## 5.3 Quantum theory, formally (continued)

In Section 4.11, we said that we were missing a key part in our formalism of quantum theory — now we can finally fill in this hole.
Our mathematical formalism of choice behind the quantum theory of composite systems is based on the **tensor product** of Hilbert spaces.

### 5.3.1 Tensor products

Let the states of some system **tensor product space** ^{89}

The tensor product operation

The tensor product of Hilbert spaces is again a Hilbert space: the inner products on ^{90}, to any number of subsystems.
Note that the bra corresponding to the tensor product state *not* change when the dagger operation is applied.

Some joint states of **separable** (or **product states**).
States that are not separable are said to be **entangled**.

A useful fact about tensor products is that

We will also need the concept of the tensor product of two operators.
If

We have described the tensor product in terms of how it acts on bases, and then extended everything by linearity, distributivity, and associativity. But there are other, more abstract approaches to defining the tensor product.

For example, given two vector spaces *quotient* of the cartesian product

But really this is hinting at the so-called **universal property** that defines the tensor product without giving a choice of explicit construction: the tensor product of *other* vector space *unique* linear map **initial object** amongst vector spaces endowed with a bilinear map from *factors through* the tensor product.

One specific reason to care about giving a definition in terms of universal property is that this guarantees (by some abstract nonsense) that the resulting object will be unique (“up to unique isomorphism”) whenever it exists, so you don’t need to worry about proving this separately.

Tensor products are much more general than just for vector spaces: they can be defined for modules (which are like vector spaces over an arbitrary commutative ring, instead of over a field), and abelian groups are, it turns out, exactly “modules over **complexes** of modules and **sheaves** of modules, and these constructions are absolutely fundamental to modern algebraic geometry.

Going even deeper still (and now far beyond the purview of this book), tensor products are generalised by the notion of **monoidal categories**.

As a final note, the universal property of the tensor product can be used to prove that we do not need to impose the postulate “the Hilbert space of a composite system is the tensor product of the Hilbert spaces of its components”, but that this actually follows “for free” from the **state** and the **measurement postulates**.
This is shown in Carcassi, Maccone, and Aidala’s “The four postulates of quantum mechanics are three”, arXiv:2003.11007.

If the bases

\{|a_i\rangle\} and\{|b_j\rangle\} are orthonormal then so too is the tensor product basis\{|a_i\rangle\otimes|b_j\rangle\} .↩︎Associativity means that

({\mathcal{H}}_a \otimes {\mathcal{H}}_b)\otimes {\mathcal{H}}_c = {\mathcal{H}}_a \otimes ({\mathcal{H}}_b\otimes {\mathcal{H}}_c) .↩︎