How is tracing out a physical operation? Suppose $\rho_{AB}$ denotes the density matrix of a bipartite system. Reduced density matrix of A ($\rho_A$) is obtained by tracing out B $$\rho_A\equiv\sum_{i}\langle i_B |\rho_{AB}|i_B\rangle$$ where $\{|i_B\rangle \}$ is the basis of the subsystem B. It is said that $\rho_A$ is the physical state of the subsystem A. What justifies this claim?   
 A: The partial trace over $B$ of the quantum state of a bipartite system $AB$ corresponds to discarding $B$: that is, the reduced density matrix $\rho_A=\mathrm{Tr}_B(\rho_{AB})$ is the complete description of the state of the system for any and all measurements that are completely local to $A$.
This can be made precise by considering an arbitrary hermitian measurement operator $\mathcal O_A$ (which includes, among other things, eigenprojectors corresponding to the measurement of some other observable), whose expectation value is
$$
\langle \mathcal O_A\rangle = \mathrm{Tr}\mathopen{}\left( \hat{\rho}_{AB} \ \hat{\mathcal O}_A\otimes \mathbb I\right)\mathclose{}.
$$
Here the trace can be decomposed as 
$$
\langle \mathcal O_A\rangle = \mathrm{Tr}_A\mathopen{}\left( \mathrm{Tr}_B\mathopen{}\left( \hat{\rho}_{AB} \ \hat{\mathcal O}_A\otimes \mathbb I\right)\mathclose{}\right)\mathclose{},
$$
and since $\hat{\mathcal O}_A$ does not act on the $B$ sector, it can be factored out of the $B$ trace, giving
$$
\langle \mathcal O_A\rangle = \mathrm{Tr}_A\mathopen{}\left( \mathrm{Tr}_B\mathopen{}\left( \hat{\rho}_{AB}\right)\mathclose{} \hat{\mathcal O}_A\right)\mathclose{},
$$
or in other words,
$$
\langle \mathcal O_A\rangle = \mathrm{Tr}_A\mathopen{}\left(  \hat{\rho}_{A} \hat{\mathcal O}_A\right)\mathclose{}.
$$
Thus, if you want to predict the results of any possible experiment that only involves $A$, then you need no more, and no less, than $\hat{\rho}_{AB}$.
