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i have problem in using deltakronker

You don't just substitute $m=n-1$. Instead, you perform the sum over $m$. The only non-zero term that you get is if $m=n-1$, so that means the second sum vanishes, and you have $$ \sum_{n=1}^{\infty}\...
DaftWullie's user avatar
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1 vote
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Are peripheral eigenvalues of a completely positive map always semisimple?

Consider $$ K:=\begin{pmatrix}1&1\\0&1\end{pmatrix} $$ as well as $\Phi:=K(\cdot)K^\dagger$. This map is completely positive (because $\Phi$ is in Kraus form) and even strictly positive ...
Frederik vom Ende's user avatar
3 votes
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Clarification about the Alberti's Theorem proof given by Watrous in his condensed lecture notes

Start from $$ \sum_{j=1}^n\sum_{i=1}^n\lambda_i\lambda_j^{-1}(u_i^\star PU_i)(u_j^\star Pu_j) $$ and split up the sum over $i$ into 3 terms: $i<j$, $i=j$ and $i>j$. $$ \sum_{j=1}^n\sum_{i<j}\...
DaftWullie's user avatar
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2 votes

For stabilizer codes, why does the error syndrome not depend on the codeword?

If the syndrome would depend on the codeword, this would imply that the syndrome measurement reveals information about the encoded qubit, which would necessarily destroy the encoded information (...
Norbert Schuch's user avatar
1 vote
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For stabilizer codes, why does the error syndrome not depend on the codeword?

You can understand the syndrome by error propagation from the source of the error to the measurements. For example you can consider how an X error at a specific location affects the syndrome ...
M. Stern's user avatar
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1 vote

Equality condition for Hölder's inequality if $p=1$

The statement in Note 2.2 of the paper is that "$p_A\|B\|_\infty = p_A|B| = |B|p_A$ [where $p_A$ is the projection onto the range subspace of $|A|$] [...] is the necessary and sufficent ...
Frederik vom Ende's user avatar
0 votes

Is every pure 1-qubit state an eigenstate of $aX + bY + cZ$?

TL;DR It is true that every pure one-qubit state is an eigenstate of some $aX + bY + cZ$. The restrictions are $a = xe^{i\phi}, b = ye^{i\phi}, c = ze^{i\phi}$ with $\phi\in[0, 2\pi)$ and $x,y,z\geq 0$...
qubitzer's user avatar
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2 votes
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Is every pure 1-qubit state an eigenstate of $aX + bY + cZ$?

This is just a rephrasing of the Bloch sphere representation of a single qubit: any pure state $|\psi\rangle\in\mathbb C^2$, $\|\psi\|=1$ can be written as $|\psi\rangle\langle\psi|=\frac12({\bf1}+aX+...
Frederik vom Ende's user avatar
3 votes
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Do all Hermiticity-preserving maps generate completely positive maps?

First a basic observation: if all Hermitian preserving $\mathcal L$ gave rise to completely positive dynamics $e^{t\mathcal L}$ for all $t\geq 0$, then so would $-\mathcal L$ (still Hermitian ...
Frederik vom Ende's user avatar
2 votes
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How to find the $+1$ eigenvectors of the stabilizers for the Shor code

For CSS stabilizers it's easy. Every X stabilizer will project the computational basis state $|k\rangle$ into $|k\rangle + |k \oplus x\rangle$ where $x$ is the bits flipped by the X gates of the ...
Craig Gidney's user avatar
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