# Tag Info

Accepted

### Is a process matrix of rank $1$ unique?

TL;DR: The elements of the process matrix with respect to an operator basis $E_i$ are just the coefficients in the expansion of the channel, viewed as an operator on $\mathcal{H}\otimes\mathcal{H}$, ...

### How to characterize the extreme points of the set of CPTP maps?

Cardinality The set $C(\mathcal{X},\mathcal{Y})$ of all channels$^{1,2}$ $\Phi:L(\mathcal{X})\to L(\mathcal{Y})$ has uncountably infinitely many extreme points. To see this, first note that every pure ...
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### Question about Nielson & Chuang Problem 9.2

I have a sketch of ideas, but haven't buttoned up all the details into a proof so use it as a set of hints. Intuition: Kraus operators have a unitary freedom, thus you can get the next set of Kraus ...

### What is known about the size of the spectral gap of unital quantum channels?

TL;DR: Spectral gap depends on the specific channel. Moreover, for any $g\in[0,1]$ there is a channel with spectral gap $g$. Non-peripheral eigenoperators are traceless We can make simple observations ...
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### Derivation of Choi-Jamiolkowski isomorphism

It's easier to work with matrix units $E_{ij} = |i\rangle \langle j|$. In particular, we have $$|\Omega\rangle \langle \Omega| = \sum_{i,j}c_i\overline{c_j} E_{ij} \otimes E_{ij}\tag{1}\,.$$ The ...
1 vote
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### How to get the Kraus operator $M_0=\sqrt{1-p}\, I$ for the depolarizing channel, from its isometric representation?

The unitary representation specifies how the isometry $U$ acts on pure states. The channel $\Phi$ is related to $U$ by $\Phi(\rho)=\operatorname{tr}_E[U\rho U^\dagger]$. Notice that on pure states ...
1 vote
Accepted

### Is there a notion of approximate entanglement breaking (EB) channels?

This work (https://journals.aps.org/pra/abstract/10.1103/PhysRevA.97.012332) may be relevant to your question. The authors have shown that "approximate additivity" holds for "...
### How can I get the process $\mathcal{E}$ from Choi matrix and Choi-Jamiolkowski isomorphism?
Given a map $\Phi$, define its Choi representation as $J(\Phi)=\sum_{ij}\Phi(E_{ij})\otimes E_{ij}$ with $E_{ij}\equiv |i\rangle\!\langle j|$. Then you can express the map from the Choi via \Phi(X) =...
As usual, the singular values of an operator $\phi$ are the square roots of the eigenvalues of the positive semi-definite operator $\phi^\dagger \phi$ (or $\phi^*\phi$ if you prefer the $*$ for the ...