Dirichlet distribution: posteriors and priors of distribution

Question

Let $|\psi\rangle \in \mathbb{C}^{2n}$ be a random quantum state such that $ |\langle z| \psi \rangle|^{2} $ is distributed according to a $\text{Dirichlet}(1, 1, \ldots, 1)$ distribution, for $z \in \{0, 1\}^{n}$.

Let $z_{1}, z_{2}, \ldots, z_{k}$ be $k$ samples from this distribution (not all unique). Choose a $z^{*}$ that appears most frequently.

1. I am trying to prove:

$$\underset{|\psi\rangle}{\mathbb{E}}\big[|\langle z^{*}| \psi \rangle|^{2}\big] = \underset{|\psi\rangle}{\mathbb{E}}\bigg[\underset{m}{\mathbb{E}}\big[|\langle z^{*}| \psi \rangle|^{2} ~| ~m\big]\bigg] = \mathbb{E}\bigg[\frac{1+m}{2^{n}+k}\bigg],$$ where $m$ is a random variable that denotes the frequency of $z^{*}$.

2. I am also trying to prove that for the collection $z_{1}, z_{2}, \ldots, z_{k}$ $$\sum_{i \neq j}\mathrm{Pr}[z_{i} = z_{j}] = {n \choose k}\frac{2}{2^{n} + 1}. $$

Basically, I am trying to trace the steps of Lemma $13$ (page 10) of this quantum paper. I realize that my questions have to deal with posteriors and priors of the chosen distribution (though I do not understand how they have been explicitly derived or used here. An explicit derivation will be helpful). Is there any resource where I can find quick formulas for calculating these for other distributions, like the Binomial distribution?

Answer 1

Let $ p_i = |\langle i | \phi \rangle|^2 \sim Dir(a_1, .., a_{2^n}) = Dir(1, .., 1) $ and $ m_i $ the occurences of outcome $ |i\rangle $ on samples $z_1, .. z_k$.

Since the Dirichlet distribution is the conjugate prior of the categorical (see here), meaning

$ \bf{p} $ $| Z, (1, .. 1), $ $\bf{m} $ $ \sim Dir(2^n, $ $\bf{m} + 1$)

and using the formula for the mean value of Dirichlet we get

$ \mathbb{E}[p_{z*} | m] = \frac{m+1}{\sum_{j=1}^{2^n} (m_j + 1)} = \frac{m+1}{2^n + k} $

For the second claim, take $ i \neq j $ and compute \begin{align*} \mathbb{P}[z_i = z_j] &= \int \mathbb{P}[z_i = z_j | (p_1, .. p_{2^n})] \cdot f(p_1, .. p_{2^n}) \\ &= \int \sum_{k=0}^{2^n} p_k^2 \cdot f(p_1, .. p_{2^n}) \\ &= \sum_{k=0}^{2^n} \mathbb{E}[p_k^2] \\ &= \sum_{k=0}^{2^n} \frac{2}{2^n(2^n + 1)} = \frac{2}{2^n + 1} \end{align*}

(the last equality holds since $ \bf{x} $ $ \sim Dir($$\bf{a}$) $ \implies \mathbb{E}[x_i^2] = \frac{a_i(a_i + 1)}{a_0(a_0 + 1)} $, $ a_0 = \sum_{i=1}^{N} a_i $.

This means that $ \sum_{i \neq j} \mathbb{P}[z_i = z_j] = {k \choose 2} \frac{2}{2^n + 1} $.