A quantum channel is defined as a completely positive, trace-preserving (CPTP) linear map between spaces of operators.
Suppose we have the following operation $$\mathcal{E}(\rho) = \sum_{j=1}^n K_j \rho K_j^{\dagger}, \tag{1}$$ where $\rho$ is a density matrix.
From what I know, since $\mathcal{E}(\cdot)$ has the Kraus operator representation, this operation is completely positive.
However, this operation is not trace-preserving because $$ \sum_{j=1}^n K_j^{\dagger}K_j = \begin{pmatrix} 1 + 2k^2dt && - 2 k \alpha dt\\ - 2 k \alpha dt && 1 + 2k^2dt \end{pmatrix} = I + O(dt).\tag{2} $$ In the above, $dt \rightarrow 0$, $k \in (0,1)$, $\alpha \in \mathbb{R}$. Here, we have $dt$ infinitesimally small but non-zero.
The eigenvalues of this matrix are very close to being $1$ given that $\alpha$ is not too large: \begin{align} \lambda_1 &= 1 + 2(-\alpha k + k^2)dt \\ \lambda_2 &= 1 + 2(\alpha k + k^2)dt \end{align}
For example, for a fixed small $dt$, a numerical representation of these eigenvalues could be $\lambda_1 = 1.0002$ and $\lambda_2 = 0.9998$. Analytically, since $dt$ tends to zero, eigenvalues should be tending to $1$.
My questions are:
- Could $\mathcal{E}(\cdot)$ be viewed as a quantum channel for $dt>0$ and $dt \rightarrow 0$?
- What if I normalize $\mathcal{E}(\rho)$ by $Tr(\mathcal{E}(\rho))$? Would that be a channel, albeit non-linear, in $\rho$.
- Perhaps, more generally, what is the meaning of an operation that is CP but not TP?
