Quantifying entanglement of Gaussian thermal states with 2- and 4-body correlation functions
Related publication: Gondret V, Lamirault C, Dias R, Leprince C, Westbrook CI, Clément D & Boiron D. Quantifying two-mode entanglement of bosonic Gaussian states from their full counting statistics. arXiv:2503.09555 (2025) PDF or HTML. See also this talk, this poster or the Chapter 2 of my thesis.
Background
A many-body quantum state is completely characterized by its equal time correlation functions of the field operators . In other words, if we consider a two-mode state with creation (annihilation) operator $\hat{a}_j^{(\dagger)}$ where j=1 or 2, one needs to measure the value of
$\langle\hat{a}_1\rangle$,
$\langle\hat{a}_2\rangle$,
$\langle\hat{a}_2\hat{a}_1\rangle$,
$\langle\hat{a}_2\hat{a}_1^\dagger\rangle$…$\langle\hat{a}_2^\dagger\hat{a}_2^3\hat{a}_1^{\dagger 2}\rangle$… and so on and so forth.
However, for some states, these many-body correlation functions can be decomposed as a sum of one- and two-fields operators: the state is said to be Gaussian. Here, it means that the measurement of ‘only’
$\langle\hat{a}_j\rangle$ and $\langle\hat{a}_j^{(\dagger)}\hat{a}_i^{(\dagger)}\rangle$ are needed.
Gaussian states are common in physics because their Gaussian nature is preserved under the evolution of any quadratic Hamiltonian.
These Hamiltonians may be simple, and the quantum field theory framework provides a clear understanding of pair creation mechanism, yet it is only recently that experimental observations have been realized.
It includes examples such as Bogoliubov pairing in an equilibrium interacting Bose gas , parametric amplification of non-interacting quasi-particles , quasi-particle creation in an analogous expanding metrics or Hawking radiations .
While the correlation signal between the two modes that was measured is clear, we aim here to discuss and emphasize how entanglement can be assessed and quantified.
(a) Two modes in a system are defined (shown by the lens here) and the number of particles in each mode is counted. (b) These counts can be histogrammed to obtain the full counting statistics of each mode. In addition, the joint probability distribution is also accessible. This information yields the n-body correlation functions of particle number operators of any order.
Number correlations versus field correlations
Nowadays quantum gases microscopes allow to detect indivdiual particles hence to resolve the full (joint) probability distribution of the state , see the figure above.
However, these experiments do not measure a field e.g. the mean value of the annihilation or creation operators, but rather any N-body correlation functions (any product of pairwise normally ordered annihilation and creation operators).
For example, they can measure the cross two-body correlation function \(G^{(2)}_{12} = \langle \hat{a}_2^\dagger\hat{a}_1^\dagger\hat{a}_2\hat{a}_1 \rangle\)
or the local two-body correlation function
\(G^{(2)}_{11}= \langle \hat{a}_1^{\dagger 2}\hat{a}_1^2 \rangle\).
When the state is Gaussian, the measurement of these many-body correlation functions can be connected to the mean value of the field and the two-field operators. For example, the cross two-body correlation function can be expanded using Wick theorem
\begin{equation}
G^{(2)}_{12} = n_1n_2+|\langle \hat{a}_2\hat{a}_1 \rangle |^2 + |\langle \hat{a}_2^\dagger\hat{a}_1 \rangle |^2.
\end{equation}
where \(n_i= \langle \hat{a}_i^\dagger \hat{a}_i\rangle\) is the mean population of the mode.
Hillery and Zubairy have shown that when the anomalous correlation exceeds the populations i.e. \(|\langle \hat{a}_2\hat{a}_1 \rangle |^2>n_1n_2\), the two-mode state is entangled.
However, to relate this inequality to the measurement of \(G^{(2)}_{12}\), one needs to assume that the coherence between the two-modes \(|\langle \hat{a}_2^\dagger\hat{a}_1 \rangle |^2\) vanishes.
Results
In this work, we propose to access this lacking information from higher order correlation functions, especially the four-body correlation function.
We show that for bosonic thermal Gaussian states, the measurement of the populations and the two- and four-body correlation functions quantify the entanglement of the two-mode state. Furthermore, when the four-body correlation function is not measured, we derive an entanglement witness only based on the (normalized) two-body correlation function. When the coherence is assumed to be zero, this witness is equal to 2. However, when this hypothesis is relaxed as in our work, the threshold value to assess entanglement is shifted to higher values depending on the population of the state.
The two-body correlation function witnesses the entanglement of the two-mode Gaussian state. The grey region corresponds to states who do not fullfil our hypothesis (e.g. non Gaussian states). In the question mark region, the measurement of the four-body correlation function is needed to discriminate wether the state is entangled or separable.
