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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.

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It is correct provided your $\rho_k^{(X)}$ denotes the partial trace of $\rho_k^{(XY)}$ with respect to $Y$:

$$\rho_k^{(X)} := Tr_Y\left[\rho_k^{(XY)} \right]\:.$$

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  • $\begingroup$ Thanks for your response. If you have a chance please see post. $\endgroup$ – user165535 Jan 18 '18 at 9:50

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