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10 votes
Accepted

How does the number of copies affect the diamond distance?

TL;DR: If $\Phi$ and $\Psi$ are quantum channels (unitary or otherwise), then things are "too good to be true". Proposition. If $\|\Phi-\Psi\|_\diamond\le\varepsilon$ and $m=\max(\|\Phi\|_\...
Adam Zalcman's user avatar
  • 22.9k
10 votes
Accepted

Is there a lower bound on the average diamond norm of two uniformly random unitaries U1 and U1 of dimension D that are sampled from haar measure?

This answer won't actually give you a bound, but will provide some information that may help you in your search. You may be able to find an answer in the random matrix theory literature if you ...
John Watrous's user avatar
  • 6,097
6 votes
Accepted

Is the diamond norm subadditive under composition?

For arbitrary linear super-operators $U_j$ and $V_j\def\D{\mathrm{Diamond}} \def\Dn#1{\lVert #1 \rVert_\diamond}\def\le{\leqslant} $, we have $$\def\D{\mathrm{Diamond}} \def\Dn#1{\lVert #1 \rVert_\...
Niel de Beaudrap's user avatar
5 votes
Accepted

Diamond norm distance bound on Stinespring dilations of channels

Yes, in fact there exists Stinespring dilations such that $$\frac{\|N_1-N_2\|_{cb}}{\sqrt{\|N_1\|_{cb}}+\sqrt{\|N_2\|_{cb}}}\leq \|V_1-V_2\|\leq \sqrt{\|N_1-N_2\|_{cb}}$$ where the distance between ...
Condo's user avatar
  • 2,048
5 votes
Accepted

Choi matrix in QETLAB

QETLAB usually deals with channels as Choi operators. You can convert your Kraus operators to the Choi matrix by providing the Kraus operators as a cell array. Example with the amplitude damping ...
user13507's user avatar
  • 126
5 votes

Is there a lower bound on the average diamond norm of two uniformly random unitaries U1 and U1 of dimension D that are sampled from haar measure?

The result you're looking for is effectively Proposition 19 of the paper: Almost all quantum channels are equidistant; which I'm rewriting here for convenience: Let $U, V \in \mathcal{U}(d)$ be two ...
keisuke.akira's user avatar
4 votes

Closeness of unitary dilations of CPTP maps

Suppose that $\Phi_i$ $i=1,2$ are CPTP maps with $\|\Phi_1-\Phi_2\|_\diamond\leq \epsilon$. Let $V_i=U_i^\dagger(A\otimes I_K)U_i$ and $\sigma=\rho\otimes |0\rangle\langle0|$, then we have that $V_i$ ...
Condo's user avatar
  • 2,048
4 votes
Accepted

Bounding diamond norm distance using probability of error in transmission of classical information

Intuition The expression $\|\mathcal{A} - \mathcal{I}\|_\diamond$ quantifies how close the channel $\mathcal{A}$ is to the identity channel $\mathcal{I}$ which is the channel that preserves quantum ...
Adam Zalcman's user avatar
  • 22.9k
3 votes

Does monotonicity of diamond distance hold for intermediate channels?

TL;DR: No. The suggested bound fails to hold for any norm. Briefly, if we choose $\mathcal{A}=\mathcal{F}$ to be an idempotent channel then the right-hand side vanishes. However, if we choose $\...
Adam Zalcman's user avatar
  • 22.9k
2 votes
Accepted

If states are close together does there always exist a channel close to the identity mapping one to the other?

We shall see that for general mixed states no such upper bound can exist in the following precise sense: For no continuous function $c:[0,2]\to[0,2]$ with $c(0)=0$ does it hold that for all $\|\rho-\...
Frederik vom Ende's user avatar
1 vote
Accepted

Diamond norm distances between some channel and the identity

You can find some useful results in these links: https://quantum-journal.org/papers/q-2021-08-09-522/, https://arxiv.org/abs/1004.4110v1, https://doi.org/10.1088/1367-2630/ab8e5c, https://arxiv.org/...
Yaron Jarach's user avatar

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