$F_{\rm LL}$ and prove a new upper bound on the kinetic energy. Section \[sec:proof\] contains the proof of our main results. Finally, in Appendix \[app:classical\] we discuss a similar bound in the classical case where the kinetic energy is dropped, extending thereby our previous result in [@LewLieSei-18].

### Acknowledgments. {#acknowledgments. .unnumbered}

The authors thank the Institut Henri Poincaré for its hospitality. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements AQUAMS No 694227 of R.S. and MDFT No 725528 of M.L.).

Main result
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The grand-canonical Levy-Lieb functional
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Let us consider a density $\rho\in L^1({\mathbb{R}}^3,{\mathbb{R}}_+)$ such that $\sqrt\rho\in H^1({\mathbb{R}}^3)$. Naturally we should assume in addition that $\int_{{\mathbb{R}}^3}\rho=N$ is an integer, but here we will work in the grand canonical ensemble where this is not needed. The *grand-canonical Levy-Lieb functional* [@Levy-79; @Lieb-83b; @LewLieSei-18] is defined by $$\begin{gathered}
F_{\rm LL}(\rho):=\\\inf_{\substack{\Gamma_n=\Gamma_n^*{\geqslant}0\\ \sum_{n=0}^{\infty}{{\rm Tr}\,}(\Gamma_n)=1\\ \sum_{n=1}^{\infty}\rho_{\Gamma_n}=\rho}} \Bigg\{\sum_{n=1}^{\infty}{{\rm Tr}\,}_{{\mathfrak{H}}^n}\Bigg(-\sum_{j=1}^n\Delta_{x_j}+\sum_{1{\leqslant}j<k{\leqslant}n}\frac{1}{|x_j-x_k|}\Bigg)\Gamma_n\Bigg\}.
\label{eq:Levy-Lieb-grand-canonical}\end{gathered}$$ Here $${\mathfrak{H}}^n:=L^2_a(({\mathbb{R}}^3\times \{1,...,q\})^n,{\mathbb{C}})$$ is the $n$-particle space of antisymmetric square-integrable functions on $({\mathbb{R}}^3\times \{1,...,q\})^n$, with $q$ spin states (for electrons $q=2$). The family of operators $\Gamma=\{\Gamma_n\}_{n{\geqslant}0}$ forms a grand-canonical mixed quantum state, that is, a state over the fermionic Fock space 