<p><strong>Background:</strong></p>

<p>In quantum information theory, a wide class of processes acting on stochastic quantum states can be described using the formalism of <em>Quantum Channels</em>:<br>
A quantum channel is a linear, completely positive, trace preserving map<br>
$\Phi:\Theta(\mathcal{H}_{in}) \to \Theta(\mathcal{H}_{out})$ <br>
where $\mathcal{H}_{in}$ and $\mathcal{H}_{out}$ are respectively the input and output Hilbert spaces, and $\Theta(\mathcal{H})$ is the set of <em>density operators</em> (i.e. stochastic states) over the Hilbert space of pure states $\mathcal{H}$.</p>

<p>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.</p>

<p>Various standard representations exist for any quantum channel. For instance, the <em>Kraus representation</em>, also known as <em>operator-sum representation</em>:<br>
$\Phi(\rho) = \sum_{k}V_{k}\rho V_{k}^{\dagger}$ <br>
where $V_{k}$ are the Kraus operators of the channel, subject to the constraint:<br>
$\sum_{k}V_{k}^{\dagger} V_{k}=I$.</p>

<p>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, <a href="http://arxiv.org/abs/quant-ph/0408063" rel="nofollow noreferrer">arXiv:quant-ph/0408063</a>, or <a href="https://cstheory.stackexchange.com/questions/4885/is-there-any-connection-between-the-diamond-norm-and-the-distance-of-the-associat">this question</a> by Joe Fitzsimons.</p>

<p><strong>Question:</strong></p>

<p>I would like to know how to define a low-dimension approximation of a finite-dimension channel. More specifically: <br></p>

<p>Let $C_{n}$ the set of all channels of finite-dimension $n$:<br>
$C_{n}\equiv \left \{ \Phi | \Phi:\Theta(\mathbb{C}^n) \right \}$</p>

<p>Given two space dimensionalities $n$ and $m &lt; n$, define an <em>Abstraction operator</em>:<br>
$\mathcal{A}:C_{n} \to C_{m}$</p>

<p>and a <em>Concretization operator</em>:<br>
$\mathcal{A}^{inv}:C_{m} \to C_{n}$.</p>

<p>such that their compositions yields an optimal approximation of any channel:<br>
$\forall \Phi \in C_{n},$<br>
Let $\Phi_{\mathcal{B},\,\mathcal{B}^{inv}} \equiv \mathcal{B}^{inv}(\mathcal{B}(\Phi))$ <br>
Let $\widetilde{\Phi} \equiv \Phi_{\mathcal{A},\,\mathcal{A}^{inv}}$<br>
then $d(\Phi\,\widetilde{\Phi}) = \min_{\mathcal{B}\,\mathcal{B}^{inv}}(d(\Phi\,\Phi_{\mathcal{B}\,\mathcal{B}^{inv}}))$<br>
where $\mathcal{B},\,\mathcal{B}^{inv}$ are arbitrary operators between channels of suitable dimension:<br>
$\mathcal{B}:C_{n} \to C_{m}\,\ \mathcal{B}^{inv}:C_{m} \to C_{n}$.</p>

<p>I'm not sure what distance measure $d(.,\,.)$ is the most appropriate to use here. I was thinking of the diamond distance.</p>

<p>Also, it would be very nice if the Abstraction operator was defined compositionally with respect to the Kraus representation. That is, given the channel:<br>
$\Phi(\rho) = \sum_{k}V_{k}\rho V_{k}^{\dagger}$</p>

<p>It should hold that:<br>
$\mathcal{A}(\Phi)(\rho) = \sum_{k}\hat{A}(V_{k})\rho \hat{A}(V_{k})^{\dagger}$ <br>
for some function $\hat{A}(.)$. (Of course this representation would not be minimal in the number of Kraus operators).</p>

CC BY-SA 3.0
2011-05-10T17:37:53.010Z
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?
Artem Kaznatcheev
CC BY-SA 3.0
2011-05-10T18:32:47.070Z
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.
Martin Schwarz
CC BY-SA 3.0
2011-05-10T19:19:25.760Z
Geometrically, another approach is to think of the operation as a (Riemannian) projection onto the lower dimensional submanifold (although this might technically be Kahlerian)
Suresh Venkat
CC BY-SA 3.0
2011-05-10T19:21:53.110Z
@Artem: yes, the abstraction and concretization operators does not depend on the specific channel, otherwise you could always find an exact approximation for any dimension. Sorry if that wasn't very clear.
Anyway, what I'm asking is indeed in some sense analogue to a lossy compression, albeit of the channel description, not of the data passing through the channel.
Antonio Valerio Miceli-Barone
CC BY-SA 3.0
2011-05-10T19:26:29.127Z
@Martin : by super-operator matrix representation do you mean the Choi-Jamiolkowski representation? Using truncated SVD may be a good idea (although I have to think about complete positivity and trace preservation), but what kind of distance does it minimize?
Antonio Valerio Miceli-Barone
CC BY-SA 3.0
2011-05-10T19:28:24.430Z
@Suresh: I'm not familiar with the terms Riemannian projection or Kahlerian (I'm neither a matematician nor a physicist). Can you point me to some entry-level reference, if there are any, please?
Antonio Valerio Miceli-Barone
CC BY-SA 3.0
2011-05-10T19:35:28.647Z
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.
Suresh Venkat
CC BY-SA 3.0
2011-05-10T19:36:32.623Z
@Antonio, the SVD-based construction is the closest (in Frobenius distance) matrix in the lower dimensional sense.
Suresh Venkat
CC BY-SA 3.0
2011-05-10T19:36:35.250Z
@Antonio: No, I don't mean Choi-Jamiolkowski (but it's related). I mean the dynamical map L as defined e.g. here: http://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).
Martin Schwarz
CC BY-SA 3.0
2011-05-10T21:49:28.873Z
@Suresh and @Martin: thanks for the references. Anyway, it's not very obvious to me how the Frobenius distance of the super-operator matrix relates to the usual distance measures of quantum channels. Do you think it might make sense to take the truncated SVD of the Kraus operators?
Antonio Valerio Miceli-Barone
CC BY-SA 3.0
2011-05-13T21:58:33.273Z
@Antonio: Since the Kraus operator representation of a channel isn't unique, it's somewhat risky to write a definition that depends on them.
Peter Shor 
CC BY-SA 3.0
2011-05-13T22:27:29.387Z
@Antonio: Another comment. It might help you get an acceptable answer if you told us why you wanted to do this.
Peter Shor 
CC BY-SA 3.0
2011-05-16T15:17:41.570Z
@Peter: I would like to define some scheme for approximate abstract interpretation of quantum programs. I defined a simple imperative language and an operational semantic to translate each program in a quantum channel (the channel models one step of execution and is fixed for a given program). Essentially, any statement is translated to one or more Kraus operators. The concrete state space is just the tensor product of all variables and a program counter.
I would like to find a sensible way to approximate this channel with a channel acting on a smaller "abstract" state space.
Antonio Valerio Miceli-Barone
<quantum-computing><it.information-theory><quantum-information>
Approximation of Quantum Channels
Antonio Valerio Miceli-Barone
# Approximation of Quantum Channels

(Asked by: username_0 on 2011-05-10)

**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](http://arxiv.org/abs/quant-ph/0408063), or [this question](https://cstheory.stackexchange.com/questions/4885/is-there-any-connection-between-the-diamond-norm-and-the-distance-of-the-associat) 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).

**Tags:** quantum-computing, it.information-theory, quantum-information
> **Comments:**
> **username_1**: 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?  
> **username_2**: 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.  
> **username_3**: Geometrically, another approach is to think of the operation as a (Riemannian) projection onto the lower dimensional submanifold (although this might technically be Kahlerian)  
> **username_0**: @Artem: yes, the abstraction and concretization operators does not depend on the specific channel, otherwise you could always find an exact approximation for any dimension. Sorry if that wasn't very clear. Anyway, what I'm asking is indeed in some sense analogue to a lossy compression, albeit of the channel description, not of the data passing through the channel.  
> **username_0**: @Martin : by super-operator matrix representation do you mean the Choi-Jamiolkowski representation? Using truncated SVD may be a good idea (although I have to think about complete positivity and trace preservation), but what kind of distance does it minimize?  
> **username_0**: @Suresh: I'm not familiar with the terms Riemannian projection or Kahlerian (I'm neither a matematician nor a physicist). Can you point me to some entry-level reference, if there are any, please?  
> **username_3**: 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.  
> **username_3**: @Antonio, the SVD-based construction is the closest (in Frobenius distance) matrix in the lower dimensional sense.  
> **username_2**: @Antonio: No, I don't mean Choi-Jamiolkowski (but it's related). I mean the dynamical map L as defined e.g. here: http://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).  
> **username_0**: @Suresh and @Martin: thanks for the references. Anyway, it's not very obvious to me how the Frobenius distance of the super-operator matrix relates to the usual distance measures of quantum channels. Do you think it might make sense to take the truncated SVD of the Kraus operators?  
> **username_4**: @Antonio: Since the Kraus operator representation of a channel isn't unique, it's somewhat risky to write a definition that depends on them.  
> **username_4**: @Antonio: Another comment. It might help you get an acceptable answer if you told us why you wanted to do this.  
> **username_0**: @Peter: I would like to define some scheme for approximate abstract interpretation of quantum programs. I defined a simple imperative language and an operational semantic to translate each program in a quantum channel (the channel models one step of execution and is fixed for a given program). Essentially, any statement is translated to one or more Kraus operators. The concrete state space is just the tensor product of all variables and a program counter. I would like to find a sensible way to approximate this channel with a channel acting on a smaller "abstract" state space.  

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## 0 answers

