Assumptions:
This post assumes the reader understands the following:
- Basics of what a linear map is
- Basics of a basis and vector space
- Basic Linear Algebra
Motivation and Preliminary Information to understanding Tensor Products:
In math and physics, we often want to represent two vector spaces as one. This, for example, is the case in quantum mechanics when we want to represent the Hilbert space of two individual systems as one combined Hilbert space. Furthermore, the tensor product can encode all possible states of both interacting and noninteracting systems.
Before we define a tensor product, you need to understand the universal property of mapping. In this case, we will explain it with a bilinear map.
- A bilinear map takes all pairs of elements from two separate vector spaces to produce a 3rd vector space.
A bilinear map T of the following form:
Satisfies the universal property if for any vector space C and any bilinear mapping of the form:
The following statement holds:
- The open circle is the same as applying the transformation of B then B hat. Follow the arrows, and you will find this true in our situation.
Definition and Examples of Tensor Product:
A tensor product can be defined and set up in the following way. Imagine a Matrix A and a Matrix B that are both 2×2. The tensor product would look the following:
We would represent this operation as:
The formal definition of the tensor product is as follows. Let V and W be two vector spaces. The tensor product of V and W is a vector space with a bilinear map which has the universal property.
We can now define a basis for this new vector space that we have made.
Definition of Tensor Product Basis:
If beta_v and beta_w are the basis for V and W, respectively, then the basis for the tensor product of V and W is defined as:
As you see, we can now represent vectors in our tensor product as the following:
- The sum is a short notation for a double sum.
That is it; you now have all knowledge to calculate and understand what a tensor product is communicating. There is more to the story, whether funky identities or partial traces, but this is the fundamental concept.