For a classical-quantum state in $\mathcal{H}_{X \otimes A}$ $$ \rho = \sum_x p(x) \vert x\rangle \langle x \vert \otimes \rho^x, $$ I can use spectral decomposition for $ \rho^x $ to get $$ \rho^x = \sum_{i_x} \lambda_{i,x} | i_x \rangle \langle i_x|, $$ and obtain $$ \rho = \sum_{x,i_x} p(x)\lambda_{i,x} \vert x\rangle \langle x \vert \otimes | i_x \rangle \langle i_x|, $$ then if we introduces new system $C$ with basis $\{|\alpha_{x,i_x}\rangle\}$, then $$ |\psi \rangle = \sum_{x,i_x} \sqrt{p(x) \lambda_{i,x}} |x \rangle | i_x \rangle | \alpha_{x,i_x} \rangle $$ is purification of $\rho$. But I wonder whether there is other purification of $\rho$ that makes use of properties of such a classical-quantum state to get a more beautiful form of purification.
Asked
Active
Viewed 128 times
