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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).

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    $\begingroup$ is it obvious that the optimal abstraction and concretization operator do not depend on the specific channel? If it is not obvious then I don't understand the definition of $A$ and $A^{inv}$ since you will define a different $A$ and $A^{inv}$ for each channel... or is that the goal? Like some sort of analogue of compression? $\endgroup$ – Artem Kaznatcheev May 10 '11 at 17:37
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    $\begingroup$ what about converting from the Kraus operator representation to the super-operator matrix representation (via vectorization), then performing an SVD on the matrix, setting all singular values other than the desired top ones to zero (standard low-rank approximation) and then converting back? I guess one still needs to verify that complete positivity is preserved, though. $\endgroup$ – Martin Schwarz May 10 '11 at 18:32
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    $\begingroup$ Let me be a little more careful. the space of real positive definite matrices forms a Riemannian manifold P(n), and its geometry is relatively well behaved. In such a manifold, you can think of restricting to a lower dimensional space, and define metrics that are based on geodesic travel on the manifold. A similar thing should be doable for the complex PD matrices, where you end up instead with a Hermitian matrix. Whether the resulting distance is meaningful is a different story. $\endgroup$ – Suresh Venkat May 10 '11 at 19:35
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    $\begingroup$ @Antonio, the SVD-based construction is the closest (in Frobenius distance) matrix in the lower dimensional sense. $\endgroup$ – Suresh Venkat May 10 '11 at 19:36
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    $\begingroup$ @Antonio: No, I don't mean Choi-Jamiolkowski (but it's related). I mean the dynamical map L as defined e.g. here: arxiv.org/abs/1009.2210v4 (section 3). The projection onto the large singular-values is the optimal low-rank approximation in Frobenius norm for any matrix (Eckart-Young theorem). $\endgroup$ – Martin Schwarz May 10 '11 at 19:36

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