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# 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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### Random quantum states and Schur-Weyl duality

Consider the following density matrix over $n$ qubits, with $C$ being a single qubit operator: $$\rho_{n} = \int_{C \sim \text{Haar}} \big(C|0\rangle\langle0|C^\dagger\big)^{\otimes n} dC.$$ Let's ...
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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 $\lvert \alpha \... • 11.6k 4 votes 1 answer 161 views ### Prove that uniformly random states have moments${\bf E}_\psi|\langle x|\psi\rangle|^{2t}\sim1/\binom d t$Im looking for the moments of Haar random states. Is it true that$\textbf{E}_{\psi\sim \text{Haar}}|\langle x| \psi\rangle|^{2t}\sim \frac{1}{\binom{d}{t}}?$How does one prove this? 3 votes 1 answer 317 views ### Show that, averaging over uniformly random unitaries,$\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 ... • 21.7k 2 votes 1 answer 186 views ### Density matrices of multiples copies of a single Haar-Random state In Pseudorandom States, Non-Cloning Theorems and Quantum Money, the authors state that: Let$\rho_\mu^m$be the density matrix of a random$|\psi\rangle^{\otimes m}$for$|\psi\rangle$chosen from ... • 4,636 9 votes 2 answers 3k views ### What is a Haar random quantum state? Can somebody please explain me what is a Haar random state? I am not able to find any friendly resource to read about it. 5 votes 2 answers 737 views ### Expected value of a Haar random quantum state multiplied by a unitary Consider a quantity $$\mathbb{E}\big[\langle z|\rho|z\rangle\big],$$ where$\rho = |\psi \rangle \langle \psi|$is a Haar-random state$n$-qubit quantum state and$z$is ... • 1,119 4 votes 1 answer 244 views ### Compute the large$n$distribution of$|\langle z_i|\psi\rangle|^2$over Haar random quantum states Let$|\psi\rangle$be a$n$qubit Haar-random quantum state. I am trying to show that in the limit of large$n$, for each$z_{i} \in \{0, 1\}^{n}$,$$|\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\... • 1,119 3 votes 2 answers 619 views ### Computing expectation value of$|\langle z|C|0^n\rangle|^2$over Haar random circuit I am trying to understand the integration on page 4 of this paper. Consider a Haar random circuit$C$and a fixed basis$z$. Each output probability of a Haar random circuit (given by$|\langle z | C |...
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Consider a Haar random quantum state $|\psi \rangle$. I was confused between two facts about $|\psi \rangle$, which appear related: Consider the output distribution of a particular $n$-qubit \$|\psi \...