Assume a composite quantum systes with state $|\psi_{AB}\rangle$ or better $\rho=|\psi_{AB}\rangle\langle\psi_{AB}|$. I want to know the state of system $A$ only, i.e. $\rho_A$.
Is there any difference if I trace out system $B$, i.e. $\rho_A=Tr_B\rho$ compared to building up $\rho_A$ from projections on the Pauli operators, i.e. $\displaystyle\rho_A=\sum_{k=1,x,y,z}Tr\big((\sigma_k\otimes 1\big)^\dagger \rho)\sigma_k$.
Some numerics indicate, that this is the same...