Suppose we have a uniform distribution $D$ over $k$-local Pauli operators $P_{q_1}\otimes \dotsc \otimes P_{q_k} $, $P_{q_i} \in \{ X, Y, Z, I \}$. Is it possible to calculate $\mathbb{E}_{P_i \sim D} \text{Tr}(P_i (\rho_1 - \rho_2))^m$ (for some $m \geq 1$) for two random pure states $\rho_1$ and $\rho_2$?

When $D$ is a distribution over Haar random $t$-design unitary operator, we can calculate the quantity. I am curious if we can calculate when $D$ is the uniform distribution over $k$-local Pauli operators.



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