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I came across the concept of average fidelity $\int f(\psi)d\,\psi$ where the integration is with respect to the uniform Haar measure on pure states. I've only seen Haar measures in connection with topological groups, so I don't understand what measure is used in this context. Could anyone provide a reference where this is explained ?

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  • $\begingroup$ I do not understand well. Are you dealing with the projective complex space (containing pure states) of a finite dimensional Hilbert space? The "Haar" measure is that induced by $U(n)$ on that projective space viewed as a quotient? $\endgroup$ – Valter Moretti Feb 12 '14 at 9:23
  • $\begingroup$ Yes it is over a finite dimensional space. Where can I find a reference about the measure induced by $U(n)$ that you mentioned? $\endgroup$ – Chris Feb 12 '14 at 9:44
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The group $U(n)$ has a unique Haar measure, both right and left invariant, since it is unimodular as it is compact. Now consider the complex projective space $$\cal{P}(\mathbb C^n)= \left({\mathbb C^n} / \sim\right) - [0]\quad \mbox{when}\quad v \sim v' \quad \mbox{iff}\quad v = cv'\:, \quad c\in \mathbb C -\{0\} \:.$$ The group $U(n)$ acts smoothly (with respect to the natural real smooth 2n-2 dimensional manifold structure of the projective space) and transitively on $\cal{P}(\mathbb C^n)$, trivially: $${\cal P}(\mathbb C^n) \ni [v] \mapsto [U\psi]\:.$$

${\cal P}(\mathbb C^n)$ can therefore be viewed as a smooth quotient of $U(n)$ and the compact subgroup $H_n$ which leaves fixed a point of ${\cal P}(\mathbb C^n)$.

Under these conditions, taking into account that both $U(n)$ and $H_n$ are compact (Lie) topological groups and thus are unimodular, there exist a $U(n)$-invariant positive Borel measure $\mu$ on ${\cal P}(\mathbb C^n) = U(n)/H_n$, unique up to the normalization. That is your measure.

These results were essentially due to Mackay. An account of the general theory can be found in Ch.4 of fundamental Barut-Raczka's textbook on the theory of group representations and applications World Scientific 1986.

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