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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?

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  • $\begingroup$ Note that in the first problem, $\rho=\left|\lambda_{\text{min}}\right\rangle\left\langle\lambda_{\text{min}}\right|$ is always a solution. In the other problem however, some constraints are impossible, such as $A_1=I$ or $A_1=-A_2$. At the very least, if there was a process to transform this problem into a problem on some $H^*$ without constraints, the transformation into $H^*$ would have to take into account the possibility that no solution exists at all. Thus, though it seems unlikely that such a transformation exists, proving it is beyond my knowledge. $\endgroup$
    – Tristan Nemoz
    Commented Oct 12, 2022 at 18:34

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