# Do the eigenvalues of the Choi matrix have any direct physical interpretation?

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?

• 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. Commented Feb 26 at 20:09
• @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
– glS
Commented Feb 29 at 8:52
• 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.
– glS
Commented Feb 29 at 9:00