Understanding Entanglement With SVD

Quantum entanglement is, as you know, a phrase that's jam-packed with meaning in physics. But what you might not know is that the linear algebra behind it is quite simple. If you're familiar with singular value decomposition (SVD), then you're 99% there. My goal for this post is to close that 1% gap. In particular, I'd like to explain something called the Schmidt rank in the hopes of helping the math of entanglement feel a little less... tangly. And to do so, I'll ask that you momentarily forget about the previous sentences. Temporarily ignore the title of this article. Forget we're having a discussion about entanglement. Forget I mentioned that word. And let's start over. Let's just chat math.

SVD is arguably one of the most important, well-known tools in linear algebra. You are likely already very familiar with it, but here's a lightening-fast recap. Every matrix $M$ can be factored as $M=UDV^\dagger$ as shown below, called the singular value decomposition of $M$. The entries of the diagonal matrix $D$ are nonnegative numbers called singular values, and the number of them is equal to the rank of $M$, say $k$. What's more, $U$ and $V$ have exactly $k$ columns, called the left and right singular vectors, respectively.
There are different ways to think about this, depending on which applications you have in mind. I like to think of singular vectors as encoding meaningful "concepts" inherent to $M$, and of singular values as indicating how important those concepts are.