# Questions tagged [haar-distribution]

Having to do with continuous distributions, which are 'uniform' in the sense of being invariant under some relevant symmetry. In the case of quantum information theory, this usually relates to unitarily invariant distributions on single-qubit states, many-qubit states, single-qubit unitary transformations, or many-qubit unitary transformations.

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### Multiplication by a Haar random unitary two times

Consider a Haar random unitary $U$. I am trying to compute the value (or put a bound on) $$\mathbb{E}\left[\left|\langle 0^{n} |U^{2} |0^{n}\rangle\right|^{2}\right].$$ The ...
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### Approximating the average of a rational function with respect to the Haar measure

Suppose I have a state $|\psi\rangle = U|0\rangle$ where $U$ is a $d$-dimensional unitary sampled uniformly with respect the Haar measure. I'm interested in computing or approximating analytically an ...
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### What is the expectation value ${\Bbb E}[\langle\psi,O\psi\rangle]$ over the Haar distribution?

What is the average $\mathbb{E}_{\text{Haar}}|\langle\psi|O\psi\rangle|$ of expectation of an arbitrary observable $O$ over the Haar distribution? Let $d$ be the dimension, i.e, the size of $O$. Do we ...
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### How to show that the integral over all Haar states vanishes: $\int|\psi\rangle\,{\rm d}\psi = 0$?

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 ...
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### Is there a lower bound on the average diamond norm of two uniformly random unitaries U1 and U1 of dimension D that are sampled from haar measure?

I could not find any lower bound on the diamond norm for two uniformly random unitaries of dimension D sampled from the haar measure.
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Consider an arbitrary $n$-qubit state $\lvert \psi \rangle$. How much do we understand about the probability distribution of the fidelity of $\lvert \psi \rangle$ with a tensor product \$\lvert \alpha \...