# Nonquantum Information Gain from Higher-order Correlation Functions

## TL;DR

This paper explores how higher-order correlation functions relate to the quantum state of light, revealing that they can provide both quantum and classical information about the photon number distribution, especially when accounting for vacuum contributions.

## Contribution

It introduces an effective higher-order correlation function that accounts for vacuum effects, broadening the application of correlation functions beyond nonclassical states.

## Key findings

- Bounds on photon number projections derived from $g^{(k)}(0)$
- Effective correlation function accounts for vacuum effects
- Information from effective correlation functions applies to classical and quantum states

## Abstract

Nonlinear correlation functions are at the heart of quantum theory. The second-order correlation function $g^{(2)}(\tau)$ has been a cornerstone of quantum optics since over half a century and a myriad of quantum and classical applications has been discovered. In contrast, higher-order correlation functions have so far only been used to reveal the nonclassical character of the emitted fields. In this paper, we study the relation between the $k$th-order correlation function $g^{(k)}(0)$ and the projection of the underlying quantum state of light onto the subspace of Fock states with photon number less than $k$. We show, that when $g^{(k)}(0)$ falls below a critical value, lower bounds for the projection on this subspace can be concluded as well as on the ratio of the subspace with one upto $k-1$ photons and $k$ to infinity. These bounds are at face value only valid for nonclassical quantum states. However, when the quantum state includes a nonzero projection on the vacuum state, the value of $g^{(k)}(0)$ is artificially enhanced, potentially covering these projections. We derive an effective $k$th-order correlation function, which accounts for the effect of vacuum. We show that the information gained from the effective correlation function is not limited to nonclassical quantum states and thus constitute a quantum- and classical application of higher-order correlation functions.