Is $\text{Tr}(\text{Tr}_\mathcal{E}(\rho)) = \text{Tr}(\rho)$?

Question

Let $\rho$ be a density matrix over some composite Hilbert space $\mathcal{H}_S \otimes \mathcal{H}_{\mathcal{E}}$. Is partial trace full trace preserving? I.e., is $$\text{Tr}(\text{Tr}_\mathcal{E}(\rho)) = \text{Tr}(\rho).$$

Answer 1

If $\rho$ is a valid density matrix, then $\text{Tr}_\epsilon(\rho)$ is also a valid density matrix, called as reduced density matrix and hence has a unit trace.

$$\text{Tr}\big( \text{Tr}_\epsilon(\rho) \big) = \text{Tr}(\rho) = 1\,.$$

Partial trace operation is a CPTP map. You can read more about it here.

Answer 2

Yes: $\mathrm{Tr}_A\circ\mathrm{Tr}_B = \mathrm{Tr}_{AB}$.