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Every probability distribution can be viewed as a quantum state & vice versa. There's a nice mathematical dictionary between the two worlds! So, what *is* a quantum state? And what's the dictionary? "A First Look at Quantum Probability, Part 2" is here! math3ma.com/blog/a-first-l… 
I’ll share a few of the ideas here, picking up where we left off in Part 1:
For motivation, we started with a simple joint probability distribution and turned it into a matrix M (and judiciously added some square roots).
Then we observed something interesting about the matrix M*M. It contains marginal probabilities along its diagonal, and conditional probabilities in its eigenvectors! Same for MM*
These matrices are the quantum analogues of marginal probability: They contain classical marginal probabilities *and more.* The “and more” relates to entanglement between the two systems you’re working with. This idea can be said more precisely…
We turned a joint probability distribution into a pure, highly entangled quantum state. The non-zero off-diagonal entries of the matrices M*M and MM*, and hence their eigenvectors, are exactly capturing that entanglement.
But what do these words mean??
But what do these words 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 is a construction in linear algebra called the partial trace.
I’ll explain...
I’ll explain...
A density operator is a linear map from a Hilbert space to itself that satisfies some properties. These properties amount to the linear-algebraic version of probability!
Sometimes a density operator is also called a “quantum state.”
Sometimes a density operator is also called a “quantum state.”
So you can think of a density operator as describing the state of a quantum particle. As a tensor diagram (aka string diagram), an operator can be drawn as a node with two edges, which represent the space being operated on.
For more on tensor diagram notation, check out my introduction here! math3ma.com/blog/matrices-…
Anyways, if you have a *bunch* of interacting particles, then the state of that large system is a density operator on the *tensor product* of the spaces associated to each particle. I like to draw such an operator as a big blob with one edge for each space.
Now here’s a thought: If you know the state of a system of interacting particles, what if you’d like to know the state of just a *few* of those particles? That is, what if you just want to glean info about a smaller subsystem?
Asked mathematically: Given a linear operator on a tensor product of spaces, can you get a linear operator on just a few of the factors?
The answer is yes! You can do it by a construction in linear algebra called the particle trace.
The answer is yes! You can do it by a construction in linear algebra called the particle trace.
I like to introduce the partial trace by citing a fact from linear algebra:
Fact: there are *no* natural linear maps from a tensor product down to each factor. E.g. you can try to project v⊗w to v, but that assignment isn't linear!
Fact: there are *no* natural linear maps from a tensor product down to each factor. E.g. you can try to project v⊗w to v, but that assignment isn't linear!
(I think this fact is really interesting. Just from a category theoretical perspective, the tensor product is a totally different beast than an actual, categorical product. It just mixes things up too much, so you can’t un-mix in a natural way to get maps down to each factor!)
So this seems bad. Happily, the problem goes away if you look at *operators* on a tensor product. Then you do get maps down to (operators on) each factor! These maps are called partial traces.
So unlike the trace, which takes a linear operator and produces a number, the partial trace takes an operator and produces another operator—one that operates on a smaller space. There are nice pictures for this, too:
The partial trace reminds me of people-watching at the park on a busy day. Lots of folks might be bustling around, but you can hone in on a few of them and learn something as they interact with their surroundings. You can do the same for density operators, i.e. quantum states.
That is, given a quantum state on a big system, the partial trace provides a way to obtain quantum states—called "reduced densities"—on smaller subsystems, *all the while* keeping track of how those subsystems interact with each other and their environment.
These subsystem interactions are captured formally by the notion of “entanglement.” That seems like a loaded word, but it’s just linear algebra! Indeed lingo like "pure," "mixed," and "entangled" are just linear algebraic notions:
The punchline is that we encoded a classical joint probability distribution as a pure, entangled quantum state. The matrices M*M and MM* are the reduced densities of that state, and their eigenvectors capture interactions between the systems in the joint distribution!
Aside from being nice mathematics, the information encoded by these eigenvectors turn out to be very useful! Another time, I’d like to share how that usefulness arises in the context of machine learning with tensor networks. [end]
For more on tensor products, check out my introduction here! math3ma.com/blog/the-tenso…
