A quirky mathematical problem in need of explanation

In my travails with the QIP Program Committee, one argument that went back-and-forth was whether or not the so-called Gell-Mann matrices (one possible representation of generators of SU(3)), when used as Kraus operators for a quantum channel, are extremal (we’re only concerned with extremality here, and not unitality). Landau and Streeter proved that a set of Kraus operators, $A_{i}$, is extremal if and only if the set

$\{A_{k}^{\dag}A_{l}\}_{k,l \ldots N}$

are linearly independent. I argued they were and Michael Ben-Or argued they were not. Oddly it appears we are somehow both correct (which only means there’s something else going on here that one or both of us missed). So I present here our two arguments and solicit your comments on this mathematical quandary.

Argument 1: They are extremal
I proved this in a straightforward way – I used Mathematica. I tried it several ways, but just to be sure I solicited suggestions in my previous blog post on how to tell Mathematica to check exactly what Landau and Streeter proposed. The result indicates that they are, indeed, linearly independent. Click here to see a PDF file export of the Mathematica notebook that includes both the input and the output (I don’t have a premium WordPress account so I can’t upload the actual notebook, but you can copy and paste from this into Mathematica if you want to tweak it yourself).

Argument 2: They are not extremal
Michael’s approach was to show that the channel constructed with the Gell-Mann matrices as Kraus operators are a convex combination of three other channels and thus can’t be extremal. He also crunched his numbers with Mathematica and here is a link to the PDF file of his notebook.

In my estimation, if Michael is correct and I didn’t make any errors in translating things to Mathematica then there’s something wrong with Landau and Streeter’s criterion for extremality. If anyone has any thoughts on this, I am eager to hear them.

3 Responses to “A quirky mathematical problem in need of explanation”

1. How can Kraus operators be extremal? And when would Gell-Mann matrices be interesting as Kraus operators?

2. quantummoxie Says:

Marsman: what I meant to say was that the channel they represent is extremal if they have the given property. I’ve replied to the comment on my Mathematica code over at MathOverflow but will note that the particular comment that is referenced above over at MathOverflow simply notes the impossibility of this based on the dimensionality of the matrices. But that should mean that my Mathematica code (if it is correct) should return a “False” for linear independence since Mathematica ought to easily pick up on this problem. Therefore, what I’d really like is for someone to tell me exactly where the problem in the code happens to be so I can fix it.

Note that one responder is incorrect about one point. Landau and Streater proved two results – one applying to extremal channels and one applying to unital extremal channels. At the moment I am only concerned with the former. Thus, all but the last two lines of my Mathematica code is correct. That means one of three things:

1. one of the last two lines of the Mathematica code is incorrect (this is the most likely scenario);

2. Mathematica’s has an inherent limitation that prevents it from correctly answering this question (at least in the stated way); or

3. Landau and Streater were wrong.