Consider the following minimization problem:
\begin{align} &\min_{\rho} \mathrm{Tr}[\rho H] \\ \text{such that:}& \\ &Tr[\rho A_i] \leq 0 \ \ \forall A_i, \ i \in \{1,2,3,...\} \end{align}
Where $H$ is a Hamiltonian and $A_i$ are Hermitian operators with both negative and positive eigenvalues, which in general may not commute with other $A_j$ operators. Without the additional constraints, the above problem would be equivalent to finding the lowest energy eigenstate of the Hamiltonian $H$. Is there a way to reformulate the above problem with additional constraints as a ground state energy problem of a new Hamiltonian $H'$ without any additional constraints?
Are there any known problems that are similar to above mentioned problem?
