You’re asking great questions, Tim. And I see only one climate science expert in the discussion of “week304”: Nathan. Maybe there are more lurking around… but I haven’t seen them posting yet.

]]>This is an opportunity to learn how me talking about climate science looks like to experts, I think that’s comparably funny :-)

]]>The main reason is to solve the Lindblad equation as a matrix eigenvalue equation using a computer program.

Yeah, there is some funny stuff there, but I still think worth the read.

]]>It seems strange to speak of ‘linearizing’ the Lindblad equation, because it’s already obviously a linear differential equation (see here).

It also seems odd, from a mathematician’s viewpoint, to define an operation ‘vec’ which takes a matrix and thinks of it as a vector, since the space of matrices is quite obviously a vector space.

It similarly seems odd to prove that the map

is linear, where are matrices, because again this is quite well-known.

But I’m sure there’s also lots of good stuff in this paper.

]]>Ladies and gents: on a related note, there is a great short review article by Nielsen which shows techniques that can be used to linearise the Lindblad equation: check out Roth’s lemma.

I think category theorists will like to see Roth’s lemma, as it’s the linear algebra behind sliding boxes around wires, operator vector duality (called map-state duality in quantum computing) and all that other great stuff. They define operations such as vec(|j>|k>) = |k>|j>, etc.

A friend of mine James Whitfield visited and learned some string diagram theory by presenting the diagrammatics behind this review paper.

]]>Eric wrote:

I wish you didn’t ask this question.

Heh. Someone knows the answer already, I’m sure. Let’s hope they tell us.

]]>I wish you didn’t ask this question. This is the type of question that gets stuck in your head (like a song you can’t kick) and you can’t rest until you solve it :)

My gut tells me it’s coming from some form of inner product.

]]>The elements called are a basis of the traceless matrices on our Hilbert space — that is, a basis of . So, presumably the relevant algebra involves . However, what’s going on with the expression

where $\rho$ is a positive matrix of trace 1? Are we dealing with some sort of representation of on the vector space of self-adjoint operators, or something?

Or maybe a representation of ?

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