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 n m dimensions.
We can quickly get vector spaces with high dimensions through this multiplicative effect. This means we need to use our algebraic intuition and tools more than our geometric ones.
The initial construction is straight linear algebra but we specialize it to quantum computing and working with multiple qubits in the next section.
Let V and W be two finite dimensional vector spaces over F. Define a new vector space V ⊗ W, pronounced ‘‘V tensor W’’ or ‘‘the tensor product of V and W,’’ to be the vector space generated by addition and scalar multiplication of...
