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### 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 ...
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### Are continuous probability distributions over quantum channels possible?

I am not an expert in the subject and apologize in advance for a strange question and (possible) abuse of the terminology. I have learned that any convex combination of quantum channels (CPTP maps, ...
483 views

### How to find the operator-sum representation of the depolarizing channel?

Consider a circuit built as follows: take two ancilla states and an operator $U$ made of a series of controlled gates which act on a pure state $\rho$ as follows: $X$ if the ancilla is in $|00\rangle$...
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1 vote
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### Does applying a random Pauli matrix to a density matrix result in the identity?

Nielsen and Chuang's textbook, Equation 8.101 (section 8.3.4 'Depolarizing Channel') shows that applying a random Pauli to a density matrix representing one qubit equals the identity (times one half): ...
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### Schur's lemma for quantum states

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 ...
1 vote
### Prove that the twirling operation on a channel gives a decomposition $\int dU\, U^\dagger\mathcal E(U\rho U^\dagger)U=\alpha P+\beta Q$
The twirled operation of a quantum channel $\mathcal E$ is defined as \begin{align} \mathcal E_T(\rho) &= \int dU U^\dagger \mathcal E(U \rho U^\dagger)U, \end{align} where the integral is over ...