How can I understand these two equations about the indirect measurement?

Question

I'm reading an article about environmental monitoring and information transfer. Suppose $S$ represents a quantum system and $E$ is the environment. Assume at time $t=0$ there are no correlations between $S$ and $E$: $\rho_{SE}(0)=\rho_{S}(0)\otimes\rho_{E}(0)$, and this composite density operator evolved under the action of $U(t) = e^{-iHt/h}$, where $H$ is the total Hamiltonian. Let $P_\alpha$ be a projective operator on $E$. Then, the probability of obtaining outcome $α$ in this measurement when $S$ is described by the density operator $\rho_s(t)$ is given as

$$ \text{Prob}(\alpha|\rho_s(t))=\text{Tr}_E (P_αρ_E(t)) $$

and the density matrix of $S$ conditioned on the particular outcome $\alpha$ is

$$ \rho_s^{\alpha}(t)= \frac{\text{Tr}_E\{(I\otimes P_\alpha)\rho_{SE}(t)(I\otimes P_\alpha)\}}{\text{Prob}(\alpha|\rho_s(t))} $$

I'm wondering how those two equations coming from? Also, since the indirect measurement aims to yield information about S without performing a projective (and thus destructive) direct measurement on S, why there's $P_\alpha$ in the equation? Thanks!!

Answer 1

The two equations are part of the measurement postulate of quantum mechanics which states that probability of the outcome $m$ in a measurement described by operators $M_m$ on a state $\rho$ is

$$ p(m) = \mathrm{tr}(M_m^\dagger M_m \rho)\tag1 $$

(c.f. $(2.159)$ in Nielsen & Chuang) and the post-measurement state is

$$ \frac{M_m\rho M_m^\dagger}{\mathrm{tr}(M_m^\dagger M_m \rho)}\tag2 $$

(c.f. $(2.160)$ in Nielsen & Chuang).

---

The first equation in the question follows from substitutions

$$ m = \alpha \\ \rho = \rho_E(t) \\ M_m = P_\alpha $$

in $(1)$. The second follows from substitutions

$$ \rho = \rho_{SE}(t) \\ M_m = I\otimes P_\alpha $$

followed by partial trace over the environment on the post-measurement state.