Thread by @math3ma

Hello friends! I’m excited to share with you the start of a mini-series on quantum probability theory. It's a *first* look at the subject, so the only prerequisites are linear algebra and basic probability. Part 1 is now on Math3ma! math3ma.com/blog/a-first-l…

Part 1 motivates the mini-series by reflecting on a thought from the world of (classical) probability theory:

*Marginal probability doesn’t have memory.*

What do I mean?

From a joint probability distribution on a product of two sets, you can get marginal probabilities by summing over, or “integrating out,” one of the variables. But marginalizing loses information—it doesn’t remember what was summed away!

I illustrate this in the blog post with a simple example: we consider a joint probability distribution on the set of bitstrings of length 3 and compute the marginal probabilities by summing along rows/columns of a table.

Alas, the marginal probabilities are forgetful! They don’t contain conditional probabilistic information about the bitstrings, which is present in the joint distribution.

That’s just how it is.

Or is it?

At the blog, I share a totally DIFFERENT way to find marginal probabilities, given a joint distribution. How? By turning your joint distribution into a matrix (in a special way). Let’s call it M.

Then both MM* and M*M contain marginal probabilities along their diagonal.

And...

…their *eigenvectors* define conditional probability distributions! That is, the information contained in the eigenvectors of these two special matrices is precisely the information that gets destroyed after computing marginal probability in the usual way.

So what’s going on??

The matrices MM* and M*M are the *quantum versions* of marginal probability distributions!

But what does that mean?

The quantum version of a probability distribution is something called a density operator. The quantum version of marginalizing corresponds to "reducing" that operator to a subsystem. This reduction is a construction in linear algebra called the partial trace.

In Part 2 of this miniseries, I'll give some intuition for the partial trace along with the definition. Then I'll explain what I meant by "quantum version." Along the way, we'll unwind the basics of quantum probability theory.

See you back in Part 2!