Random Real Valued and Complex Valued States Cannot be Efficiently Distinguished

In this short note we show that the ensemble $\{O\mathrm{\mid }0⟩⟨0\mathrm{\mid }{O}^{\mathrm{\top }}\mathrm{\mid }O\in \mathbb{O}(\mathbb{d})\}$, where $O$ is drawn from the Haar measure on $\mathbb{O}(d)$ cannot be distinguished from $t$ copies of a Haar random state unless $t=\mathrm{\Omega }(\sqrt{d})$. Our proof has the benefit of exactly computing the trace distance, which scales as $\mathrm{\Theta }(t^2\mathrm{/}d)$ 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 $t>3$.