Closest N-representable Density Matrix (Semidefinite and Equality Constraints)

[AlexandreDeCamargo]

Semidefinite Constraints

In the Semidefinite Constraints the first step is to map the the density matrix, $\Gamma$ , to a positive semidefinite constraint. Do we have the definition of $\mathcal{L}(\Gamma)$ ?

Equality Constraints

In the Equality Constraints part if the equality is not satisfied do we need to check if $Tr[\mathbf{B}_m \Gamma] > 0$. If it's not, should we then replace $\mathbf{B}_m$ with $-\mathbf{B}_m$? After this adjustment, is the projection based on the constraint $Tr[\mathbf{B}_m \Gamma] \leq 0$?"

@PaulWAyers

[PaulWAyers]

With equality constraints, there are three cases:

$$ \begin{split} \text{Tr}[\mathbf{B}_m \Gamma] &> \epsilon \ \text{Tr}[\mathbf{B}_m \Gamma] &< \epsilon \ \left|\text{Tr}[\mathbf{B}_m \Gamma]\right| &\leq \epsilon \end{split} $$

Here $\epsilon$ is a threshhold to determine how tightly the constraint is required to hold.

If case 3 then there is no need to worry about the constraint.

If case 1 you use that inequality constraint; if case 2 you multiply $\mathbf{B}_m$ by $-1$ and it becomes case 1.

Does this help?