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Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map, and let $J(\Phi)$ be its Choi representation. As is well known, any such map can be written in a Kraus representation of the form $$\Phi(X)=\sum_a p_a A_a X A_a^\dagger,\tag A$$ where $p_a\ge0$ and $\operatorname{Tr}(A_a^\dagger A_b)=\delta_{ab}$ (as mentioned for example in this other question of mine). The positive numbers $p_a$ appearing here can be seen to be the eigenvalues of $J(\Phi)$.

Is there a way to directly relate the eigenvalues $p_a$ to some properties of the channel, or to how the channel acts on states?

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  • $\begingroup$ More or less the probabilities of applying the Kraus operator $A_a$? (Maybe not quite exactly, but to some extent?) At least, changing them has this meaning. $\endgroup$ Commented Feb 26 at 20:09
  • $\begingroup$ @NorbertSchuch I guess it depends on what exactly you mean by that? I can definitely interpret $p_a$ as probability of the "$a$-th noise" occurring, meaning probability of the $a$-th outcome in the ancillary dof in the dilation isometry view. But then again, that's not unique: different dilations, corresponding to different Kraus ops, would give different probabilities; corresponding to the different ways to decompose the Choi as convex combination of rank-1 ops. I guess you could single-out the eigvals as decomposition with smaller entropy, but it doesn't seem a very "direct" interpretation $\endgroup$
    – glS
    Commented Feb 29 at 8:52
  • $\begingroup$ maybe one could make an argument about the eigenvalues somehow quantifying information loss. In the dilation view, any isometric implementation needs to "leak out" an amount of information at least equal to the information you'd get measuring the env dof. Not sure what'd be the best way to quantify this though. From this perspective, maybe the eigenvalues can be tied to some property of complementary channels? Problem being of course that this wouldn't take into account information (about input state) hidden in the correlations between the 2 output registers. $\endgroup$
    – glS
    Commented Feb 29 at 9:00

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When I was doing some work on quantum cloning (so, slightly different applications to the channels you were asking about), I basically ended up setting up the Choi matrix as a description of the action that I'd like to achieve (perfect cloning), averaged over all possible inputs. In that case, the maximum eigenvalue tells you quite a lot - up to a scale factor of the size of the Hilbert space dimension of the input state, it gives you an upper bound on the fidelity that you can achieve that operation with. Moreover, if the maximal eigenvector is maximally entangled, that upper bound in the fidelity can be achieved (you can extend that condition to deal with degeneracy in the maximum eigenvalue).

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    $\begingroup$ what do you mean with "setting up the Choi matrix as a description of the action that I'd like to achieve, averaged over all possible inputs"? Isn't the Choi matrix just another way to express the channel? So isn't this the same as saying "set up a channel as description of the action I want to achieve", which would be sort of a tautology given that a channel is generally a description of some operation? $\endgroup$
    – glS
    Commented Aug 12, 2019 at 13:36
  • $\begingroup$ @glS but the issue is that the channel I want is an unphysical channel. The Choi matrix gives me a way to access the closest physical channel (corresponding to the projector onto the maximum eigenvector of the matrix, if the required conditions are satisfied) $\endgroup$
    – DaftWullie
    Commented Aug 12, 2019 at 13:52

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