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Results tagged with haar-distribution
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user 1351
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.
6
votes
1
answer
255
views
How close or far apart are the distributions generated by two Haar random states?
Consider two $n$ qubit Haar-random quantum states $|\psi\rangle$ and $|\phi\rangle$. Let $D_{|\psi\rangle}$ and $D_{|\phi\rangle}$ be the two probability distributions (over $n$-bit strings) obtained …
- 1,515
5
votes
1
answer
408
views
Multiplication by a Haar random unitary two times
Consider a Haar random unitary $U$.
I am trying to compute the value (or put a bound on)
\begin{equation}
\mathbb{E}\left[\left|\langle 0^{n} |U^{2} |0^{n}\rangle\right|^{2}\right].
\end{equation}
The …
- 1,515
5
votes
2
answers
1k
views
Expected value of a Haar random quantum state multiplied by a unitary
Consider a quantity
\begin{equation}
\mathbb{E}\big[\langle z|\rho|z\rangle\big],
\end{equation}
where $\rho = |\psi \rangle \langle \psi|$ is a Haar-random state $n$-qubit quantum state and $z$ is th …
- 1,515
2
votes
1
answer
86
views
Conditional expectation for Haar random states
Let $U$ be an $n$ qubit Haar random circuit applied to $|0^n \rangle$. Thereafter, the state is measured in the standard basis. Let $p_0$ be the probability of getting $0$ in the first qubit. We know …
- 1,515
2
votes
2
answers
145
views
Computing a ratio involving Haar random unitaries
Consider an $n$-qubit Haar random unitary $U$.
I am trying to compute the expression
\begin{equation}
\mathbb{E}\left[ \frac{\text{Tr}\left(|0^n\rangle \langle 0^n | ~U\rho U^*\right)}{\text{Tr}\left( …
- 1,515
2
votes
0
answers
262
views
Spreading of entanglement with depth for Haar random states
Consider a Haar random quantum state of depth $d$. Consider any bipartition of this state. According to this paper (page $2$):
Haar-random states on $n$ qudits are nearly maximally entangled across
a …
- 1,515
5
votes
0
answers
326
views
Reduced density matrix of a Haar random state and its Schmidt decomposition
Consider a Haar random quantum state $|\psi\rangle$. Note that
$$\rho =\mathbb{E}[|\psi\rangle\langle \psi|] = \frac{\mathbb{I}_{n}}{2^{n}}.$$
$\mathbb{I}_n$ is the identity operator on $n$ qubits. No …
- 1,515
4
votes
1
answer
284
views
Compute the large $n$ distribution of $|\langle z_i|\psi\rangle|^2$ over Haar random quantum...
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 $z_{i} \in \{0, 1\}^{n}$,
$$ |\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\rang …
- 1,515
1
vote
1
answer
73
views
Expected trace distance between two types of random ensembles
Consider a Haar random state on $n$ qubits, and denote it by $|\psi\rangle$. Now consider the following state
$$|\phi\rangle = \frac{1}{\sqrt{k}} \sum_{i=1}^{k} |\phi_{1, i} \rangle \otimes |\phi_{2, …
- 1,515
1
vote
1
answer
109
views
Two qubit Pauli expectation value of $\underset{U}{\mathbb{E}}[U^{\otimes 2} (P_1 \otimes P_...
I want to find a value for the expression:
$$\underset{U}{\mathbb{E}}[U^{\otimes 2} (P_1 \otimes P_2)^{\otimes 2} U^{*\otimes 2}],$$
where $U$ is a two-qubit unitary operator chosen Haar randomly, $P_ …
- 1,515
1
vote
1
answer
110
views
Matrix representation of the symmetric subspace for two copies
Consider two copies of an $n$ qubit Haar random state, given by:
\begin{equation}
\rho = \mathbb{E}_{U \sim \mathsf{Haar}}\left[U |0^n\rangle \langle 0^n| U^{*}\otimes U |0^n\rangle \langle 0^n| U^{*} …
- 1,515
1
vote
1
answer
66
views
Property of Haar random state
Let $|\psi\rangle$ be a Haar random state and let $|\psi^{\perp}\rangle$ be any state that is perpendicular to $|\psi\rangle$. Let us define
$$p_x = |\langle x| \psi \rangle|^2,$$
and $$q_x = |\langl …
- 1,515
4
votes
2
answers
780
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 …
- 1,515
1
vote
0
answers
96
views
Optimality of the SWAP test versus weak Schur sampling for testing unitarily invariant prope...
Consider the following setting.
I am either given the density matrix $|\psi\rangle \langle \psi|^{\otimes k}$ or the density matrix $\frac{\mathbb{I}^{\otimes k}}{2^{nk}}$, where $\mathbb{I}$ is the $ …
- 1,515
5
votes
2
answers
222
views
Quantum hardness of XQUATH conjecture
Consider the XQUATH conjectures, as defined here (https://arxiv.org/abs/1910.12085, Definition 1).
(XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There
is no polynomial-time classica …
- 1,515