The real Haar state twirl is diagonalized exactly, yielding the trace distance between real and complex Haar random states and new lower bounds on real-valued t-designs and imaginarity testing.
Spherical Harmonics
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abstract
This is Chapter 1 of the book {\it Approximation Theory and Harmonic Analysis on Spheres and Balls} by the authors. It provides a self-contained introduction to spherical harmonics. The book will be published as a title in {\it Springer Monographs in Mathematics} by Springer in 2013. The table of contents of the book is attached at the end of this file.
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Exact distinguishability between real-valued and complex-valued Haar random quantum states
The real Haar state twirl is diagonalized exactly, yielding the trace distance between real and complex Haar random states and new lower bounds on real-valued t-designs and imaginarity testing.