Random Real Valued and Complex Valued States Cannot Be Efficiently Distinguished

In this short note we show that the ensemble {O | 0⟩⟨ 0 | O^⊤ | O ∈𝕆(𝕕)}, where O is drawn from the Haar measure on 𝕆(d) cannot be distinguished from t copies of a Haar random state unless t = Ω(√(d)). Our proof has the benefit of exactly computing the trace distance, which scales as Θ(t^2/d) for t = O(√(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.