# 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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### Moments of Pauli coefficients of Haar-random states

I want to evaluate the quantity $\sum_{P\in \rm{P}^n}\text{Tr}^{\alpha}(\rho P)$, where $P$ is an element of n-qubit Pauli group $\rm{P}^n$ and $\rho$ is a density matrix of a Haar random state. It is ...
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### What is the expectation value of the overlap of two uniformly random pure states? [duplicate]

Let $\psi$ and $\phi$ be two uniformly random pure state $\psi, \phi \sim\mathbb{C}^d$. The the following equality holds \begin{align} \mathbb{E}_{\psi, \phi \sim \mathbb{C}^d} {\rm Tr}[\vert \phi \...
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### A question on a subset of projectors onto symmetric subspace

Use $\text{perm}_t$ to denote the set of all permutations among $t$ items. For any particular subset $S\subseteq\{0,1\}^n$ and any $\sigma\in \text{perm}_t$, we define \begin{align} P_S(\sigma) = \...
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### How to calculate the Haar measure for the middle SU(2), in an SU(3) factorization?

I read this blog https://pennylane.ai/qml/demos/tutorial_haar_measure#deguise2018 regarding a basic introduction to haar measure. In the "show me more math" section, they said $SU(3)$ can be ...
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### Why does Schur's lemma imply that $\int \sigma^{\otimes n}_{HK} d(\sigma)$ must be a multiple of the identity on the symmetric subspace?

I am trying to understand Lemma 2 in this paper. Consider a state $\tau_{H^n}=\int \sigma^{\otimes n}_{H} \mu(\sigma)$ where $\mu(\sigma)$ is the measure on the space of density operators on a single ...
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### Does integrating w.r.t. the Haar measure commute with taking partial trace?

Consider a density matrix $\rho(U)$ which depends on $U \in SU(2^n)$, corresponding to a state of a composite, finite-dimensional Hilbert space $\mathcal{H} \cong \bigotimes_{i=1}^{2^n} \mathbb{C}^2$ ...
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### Measure on the unitary space and complexity

I'm currently studying various quantum supremacy protocols and i'm struggling to have a clear and well defined view on the rôle of approximating the Haar-measure (through k-designs ...) and the ...
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### What is the expectation value of $|\langle \psi|U|\psi \rangle|$ over Haar random states $|\psi\rangle$?

We know the average unitary fidelity, $\int |\langle \psi|U|\psi \rangle|^2 d\psi$, has a nice closed-form solution: $\frac{1+\frac{1}{d}|Tr (U)|^2}{1+d}$, thanks to Horodecki and Nielsen. However, I ...
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### Werner Twirling Channel - How to Retrieve Prefactors?

In Watrous' Theory of Quantum Information, Example 7.25 discusses the Werner Twirling Channel: $$\Xi(X) = \int (U \otimes U) X (U \otimes U)^* \mathrm{d}\eta(U)$$ where $\eta$ denotes the Haar measure ...
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### Optimal estimation of quantum state overlap - Circuit implementation?

I've been reading this paper, but don't understand what their optimal method really is, and how it can be realized as a quantum circuit. The paper mentions the "Schur transform" which has a ...
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### Integral over Haar measure of squared density matrix of Haar random state is proportional to the identity plus swap operator

I am having some trouble understanding why $\int d\psi (| \psi \rangle \langle \psi | )^{\otimes ^2}\propto \ I+$ SWAP , where $|\psi \rangle =U|\psi _0\rangle$ are Haar random states and $d\psi$ is ...
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### Generating random, but non-uniform state

I would like an algorithm that generates a random state, sampled according to some probability distribution which is not uniform in Hilbert space. Assume though that I have at my disposal a uniform (...
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### Independence in state prepared by independently drawn Haar random gates

Consider independently drawn $2 \times 2$ Haar random unitaries $U_1, U_2, \ldots, U_n$ and $$V = U_1 \otimes U_2 \otimes \cdots U_n.$$ Consider the state $\sigma$ given by $$\sigma = V \rho V^{*},$$ ...
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### Computing a ratio involving Haar random unitaries

Consider an $n$-qubit Haar random unitary $U$. I am trying to compute the expression \mathbb{E}\left[ \frac{\text{Tr}\left(|0^n\rangle \langle 0^n | ~U\rho U^*\right)}{\text{Tr}\left(\...
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