# Discarding a quantum system in joint state

If you have a joint quantum state given by the density operator $$\rho^{(XYZ)} = \sum_{k}p_k\rho_{k}^{(XY)} \otimes |k^{(Z)} \rangle \langle k^{(Z)}|$$ then am I correct in stating that if we want to discard the quantum system $Y$ and obtain the joint system $\rho^{(XZ)}$ then we do this by taking the partial trace over $Y$ hence obtaining $$\rho^{(XZ)} = Tr_{Y}[ \rho^{(XYZ)}] = \sum_{k}p_k \rho_{k}^{(X)} \otimes |k^{(Z)} \rangle \langle k^{(Z)}|$$

Is this correct?

Thanks.

It is correct provided your $\rho_k^{(X)}$ denotes the partial trace of $\rho_k^{(XY)}$ with respect to $Y$:
$$\rho_k^{(X)} := Tr_Y\left[\rho_k^{(XY)} \right]\:.$$