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.
Questions tagged [haar-distribution]
62 questions
5
votes
3 answers
Show that, averaging over uniformly random unitaries, $\mathbb{E}[UXU^\dagger]=\operatorname{Tr}(X)\frac{I}{d}$
As mentioned e.g. in this answer, if we compute the average
$$\Phi(X)\equiv \mathbb{E}_U[UXU^\dagger] \equiv \int_{\mathbf U(d)} d\mu(U) UXU^\dagger,$$
where $d\mu(U)$ is the Haar measure over the unitary group $\mathbf U(d)$, we get…
glS
- 24,708
- 5
- 34
- 108
4
votes
0 answers
How do I calculate the expectation of the rational function, in the sense of the Haar measure?
I want to know the analytical solution of $\mathbb{E}_{\psi}\frac{\langle \psi |A|\psi\rangle}{\langle \psi |A^2|\psi\rangle}$. I see similar questions before approximate average, but it does not provide a very specific calculation method.
So I want…
Dan David
- 41
- 3
4
votes
1 answer
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 have something similar to…
doug doug
- 41
- 2
2
votes
1 answer
How to compute k-moment of Haar averaging with n qubits
Let us consider the following Haar averaging over $k$ copies of Pauli strings of $n$ qubits:
$\mathbb{E}_U \left[ U^{\otimes k}\sigma_{q_1} \otimes … \otimes \sigma_{q_k} (U^{\dagger})^{\otimes k}\right]$
where the averaging is done with respect to…
Emma
- 31
- 2
1
vote
0 answers
How to calculate the volume of a point set with parameters go over the Haar distribution?
Speically, how to calculate the volume of the set $\{(|\langle\psi|M_1|\psi\rangle|^2,...,|\langle\psi|M_s|\psi\rangle|^2)|\rho \in \mathbb{H}^n\}$ in the space $\mathbb{R}^{s}$, in which $\mathbb{H}^n$ is the set of all $n$-qubit pure states, and…
Atian
- 11
- 1