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Two qubit Pauli expectation value of $\underset{U}{\mathbb{E}}[U^{\otimes 2} (P_1 \otimes P_2)^{\otimes 2} U^{*\otimes 2}]$

We can prove a slightly more general statement, which works for any Hermitian "sandwiched" matrix $P$ in any dimension $d$... but only for $t=2$ in the number of copies :'( Let us call $V$ ...
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