained states. Subsequently, we investigate states with a large number of modes and provide a closed-form expression for the Haar averaged GGM in terms of the average energy per mode. Furthermore, we demonstrate that typical states closely approximate their Haar averaged GGM value, with deviation probabilities bounded by an exponentially suppressed limit. We then analyze the GGM content of typical states with a finite number of modes and present the distribution of GGM. Our findings indicate that as the number of modes increases, the distribution shifts towards higher entanglement values and becomes more concentrated. We quantify these features by computing the Haar averaged GGM and the standard deviation of the GGM distribution, revealing that the former increases while the latter decreases with the number of modes.