Exact distinguishability between real-valued and complex-valued Haar random quantum states

Quantum 10, 2120 (2026). https://doi.org/10.22331/q-2026-05-29-2120 Haar random states are fundamental objects in quantum information theory and quantum computing. We study the density matrix resulting from sampling $t$ copies of a $d$-dimensional quantum state according to the Haar measure on the orthogonal group. In particular, we analytically compute its spectral decomposition. This allows us to compute exactly the trace distance between $t$-copies of a real Haar random state and $t$-copies o