October 2023: two nodes and an edge

I’m trying to win Substack’s “Dullest Social Media Preview Image” competition

As expected, this month is yet more physics. There’s also a longish section at the end about what I’ve been reading this month, but – surprise! – that’s mostly physics too.

Last time I talked about a simple model of some points in space which has some states that behave a bit like qubit states if you squint hard. But I was feeling like I’ve taken that as far as I know how to go with it:

… the model is pretty unmotivated mathematically. It’s just something simple I made up, and it was hard to see how to extend in any very satisfying way. I’m currently doing something with the graph Laplacian for two points with an edge between them, and that seems kind of promising, so maybe I’ll write it up next time.

The trouble with “next time” is that suddenly it becomes “now” and you have to do the thing you said you were going to do. I was hoping to get further this month and have something solid to write down, but I didn’t, so it’s still at the numerology-and-vibes stage. I think writing down the scraps that I have will still be helpful, though.

So, first, here’s a condensed version of the chain of thought that got me wondering about the graph Laplacian. The previous example is just points and lines, which looks like a graph, right? And also the states kind of look like eigenfunctions. For example, if you take the left and right hand values of x to both be 0, then there are two solutions, one where the middle value is also 0 that looks like a sort of ground state and one where the middle value is 1 that looks like a first harmonic. So if I think about graphs and boundary values and eigenfunctions, the thing that comes to mind is the graph Laplacian.

I’m more familiar with the continuous Laplacian, which governs stuff like vibrations of a string or drum head. The graph Laplacian is a discretised version of the continuous one: this blog post is really nice for explaining how that works. It’s a matrix, rather than a differential operator.

I played with some examples, and the one that seems interesting to me is this really simple graph with just two nodes and an edge between them:

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In this case, the Laplacian is just a 2 x 2 matrix:

\(L = \left(\begin{matrix} 1 & -1 \\ -1 & 1 \end{matrix}\right).\)

This actually looks very familiar to me already! If you put a factor of ½ in front of it (let’s call that version L’), it’s the density matrix for one of the standard qubit states:

\(L' = | - \rangle \langle - |.\)

Also, it’s an operator that comes up when you move to qubit phase space:

\(L' = \frac{1}{2}\left(I - \sigma_x\right).\)

where the sigma_x is one of the Pauli matrices. You take the expectation value of L’ with your qubit state to get a measurement for one of your coordinates of phase space (I called the value of this measurement q_x in a previous post):

\(q_x = \langle \psi | L' | \psi \rangle .\)

Now, I’d expect L (or L’) to appear in an eigenvalue equation, like this:

\(L|\psi\rangle = \lambda |\psi\rangle.\)

The eigenvectors of L are the qubit states |+> and |->¹. We can get those from the standard basis states |0> and |1> by operating on them with the discrete Fourier transform matrix:

\(W = \frac{1}{\sqrt{2}}\left(\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix}\right) \)

And then if we write out L’ in the |+> and |-> basis instead, we get

\(\tilde{L'} = \left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right) = \frac{1}{2}(I - \sigma_z),\)

which is the operator that we want for the other coordinate of qubit phase space.

So this all seems like useful stuff. But I need to put the pieces together. I did search for existing work, and I found a couple of papers like this one on connections between graph Laplacians and density matrices, but they mainly seemed to be coming from the perspective of “let’s represent lots of states as graphs” rather than investigating what it means conceptually in simple cases.

Maybe it’ll become clearer over the next month, but I’m wary of making big claims about “next time”.

Other things

• I found this post on galant style in classical music, which did a great job of explaining why I like it so much: “It’s not about following retrograde canons at the fifth or straining our grey matter to take in all at once the contrapuntal weavings of four or five independent voices.  We’re handed a pretty melody on a plate and told, “Enjoy!”  What’s not to like about that?” Looks like there are lots more music recs on the site too. This is exactly the kind of weird website I want Google to find for me, rather than its usual load of SEO-optimised crap.

• I read Tom Wolfe’s The ‘Me’ Decade, which was an experience. First Tom Wolfe I’ve ever read.

• I’m suddenly reading a lot of physics-related books. Well, starting a lot of physics-related books. Here’s the current stack:

  • Lee Smolin’s The Trouble with Physics. This is a reread of a book that came out at the height of the String Wars (remember those?) when I was an undergrad, and that made a big impact on me. It’s only partly about string theory; the rest is about dysfunction in academic physics more generally, mixed in with lots of stories of Smolin’s own experiences of doing research. I think it was also the first place I saw the “two types of mathematician” split that I later got a bit obsessed with (I haven’t got to that bit yet though, because I keep starting more books.)

  • David Deutsch’s The Beginning of Infinity. His The Fabric of Reality also made some sort of impression on me around the same time, but I tried to reread it a few years ago and just found it weird. Getting the same feeling from this one so I might drop it soon.

  • Abraham Pais’s Subtle is the Lord, a hefty scientific biography of Einstein by a physicist who overlapped with him at the Institute for Advanced Study. I’m mainly interested in the development of general relativity, because I know it’s a very convoluted story and I’d like to know the details better. But the whole book is very well written so for now I’m just reading from the beginning forwards.

  • An early draft of Carlo Rovelli’s Quantum Gravity textbook (pdf). Not planning to read very much of this one, but I remember some of the early conceptual stuff on general relativity being good, so I’m reading that.

• I really liked this piece (pdf) by Sanjoy Mahajan on how to roughly estimate the deflection of light by gravity with dimensional analysis. Fun writing style and lots of the sort of insights on how to think about problems that normally get left out in written treatments. This example also appears in his MIT course on The Art Of Approximation In Science And Engineering, which looks great, and maybe in his book which sounds related.

• All this physics reading seems to be part of some kind of reorientation, though I don’t know exactly what I’m orienting to. I seem to want to go back to old topics that were important to me and maybe reintegrate them somehow. Or maybe I’m just procrastinating on the graph Laplacian idea. It’ll probably get clearer as I do it more.

• I think there’s some connection between this reorientation and being off twitter. It’s like I’ve finally got my head clear of my normal soup of takes, and have managed to fit something else in there. So it looks like I need to be off there for at least another couple of months. I do miss twitter though!

Next month I’ll probably keep going with the physics reading, and maybe even finish something. I’ll also probably keep banging my head against the qubit / graph Laplacian connection.

1. I’m definitely missing inline LaTeX. But if John Baez could write 300 editions of This Week’s Finds in Mathematical Physics in ASCII, I should be able to put up with it for a few newsletter posts.