Let $\psi$ and $\phi$ be two uniformly random pure state $\psi, \phi \sim\mathbb{C}^d$. The the following equality holds \begin{align} \mathbb{E}_{\psi, \phi \sim \mathbb{C}^d} {\rm Tr}[\vert \phi \rangle \phi \vert \psi \rangle \langle \psi \vert] = \mathbb{E}_{\psi, \phi \sim \mathbb{C}^d} \vert \langle \phi \vert\psi \rangle\vert^2 = \frac{1}{d} \tag{1}. \end{align} How to prove Eq. (1)?
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