Random Real Valued and Complex Valued States Cannot be Efficiently Distinguished

Abstract

In this short note we show that the ensemble $\{O|0⟩⟨0|O^{\top }|O\in \mathbb{O}(d)\}$, where $O$ is drawn from the Haar measure on $\mathbb{O}(d)$ cannot be distinguished from $t$ copies of a Haar random state unless $t=\hat{I}\text{c◯}(\sqrt{d})$. Our proof has the benefit of exactly computing the trace distance, which scales as $\hat{I}˜(t^2/d)$ for $t=O(\sqrt{d})$, between the moments as well as being surprisingly short. Lastly, we show that twirling certain states with orthogonal matrices yields exact $t=3$ designs, yet the same cannot be true for $t>3$.

5 pages, 1 figure. Comments welcome!