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Questions tagged [diamond-norm]

In quantum information, the diamond norm is a distance metric between quantum operations. (Wikipedia)

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

Question: Given states $\rho,\omega\in\mathbb C^{n\times n}$ and $\varepsilon>0$ such that $\rho$ and $\omega$ are $\varepsilon$-close in trace norm does there exist a channel $\Phi$ with $\Phi(\...
Frederik vom Ende's user avatar
5 votes
1 answer
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Does monotonicity of diamond distance hold for intermediate channels?

It is well known that $\|\mathcal{E} \circ \mathcal{F} - \mathcal{E}\|_\lozenge \leq \|\mathcal{F} - \mathcal{I}\|_\lozenge$. What if I have $\|\mathcal{A} \circ \mathcal{E} \circ \mathcal{F} - \...
usermm's user avatar
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2 votes
1 answer
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Diamond norm distances between some channel and the identity

I'm currently working with the continuity result by Kretschmann-Schlingemann-Werner (arXiv version) for Stinespring isometries (more precisely, the following corollary to their result, cf. Appendix C ...
Frederik vom Ende's user avatar
6 votes
0 answers
163 views

Is it possible to obtain a closed-form expression of the diamond distance between two single-qubit channels?

I would like to compute the diamond norm of the difference of two single-qubit channels $\Phi_1$ and $\Phi_2$. This difference is equal to, for any $2\times2$ complex matrix $\rho$: $$\...
Tristan Nemoz's user avatar
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10 votes
1 answer
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How does the number of copies affect the diamond distance?

Suppose we are given two maps $\Phi$ and $\Psi$ such that $$\|\Phi-\Psi\|_{\diamond}\leqslant\varepsilon.$$ What can we say about $\left\|\Phi^{\otimes t}-\Psi^{\otimes t}\right\|_{\diamond}$? Is it ...
Tristan Nemoz's user avatar
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4 votes
1 answer
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Closeness of unitary dilations of CPTP maps

Let $\Phi_1,\Phi_2 \colon S(\mathcal{H}) \to S(\mathcal{H})$ be CPTP maps on the same Hilbert space $\mathcal{H}$ which are $\varepsilon$-close in diamond norm, and let $U_1,U_2$ be respective unitary ...
nickspoon's user avatar
3 votes
2 answers
353 views

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

Let us consider an encode, noisy channel and a decoder such that classical messages $m\in\mathcal{M}$ can be transmitted with some small error. That is, for a message $m$ that is sent by Alice, Bob ...
user1936752's user avatar
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4 votes
1 answer
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Diamond norm distance bound on Stinespring dilations of channels

The diamond distance between two channels $\Phi_0$ and $\Phi_1$ is defined in this answer. $$ \| \Phi_0 - \Phi_1 \|_{\diamond} = \sup_{\rho} \: \| (\Phi_0 \otimes \operatorname{Id}_k)(\rho) - (\Phi_1 ...
user1936752's user avatar
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3 votes
1 answer
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Choi matrix in QETLAB

I am using QETLAB, a package for working with quantum information theory in Matlab and I have some doubts. I am trying to calculate diamond norms using such for some quantum channels. However, when ...
Josu Etxezarreta Martinez's user avatar
4 votes
2 answers
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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?

I could not find any lower bound on the diamond norm for two uniformly random unitaries of dimension D sampled from the haar measure.
Niraj Kumar's user avatar
8 votes
0 answers
553 views

What are the "nice" properties of the diamond norm and why is it used?

I have heard about the diamond norm, and from what I understood it is a "nice" tool to quantify quality of quantum gates in the NISQ era. I would like to know a little more before going in detail in ...
Marco Fellous-Asiani's user avatar
7 votes
1 answer
602 views

Is the diamond norm subadditive under composition?

The diamond norm distance between two operations is the maximum trace distance between their outputs for any input (including inputs entangled with qubits not being operated on). Is it the case that ...
Craig Gidney's user avatar
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