Talk:Quantum operation

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[edit] Kraus operators

The theorem relating the Kraus operators is unclear. In the finite dimensional case, the corresponding result usually involves undoing the Vec or Row operation on the Kraus matrices. The theorem in the article probably means to say something similar. Can whoever wrote it provide a reference? Mct mht 11:13, 12 April 2006 (UTC)

I thought I had convinced myself it was true in general. But I changed it back to the finite dimensional case. --CSTAR 14:32, 12 April 2006 (UTC)

Another question, regarding the following sentence in the article:

For the general case, S is replaced by a trace class operator and {Bi} by a sequence of bounded operators.

Can the person who added it outline an argument? Choi's finite dim argument seems to not work here, unless the range of the CP map is a finite dim C* algebra. If the Choi matrix M has Cholesky factorization M = BB ^* , each column of B then corresponds to a Kraus operator via the Vec operation. This doesn't make sense in general. Next one might try to prove it from Stinespring's theorem. So if (π,V,K) is a Stinespring representation of Φ, which takes value in B(H). Assuming both the domain and range of Φ are separable, then K is also. I guess one might identify K with the direct sum of countably many copies of H, then take B_i = P_iV where P_i is projection onto the i-th copy. Anyhow, is there a reference where the argument is given explicitly? Mct mht 18:28, 17 May 2006 (UTC)

I added it. One can either use Choi and a weak limit argument or Stinespring. This is completely standard.--CSTAR 18:36, 17 May 2006 (UTC)

ok, can you give me a reference? thanks. Mct mht 18:43, 17 May 2006 (UTC)

Good God. I dunno, some book, maybe Arveson's on CP semigroups? At the beginning there is some stuff on generalities.--CSTAR 18:50, 17 May 2006 (UTC)

haha, after pressing the save button, all i saw was your stuff. the following is what i typed that didn't get saved:

"ok, taking the sequence of upper-left finite blocks of the Choi matrix then the weak limit seems to be believable. a reference would still be nice though." Mct mht 19:00, 17 May 2006 (UTC)

So this is a folklore result, heh. If so, that's fair enough. Mct mht 19:01, 17 May 2006 (UTC)

[edit] historical reference accurate?

The accuracy of following sentence in the article seems to be debatable:

The quantum operation formalism emerged around 1983 from work of K. Kraus, who relied on the earlier mathematical work of M. D. Choi.

There is at least one paper(E. C. G. Sudarshan, et al, Phy Rev, 1961) that seems to have already contained the idea that quantum operations are CP maps. Mct mht 17:56, 17 May 2006 (UTC)

Hmmm. You may be right. Although Sudarshan now seems to be having second thoughts.--CSTAR 17:58, 17 May 2006 (UTC)