6 problems

Carlen–Lieb–Reuvers conjecture. The Hilbert–Schmidt norm of the $k$-particle reduced density matrix is maximised by Slater determinants, equivalently,

Let $\hat{\rho}$ be a density matrix for a system with $L$ sites and $N$ flavors, and let $\hat{\rho}_n$ be its $n$-particle reduced density matrix. Write $\lambda_n$ for the maxim…

A natural orbital is an eigenstate of the one-body reduced density matrix, whose eigenvalues are the natural occupation numbers. Löwdin's conjecture. The natural orbitals would be…

Let $\mathcal{H}_N$ be the Hilbert space of $N$ fermionic particles, let $\Psi_N\in\mathcal{H}_N$ be normalized, and set $\rho_N=|\Psi_N\rangle\langle\Psi_N|$. Denote by $\gamma_2$…

Let $\varrho$ be a pure fermionic density operator, let $\varrho_p$ denote its conventional $p$-particle reduced density matrix, and let … be the particle-reduction operation, with…

Let an $N$-particle fermionic state have a reduced $2$-particle density matrix. A pure-state Slater determinant is an antisymmetrized product of $N$ one-particle states, and its re…