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, , is extremal if and only if the set
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.