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2 votes
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How to get the Kraus decomposition of the amplitude damping channel from its Choi?

Decomposing $C_{N_\gamma}$ amounts to finding all non-zero eigenvalues as well as corresponding eigenvectors: Assume for now that we already found all eigenvalues $\lambda_j>0$ as well as ...
Frederik vom Ende's user avatar
6 votes

How can quantum error correction correct small rotations/continuous errors?

The trick is to rewrite your continuous rotations as a perturbative sum. For example, consider applying $R_Z(\theta)$ to all data qubits of an $n$-qubit code. You can rewrite: $$R_Z(\theta) = I \cos \...
Craig Gidney's user avatar
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3 votes
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How can quantum error correction correct small rotations/continuous errors?

When you encode in an error correcting code, you select a subspace that your encoded qubit sits in. You can identify this with some projector $P$. For instance, if you know your logical 0 and logical ...
DaftWullie's user avatar
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3 votes

Does measurement in different bases allow for FTL communication?

If Alice measures in the $|+ \rangle$, $| - \rangle$ basis, then Bob's qubit will indeed be in either the $|+\rangle$ or $|-\rangle$ state depending on the result of Alice's measurement. But your ...
Nick Mertes's user avatar
1 vote

Does the quantum relative entropy have a direct operational interpretation?

I would argue yes, in the context of recoverability. Given a quantum channel $\mathcal{N}: L(A) \to L(B)$ and a state $\sigma$ on $A$, we say that a quantum channel $\mathcal R: L(B) \to L(A)$ is a $(...
Rammus's user avatar
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1 vote

Which quantum entropies are meaningful with respect to continuous distributions of states?

I've found a partial answer for the case of conditional min-entropy, due to Ref. [1] (Appendix IV.B): Consider a fixed ensemble $\{(\rho_B(x), p(x))\}_{x \in \Sigma}$, where $p(x)$ is a probability ...
forky40's user avatar
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3 votes
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Are all extremal points of the feasible set of an arbitrary affine equation pure states?

A simple counterexample but perhaps I'm misinterpreting "one affine equation". Take the map $\Lambda$ to be the identity map and let $Y$ be any mixed state. Then the set of states satisfying ...
Rammus's user avatar
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1 vote

Are all extremal points of the feasible set of an arbitrary affine equation pure states?

This is generally not the case. One example is the set of CPTP maps. Under the Choi-Jamiołkowski isomorphism, this set corresponds to a set of quantum states intersected with an affine hyperplane ...
Markus Heinrich's user avatar
3 votes

Deriving the choi matrix definition of the quantum depolarizing channel

For $\mathcal E(X)=p\mathrm{tr}(X)\,\frac{\mathbb I}{2}+(1-p)X$, then $$\begin{align} \sigma &= (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle \Omega|)\\ &= \sum_{ij} \mathcal E(|i\...
HerrWarum's user avatar
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