Lemma 9, p. 28 — the level-k Lasserre value is a relaxation: νk(H) ≥ λmax(H)

Let $H$ be a Hermitian operator on $n$ qubits and $k\ge0$. Let $\widetilde\rho$ be an optimal solution of level $k$ of the quantum Lasserre hierarchy, i.e. a $k$-positive pseudo-density maximizing $\operatorname{Tr}(H\widetilde\rho)$, so that $\nu_k(H)=\operatorname{Tr}(H\widetilde\rho)$. Then for every density matrix $\sigma$ (positive semidefinite, trace one)

Since $\lambda_{\max}(H)=\max_\sigma\operatorname{Tr}(H\sigma)$, this is $\nu_k(H)\ge\lambda_{\max}(H)$: the hierarchy is a relaxation of the largest eigenvalue.

This is the inequality in equation (1) and the second inequality in the proof of Proposition 1.

Formalization Note $\lambda_{\max}(H)$ is expressed through comparison with every density matrix, and $\nu_k(H)$ through an optimal pseudo-density (equation (1) writes $\nu_k$ as a maximum). The paper proves the lemma in the equivalent vector-program form (Definition 17); the pseudo-density form is equation (1). Real parts of traces are taken; they are the full traces for Hermitian arguments.