In a paper of last year [1] we used an inequality found 15 years ago by Eisert and collaborators [2]. In the most interesting version of this inequality one supposes a pure Gaussian state on $N$ bosonic modes, and considers the reduced density matrices of each mode. Those reduced density matrices are also Gaussian, and are thermal equilibrium states, each one characterized by its own "mode temperature". Which sets of mode temperatures are compatible with some global pure Gaussian state?
The answer is given by auxiliary quantities $b_j = \coth\frac{\hbar\omega_j}{2k_B T_j} - 1$, where $\omega_j$ is the frequency of mode $j$, and $T_j$ is its temperature: there is a global Gaussian pure state with such marginals if and only if $b_k \leq \sum_{j\neq k} b_j$, for all $k$.
In this talk I will try to prove the Eisert et al inequality, following [2] and the earlier papers and books the authors refer to, at times in several layers. I will also outline what I think is an interesting related problem in the limit of very large $N$.
[1] "Hawking radiation and the quantum marginal problem" EA, M Eckstein & P Horodecki, J Cosmology and Astroparticle Physics (01), 014 (2022)
[2] "Gaussian Quantum Marginal Problem", Eisert et al, Comm Math Phys 280, 263–280 (2008)