September 2023: more numbers in boxes

A toy model that looks a bit like a qubit if you squint a lot

Hi again. More physics. This time I’m writing up some missing bit of my thinking that’s been sitting in the drafts of my old Wordpress blog since 2020. I don’t know that this is going to be especially exciting to anyone else, but it’s good to finally get this out in some form.

(I was sort of hoping that this newsletter experiment would get me writing more generally, but apparently at the moment it’s physics or nothing. There is an “other stuff I’ve been reading or thinking about” section at the end if you don’t care about any of this.)

So, last time I talked about qubit states in phase space. This time I’m going to talk about a simple toy model that has a bit of a resemblance to that if you squint a lot. It’s nothing to do with quantum physics in itself, just three points that can take some values. I’ve been interested for a while in the idea of spatial systems that have analogies to quantum systems, and this is a sort of sketch towards what I’m imagining. It’s very limited – it only reproduces a few qubit states, and they’re not even the interesting ones with negative probabilities – but it’s simple to explain, and does suggest that there’s potential here for either extending this model or finding something better in the same general area (probably I’ll talk about my idea for that in a later newsletter).

OK, so this toy model consists of three points, x_L, x_M and x_R. (The subscripts are for left, middle and right. And, unfortunately, they’re going to be ugly underscores rather than proper subscripts, because Substack doesn’t do inline LaTeX.) Each point can take one of two values, 0 or 1. Here's an example state:

The other part of the model is a rule saying that the gradient of the line between points has to be constant. We’ll use a slightly funny definition of “constant” that treats a line sloping up from 0 to 1 as equivalent to a line sloping down from 1 to 0, so the example above counts as an allowable state. What it does rule out is states like this, with a sloping line between x_L and x_M and a constant line between x_M and x_R:

This constraint leaves us with the following four options:

As shorthand notation I'll refer to these four states as [000], [010], [101] and [111].

Next, I'll introduce two new sets of variables to describe the system. First up are the gradient variables, which I'll call {v_L, v_M, v_R}. The gradient is constant, so these will either be all 0 (for the [000] and [111] states) or all 1 (for the [010] and [101] states).

The other set of variables, which I'll call {+_L, +_M, +_R}, is just the sum of the first two variables, taken mod 2. E.g +_L = x_L + v_L. Putting all these variables into a table, we get (sorry this is a screenshotted image from my Wordpress drafts, because Substack doesn’t support tables):

OK, now we have four simple states and an unnecessarily large number of variables to describe them. So what’s the point?

To explain that, we need to introduce one more ingredient. We’ll condition on knowing the value of one variable on the left hand side, and then see which states are compatible with that. There are six options we could choose for this boundary condition: we can set one of the variables x_L, v_L or +_L to either 0 or 1.

As an example, take x_L = 0. This tells us that the system must be in either the [000] state or the [010] state, but we don't know which, so we assign them both the same probability. Either way, we find that the corresponding value of x_R will always be 0. The value of v_R could be 0 or 1 with equal probability, depending on what the underlying state is. Same for +_R. I've drawn this on a diagram below - it's a little cluttered, because I tried to make everything explicit:

The numbers in the boxes are the probabilities of being in that state: there's a ½ chance of being in the [000] state with x_R=0 and v_R=0, and a ½ chance of being in the [010] state with x_R=0 and v_R=1. To find the probability of, say, p(v_R =0), you add up the two boxes (x_R=0, v_R=0), (x_R=1, v_R=0) that are consistent with that, as indicated by the bottom horizontal red arrow. The other arrows work in the same way.

Here are the diagrams for all six states (I won't draw out all the clutter with arrows and labels, just the boxes with numbers in):

These boxes should look a lot like the ones from last month when I talked about qubit phase space.

If you’re familiar with some of the quantum foundations literature (or have just heard me rant on before), they might also remind you of something else: the epistemic states of the Spekkens toy model. The Spekkens toy model gets these states with a “knowledge balance principle” which says that you can only get half the information about the system at once. This model makes this much more concrete, by telling you why you can only get half the information. As with the Spekkens model, there are four underlying “ontic states” (the allowed states [000], [010], [101], [111]), but you’re constrained to only know the left hand value.

(Aside: this is an artificial constraint for a spatial system, but seems completely normal when the points are arranged in time instead of space. We’re used to only having knowledge of the past state of a system, and not the future one. The underlying generator for pretty much all my fiddling around with models like this is: quantum physics looks very much like something time-symmetric where we only have knowledge of the past, in the same way that this model is spatially symmetric where we only have knowledge of the left. This idea is subtle and hard to think about carefully, so it really needs much more expansion than this short paragraph, but that'll have to wait for another time.)

Now, what I’d really like is something with a lot more than six states! And that maybe includes some of those negative probabilities I like so much. Last time I thought about this in 2020 I ran into a dead end because (covid, and also) 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.

Other stuff this month

• I’m still into music theory. This four-part harmony video helped me think about why I’m enjoying it so much. It exactly demonstrates the kind of "half-structured-half-vibey" domain I like. There isn't one single solution, sudoku-style, and at some point you are going to have to use your own sense of taste to decide what to do, but if you're trying to keep within standard rules of harmony and voice leading you are fairly constrained at each step. This is also what technical writing to a style guide is like! Very structured subset of writing, sometimes the rules pretty much funnel you into one way of expressing the thing, sometimes not. It's a very satisfying kind of material to work with.

• I discovered this composer Johann Schobert who died in an awful way:
  ”In 1767, Schobert went mushroom picking with his family in Le Pré-Saint-Gervais near Paris. He tried to have a local chef prepare them, but was told they were poisonous. After unsuccessfully trying again at a restaurant at Bois de Boulogne, and being incorrectly told by a doctor acquaintance of his that the mushrooms were edible, he decided to use them to make a soup at home. Schobert, his wife, all but one of their children, and his doctor friend died.”

• My big internet timewasting hole this month has been watching far too many dog grooming videos on Youtube. There’s just something really satisfying about seeing big floofy dogs have insane amounts of undercoat combed out of them. Also sometimes there are amusingly pissed off cats wearing something called a happy hoodie, which does seem to take the edge off their murderous rage but “happy” is pushing it.

• I read Matthew Crawford’s Shop Class as Soulcraft and was surprised by how much I enjoyed it. A few years ago I wrote about his other book The World Beyond Your Head, which I found to be some confusing mix of annoying and insightful. Shop Class is annoying in exactly the same ways as the other one, so there was no new kind of annoyingness to get used to, and the material is much more concretely grounded in actual jobs he’s worked. Also, it fitted in well with my current compulsion to read and watch lots of practical stuff (I guess the dog grooming is the same thing).

That’s it for now. I’d like to pretend I’ll write about something different next month, but it’s probably going to be more physics isn’t it.