michael_nielsen

Hmm, I guess not, actually. If that were the case, UHU* + VKV* would be diagonal, with eigenvalues lambda_i + mu_i, if lambda_i are the eigenvalues of H and mu_i are the eigenvalues of K. So the rhs above should be diagonal with e^{i(lambda_i + mu_i)} entries on the diagonal.

That's right. When H and K don't share a common basis there's no obvious reason to expect there to be any relationship between the spectrum of exp(iH) exp(iK) and H and K. Thompson tells you there's a really strong relationship.

Interesting. Is there a short answer to what U and V are in this case?

Not as far as I know. In fact, I'm not even certain there's a constructive proof known - unless the situation has changed, Thompson's result depends on the proof of Horn's Conjecture by Allen Knutson, Terry Tao, and Alexander Klyachko, and I've never mastered that.

Whoa! Now I feel downright sheepish about my initial take. I'll have to think more on this... It feels like U and V have to bear some relation to the relevant diagonalizing, but maybe that's just me being grounded too stubbornly in certain intuitions.

Some relationship, yes. But a priori, to me, it seems very complicated to say what exactly.

Hmm. Actually, upon reflection I like your response: I'll bet there is a reasonably simple proof of Thompson's theorem possible. But so far as I know the best way known today is still through the proof of Horn's conjecture, and that is somewhat hairy.

Well, in either case, it seems worth meditating on for a bit. Is there a nice use for this result in QM/QC?