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The two-edge theorem (Theorem \[twisted\_reduction\]) also provides a criteria to determine if the joint distribution of a collection of independent unitaries follows a twisted Haar measure of the form. In fact, for each $n$, we can find a collection of unitaries in $\cal U(2^n)$ such that their joint distribution follows a Haar measure which is not twisted. This leads to a natural question.

Does there exist an $n$ such that the Haar measure on $\cal U(2^n)$ is not twisted?

In the following, we will show that the answer to this question is ‘yes’.

Let $p_m=\frac{1}{2}(1+\eta_m)$, where $\eta_m$ is a Rademacher sequence. By Proposition \[twisted\_measure\], the Haar measure on $\cal U(2^n)$ is not twisted if and only if $p_m=1$ for infinitely many $m$’s.

We first provide an upper bound on $p_m$’s. In fact, we can show that $p_m\le 2^{ -n}$. To see this, note that by Hölder’s inequality, $$\begin{aligned}
\E\left(\frac{1+\eta_m}{2}\right)&\le \left(\E\left(\frac{1+\eta_m}{2^n}\right)\right)^{2/n}.\end{aligned}$$ By Lemma \[Rademacher\_approximation\], there is a function $h(x)$ such that $$\begin{aligned}
\E\left(\frac{1+\eta_m}{2^n}\right)&=\frac{1}{2}+\frac{1}{2^{2n}}\sum_{l=1}^n\E(h(x_l)).
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