Let's say we have a single system $A$, with density matrix $\rho_A$, and we perform the projective measurement $\{P_i\}$. The statistics will be given as $p_i =$ Tr$\{ P_i \rho \}$.

Now, say we have two systems, with Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$, and we have a joint space $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$. I wish to perform a projective measurement only on sybsystem $A$, given $\rho$ which belongs to the tensor product space. The projectors $\{ P_i^{A} \}$ are defined on the sybsystem $A$.

To find the statistics $p_i$, I'm confused between the following two possibilities:

1. $p_i =$ Tr$\{ (P_i^{A} \otimes I^B) \; \rho \}$, here the trace is taken over the entire (composite) Hilbert space.
2. $p_i = $ Tr$\{ P_i^{A} \; \rho_A \}$ (here $\rho_A = Tr_B \{ \rho \}$, that is, a partial trace over the subsystem $B$). Of course, the trace in $p_i$ is only taken over the subsystem $A$

Which of the above is correct? 1 or 2? Both? Neither?

Let's say we have a single system $A$, with density matrix $\rho_A$, and we perform the projective measurement $\{P_i\}$. The statistics will be given as $p_i = \mathrm{Tr}\{ P_i \rho \}$.

Now, say we have two systems, with Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$, and we have a joint space $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$. I wish to perform a projective measurement only on sybsystem $A$, given $\rho$ which belongs to the tensor product space. The projectors $\{ P_i^{A} \}$ are defined on the subsystem $A$.

To find the statistics $p_i$, I'm confused between the following two possibilities:

1. $p_i = \mathrm{Tr}\{ (P_i^{A} \otimes I^B) \; \rho \}$, here the trace is taken over the entire (composite) Hilbert space.
2. $p_i = \mathrm{Tr}\{ P_i^{A} \; \rho_A \}$ (here $\rho_A = \mathrm{Tr}_B \{ \rho \}$, that is, a partial trace over the subsystem $B$). Of course, the trace in $p_i$ is only taken over the subsystem $A$

Which of the above is correct? 1 or 2? Both? Neither?