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Can we show that the integral over all Haar states $|\psi \rangle $ is $$ \int |\psi \rangle \, \mathrm{d}\psi = 0~. $$

This is an integral over Haar vectors

Reference to a post about what is Haar state

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Yes, we can show this using the unitary invariance of the Haar measure on states. In more detail, we have $$ U \int |\psi\rangle\, \mathrm{d}\psi = \int U|\psi\rangle\, \mathrm{d}\psi = \int |\psi\rangle\, \mathrm{d}(U^\dagger\psi) = \int |\psi\rangle\, \mathrm{d}\psi. $$ Hence, the integral is a fixed point of any $U\in U(d)$. However, the only vector fixed by all unitaries is the zero vector (since $U(d)$ acts irreducibly), hence $$ \int |\psi\rangle\, \mathrm{d}\psi = 0. $$

PS: Geometrically, this is basically an integral over the complex sphere in $\mathbb C^d$, so it might be intuitively clear that it has to be zero by symmetry. This is exactly the unitary symmetry I have used above. I say "basically" because $\psi$ is actually a ray in $\mathbb C^d$, so it should live in complex projective space $P\mathbb C^{d-1}$. However, this detail does not matter here.

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  • $\begingroup$ This is beautiful. $\endgroup$
    – R.W
    Mar 10, 2022 at 10:28

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