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4 votes
Accepted

Prove that $A\preceq B$ implies $A=\Psi(B)$ for some channel $\Psi$

Let $\rho_{d}, \sigma_{d}$ be the (simultaneously diagonal) density matrices whose eigenvalues are $\{ p_{j} \}, \{ q_{j} \}$, respectively (represented as probability vectors below). Then, if $\vec{...
keisuke.akira's user avatar
2 votes
Accepted

Why can we write $\rho=\sum_\mu q_\mu|\varphi_\mu\rangle\!\langle\varphi_\mu|$ iff $q\preceq \mathrm{spec}(\rho)$?

We have $\sqrt{q_\mu}\langle\alpha_j|\varphi_\mu\rangle = \sum_i \sqrt{p_i} V_{\mu i}\langle\alpha_j|\alpha_i\rangle = \sqrt{p_j} V_{\mu j}$. Thus $q_\mu |\langle\alpha_j|\varphi_\mu\rangle|^2 = p_j |...
Danylo Y's user avatar
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