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

Haar random states are fundamental objects in quantum information theory and quantum computing. We study the density matrix resulting from sampling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> </mml:math> copies of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> </mml:math> -copies of a real Haar random state and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> </mml:math> -copies of a complex Haar random state. Using this we show a lower-bound on the approximation parameter of real-valued state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> </mml:math> -designs and improve the lower-bound on the number of copies required for imaginarity testing.