TITLE: The physical meaning of the tensor product
QUESTION [17 upvotes]: I have come across tensor products many times in physics, namely for matrices, vector-space elements, Hilbert-space elements (quantum states), and representations of groups and algebras. However, the abstract meaning of the tensor product still alludes me. Although I can understand its context in physics, I cannot see that its physical description and its abstract definitions, for example those given on Wikipedia, are equivalent. Could someone provide me with a geometrical, as well as an algebraic, interpretation of the tensor product and also explain why it is required?

REPLY [3 votes]: This is a really insightful question! You're pointing out that, in quantum mechanics, there are two different notions of "tensor product":

The composition of physical systems (a physical notion).
The tensor product of Hilbert spaces (a mathematical notion).

You say you can't see how these two notions are equivalent, and you're right! They aren't.
In different kinds of physical theories, the composition of physical systems is represented mathematically in different ways. For example:

In statistical mechanics, each type of physical system is described by a measurable space (a space that can carry probability distributions), and the composition of systems is described by the Cartesian product of measurable spaces.
In classical mechanics, each type of physical system is described by a symplectic manifold (a space where Hamilton's equations make sense), and composition is described by the symplectic product of symplectic manifolds.

There's a mathematical idealization that perfectly captures the physical idea of composition of systems: the "monoidal tensor product" in a monoidal category. The tensor product of vector spaces, the Cartesian product of measurable spaces, and the symplectic product of symplectic manifolds are all examples of monoidal tensor products.
To learn more about monoidal categories, and their deep relationship with physics, I recommend Bob Coecke's "Introducing categories to the practicing physicist." You might also want to check out a classic but more background-intensive article by John Baez, called "Quantum Quandaries: A Category-Theoretic Perspective."

p.s.
As I've been saying, the use of the tensor product of Hilbert spaces to represent the composition of physical systems is not requried—in fact, even the use of Hilbert spaces to represent types of physical systems is not required! So, why do we always use the mathematical machinery of Hilbert spaces, linear maps, and tensor products to build our quantum theories?
If you learn enough category theory, you can find a really cool answer in Chris Heunen's thesis, "Categorical quantum models and logics." Heunen shows that the bundle of mathematical machinery I mentioned above (usually called "the category of Hilbert spaces," or Hilb for short) is sophisticated enough to build any theory that satisfies certain conditions (see Definitions 3.7.1 and 3.6.1 and Theorem 3.7.8). The conditions look pretty technical, but many of them—maybe all of them!—can be thought of in an intuitive physical way. Roughly speaking, I'm pretty sure that...

"It has a dagger" means your theory has a notion of time-reversal. The adjective "dagger" that shows up in the other conditions means "compatible with time reversal."
"It is symmetric dagger monoidal" means your theory has a notion of composition of physical systems.
"It has finite dagger biproducts" means that if your theory can describe systems of type proton and systems of type neutron, it can also describe systems of type proton or neutron, but I don't know which, and that you can always do an experiment to determine whether a system of this type is a proton or a neutron.
"Every dagger mono is a dagger kernel" means that if the type car has a subtype awesome car, there's also a subtype not-awesome car.
"It has finite dagger equalisers" means that for any two experiments that can be done on the same type of system, there's a subtype describing systems for which the outcomes of the experiments are always the same.