Background:
In quantum information theory, a wide class of processes acting on stochastic quantum states can be described using the formalism of Quantum Channels:
A quantum channel is a linear, completely positive, trace preserving map
$\Phi:\Theta(\mathcal{H}_{in}) \to \Theta(\mathcal{H}_{out})$
where $\mathcal{H}_{in}$ and $\mathcal{H}_{out}$ are respectively the input and output Hilbert spaces, and $\Theta(\mathcal{H})$ is the set of density operators (i.e. stochastic states) over the Hilbert space of pure states $\mathcal{H}$.
In particular, if the input and output spaces are finte dimensional, and they concide, then $\mathcal{H}_{in} \equiv \mathcal{H}_{out} \equiv \mathcal{H} \equiv \mathbb{C}^n$, and $\Theta(\mathcal{H})$ is the set of $\mathbb{C}^{n\times n}$ Hermitian, positive semi-definite matrices of trace one.
Various standard representations exist for any quantum channel. For instance, the Kraus representation, also known as operator-sum representation:
$\Phi(\rho) = \sum_{k}V_{k}\rho V_{k}^{\dagger}$
where $V_{k}$ are the Kraus operators of the channel, subject to the constraint:
$\sum_{k}V_{k}^{\dagger} V_{k}=I$.
Given two quantum channels $\Phi_1,\,\Phi_2$ defined on the same space, various distance measures $d(\Phi_1,\,\Phi_2)$ can be defined. See, for instance, arXiv:quant-ph/0408063, or this question by Joe Fitzsimons.
Question:
I would like to know how to define a low-dimension approximation of a finite-dimension channel. More specifically:
Let $C_{n}$ the set of all channels of finite-dimension $n$:
$C_{n}\equiv \left \{ \Phi | \Phi:\Theta(\mathbb{C}^n) \right \}$
Given two space dimensionalities $n$ and $m < n$, define an Abstraction operator:
$\mathcal{A}:C_{n} \to C_{m}$
and a Concretization operator:
$\mathcal{A}^{inv}:C_{m} \to C_{n}$.
such that their compositions yields an optimal approximation of any channel:
$\forall \Phi \in C_{n},$
Let $\Phi_{\mathcal{B},\,\mathcal{B}^{inv}} \equiv \mathcal{B}^{inv}(\mathcal{B}(\Phi))$
Let $\widetilde{\Phi} \equiv \Phi_{\mathcal{A},\,\mathcal{A}^{inv}}$
then $d(\Phi\,\widetilde{\Phi}) = \min_{\mathcal{B}\,\mathcal{B}^{inv}}(d(\Phi\,\Phi_{\mathcal{B}\,\mathcal{B}^{inv}}))$
where $\mathcal{B},\,\mathcal{B}^{inv}$ are arbitrary operators between channels of suitable dimension:
$\mathcal{B}:C_{n} \to C_{m}\,\ \mathcal{B}^{inv}:C_{m} \to C_{n}$.
I'm not sure what distance measure $d(.,\,.)$ is the most appropriate to use here. I was thinking of the diamond distance.
Also, it would be very nice if the Abstraction operator was defined compositionally with respect to the Kraus representation. That is, given the channel:
$\Phi(\rho) = \sum_{k}V_{k}\rho V_{k}^{\dagger}$
It should hold that:
$\mathcal{A}(\Phi)(\rho) = \sum_{k}\hat{A}(V_{k})\rho \hat{A}(V_{k})^{\dagger}$
for some function $\hat{A}(.)$. (Of course this representation would not be minimal in the number of Kraus operators).
