Special properties of a channel whose Kraus decomposition contains Identity

Question

I would like to know if there are any special properties of channels that permit a Kraus representation that includes an identity? That is, if I am given a Kraus representation of a CPTP map $\Phi$ for which one Kraus operator is $\sqrt{p} \,\mathbb{I}$ with $p > 0$ and no other Kraus operator is proportional to the identity, then I can express $\Phi$ as

$$ \Phi(\rho) = p \rho + \sum_{a} A_a \rho A_a^* \tag{1} $$ with $\sum_a A_a^* A_a = (1-p) \,\mathbb{I}$. Then can I make any interesting comments about $\Phi$?

This kind of noise seems interesting because $\Phi (\rho)$ written as a mixture with a term proportional to $\rho$ in its output as in Equation $(1)$ and so we can interpret the effect of the channel as "with probability $p$, nothing happened, otherwise with probability $(1-p)$ something nontrivial happened".

Certainly not all channels have this form, but maybe channels that do have this form share some other properties in common?

Answer 1

We can view $\Phi$ as a convex combination $\Phi = pI + (1-p)\Psi$ of the identity channel $I(\rho)=\rho$ and the channel

$$ \Psi(\rho)=\sum_a B_a\rho B_a^*\quad\text{where}\quad B_a=\frac{A_a}{\sqrt{1-p}}. $$

This indicates that some properties of $\Phi$ will depend on corresponding properties of $\Psi$. For example, it is not hard to show that $\Phi$ is unital$^1$ if and only if $\Psi$ is unital. More generally, $\Phi$ has the same fixed-points$^2$ as $\Psi$.

Other properties need to be checked for $\Phi$ independently. For example, $\Phi$ may preserve entanglement even if $\Psi$ is an entanglement-breaking channel. Also, it is easy to see that there is a floor for the average fidelity$^3$ of $\Phi$ regardless of $\Psi$, namely$^4$ $\overline{F}(\Phi)\ge p$.

Note that any channel $\Xi$ that cannot be written in the form $(1)$ can be approximated arbitrarily well by

$$ \Xi' = \epsilon I + (1-\epsilon)\Xi $$

which is of the form $(1)$, where $\epsilon$ is a sufficiently small positive number. In other words, the requirement $(1)$ defines a set which is not closed. Therefore, no property corresponding to a closed set of channels holds exclusively for channels of the form $(1)$.

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^{$^1$ Channel is unital if it maps identity to identity. For example, depolarizing channel is unital and amplitude damping channel generally is not.
$^2$ All quantum channels have at least one fixed point by the Schauder fixed-point theorem.
$^3$ Defined as $\overline{F}(\Phi)=\int \langle\psi|\Phi\left(|\psi\rangle\langle\psi|\right)|\psi\rangle d\psi$.
$^4$ This naive bound can actually be improved using $\overline{F}(\Phi)=\frac{d F_e(\Phi)+1}{d+1}$ where $F_e$ is the entanglement fidelity and $d$ the dimension of the underlying Hilbert space.}