Linked Questions
12 questions linked to/from Why does the twirl of a quantum channel give a depolarizing channel?
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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 ...
glS♦
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2
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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, ...
- 141
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Quantum channels that commute with any unitary channel
Consider a quantum channel $\Phi$ that maps from density operators $\mathcal{S}(\mathcal{H}_A)$ to itself, that commutes with any unitary channel $\mathcal{U}$ on $\mathcal{S}(\mathcal{H}_A)$, i.e. $\...
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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 $|00\rangle$...
- 143
3
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1
answer
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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 ...
- 7,302
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1
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Averaging over a single Haar-random unitary applied $t$ times
I'm trying to compute the following integral:
$$\int U^{\otimes t}\left|x_1,\cdots,x_t\middle\rangle\middle\langle x'_1,\cdots,x_n'\right|\left(U^\dagger\right)^{\otimes t}\,\mathrm{d}\mu(U)$$
Where $\...
- 7,302
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1
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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?
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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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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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Is the "unitary twirling operation" physically realizable?
In this neat answer by Markus Heinrich, it is shown that twirling an arbitrary quantum channel $\Lambda$ over the unitary group $U(d)$ yields a depolarizing channel $\tilde{\Lambda}$ given by
$$
\...
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1
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What is the definition of twirled superoperator?
I am trying find the definition of twirled (super)operator. One such is Definition 2.3.16 on p. 33 of Christoph Dankert, Efficient Simulation of Random Quantum States and Operators. However, the ...
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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 ...
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