Is a quantum channel well behaved under a perturbation of its Choi matrix? Every completely positive trace preserving quantum channel can be associated with a unique quantum state. Supposing one perturbs the quantum state into a new state. Is there some sense in which one should expect the corresponding "perturbed" channel to be close to the original channel? If our perturbed state is only considered to nth order, is there some sense in which the corresponding "perturbed" channel is well formed up to nth order?
I have been playing with this and am finding that it appears to not be the case, namely, I have been perturbing a quantum state to a new state, and observing the corresponding new channel, and although both states are positive semidefinite to first order in my perturbation, the corresponding channels are not trace preserving to first order in the same parameter. Nevertheless my intuition tells me that the opposite should be true. Can anyone enlighten me?
 A: There are essentially two questions here:


*

*(mathematical) stability under perturbation (i.e. changing $\tau$ to $\tau+\varepsilon \eta$ for some small $\varepsilon$).

*channel stability under perturbation. 


Let's fix notation: We define the Choi matrix $\tau$ of a completely positive map $T$ from $n\times n$ matrices to $n\times n$ matrices via 
$$\tau:=(T\otimes 1_n)(|\Omega\rangle\langle \Omega|)$$
where $|\Omega\rangle$ is the maximally entangled state.
Now lets deal with the second question: you have observed that a perturbation will result in the channel not being trace-preserving. This is in general correct, because being trace-preserving is essentially a very fragile property (like any kind of "normalization" in physics): For the Choi matrix, it means that the partial trace over the second system is the identity, i.e. $\operatorname{tr}_2(\tau)=1_n$. Since an arbitrary perturbation will most likely also affect $\operatorname{tr}_2(\tau)$, the resulting Choi matrix will not belong to a channel anymore. This means you'll either have to renormalize or restrict the class of admissible perturbations.
Now for the first question: Is the map stable under perturbations. This is in general yes, because the Choi-Jamiolkowski isomorphism has an inverse given by
$$\operatorname{tr}[AT(B)]=n\operatorname{tr}[\tau A\otimes B^T]$$
Now, if you plug in $\tau+\varepsilon \eta$ with some Hermitian $\eta$ and small $\varepsilon$, the new state will still be a state (i.e. $T$ will be completely positive) and $T$ will be close to $\tau$ in any norm, essentially because the trace is continuous. Mathematically, this is what I'd call "stability".
