How are mixed states given to a quantum algorithm?

Question

I've been reading this paper about quantum fidelity estimation, but really have no idea what's going on when it comes to density matrix notation. In the abstract, they have the following quote:

provided that the purifications of ρ and σ are prepared by quantum oracles.

Where $\rho$ and $\sigma$ are the two mixed quantum states.

But I'm not really sure how these states are meant to be given. It says that they are "quantum oracles" but oracles must be unitary, wheras density matrices are Hermitian. So how are these states given?

Answer 1

Given any density matrix $\rho$, we can find a (non-unique) pure state $|\rho\rangle$ that purifies it. For instance, if $$ \rho = (|0\rangle\langle 0| + |1\rangle\langle 1|)/2, $$ then one choice of $$ |\rho\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle). $$ The circuit to prepare this last state is just $CX_{01}H_{0}$.

For this paper, refer to Section 3

They assume that there are two boxes that output, respectively, one copy of states $|\rho\rangle$ and $|\sigma \rangle$ each time they are queried. Each box is just a quantum circuit that acts on all-zero state to produce the purified state.