8.1 Tensor products

In this section, I introduce the linear algebra construction of a tensor product. If the direct sum seems to concatenate two vector spaces, then the tensor product interleaves them. In the first case, if we start with dimensions n and m, we end up with a new vector space of n + m dimensions. For the tensor product, we get nm dimensions. linear$algebra product$tensor product$Kronecker Kronecker$product tensor product vector space$tensor product vector space$Kronecker product

We can quickly get vector spaces with high dimensions through this multiplicative effect. We must use our algebraic intuition and tools more than our geometric ones.

The initial construction is straight linear algebra, but we specialize it later to quantum computing and working with multiple qubits.

Vector spaces

Let V and W be two finite-dimensional vector spaces over F. We define a new vector space V ⊗ W, pronounced “V tensor W...