Tensor Product of Vectors

1. The Kronecker Product: A Concrete Recipe

The tensor product of two column vectors is calculated using an operation called the Kronecker product, also denoted by the symbol $\otimes$. The procedure is straightforward and best understood as a "scaling and stacking" operation.

Let's say we have two vectors, $\mathbf{v}\in {\mathbb{R}}^n$ and $\mathbf{w}\in {\mathbb{R}}^m$. To compute their Kronecker product $\mathbf{v}\otimes \mathbf{w}$, which will be a vector in ${\mathbb{R}}^{nm}$, you follow this recipe:

1. Take the first scalar component of $\mathbf{v}$.
2. Multiply this scalar by the entire vector $\mathbf{w}$. This gives you the first $m$ entries of your new vector.
3. Take the second scalar component of $\mathbf{v}$.
4. Multiply it by the entire vector $\mathbf{w}$. This gives you the next $m$ entries.
5. Continue this process until you've used every component of $\mathbf{v}$.

In mathematical notation, it looks like this:
If $\mathbf{v}=\left(\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{c}{w}_1\\ w_2\\ ⋮\\ w_m\end{array}\right)$, then:

This article snippet provides a clear visual explanation of this procedure.

The image below shows this rule applied to vectors of various sizes. Notice how the dimension of the resulting vector is always the product of the input dimensions.

2. A Step-by-Step Video Demonstration

Now, let's watch a video that walks through this calculation in detail. Repetition and seeing it from a slightly different perspective will help solidify the process.

3. Application: Building Multi-Qubit States

This mathematical tool is not just an abstract exercise; it's the language of composite quantum systems. A single quantum bit (qubit) can be described by a vector in a 2D complex vector space, ${\mathbb{C}}^2$. The standard basis states are:

• The "state 0": $|0⟩=\left(\begin{array}{c}1\\ 0\end{array}\right)$
• The "state 1": $|1⟩=\left(\begin{array}{c}0\\ 1\end{array}\right)$

To describe a system of two qubits, we use the tensor product. The combined system lives in a $2\times 2=4$-dimensional space, ${\mathbb{C}}^4$. The four basis states of this new space, denoted $|00⟩,|01⟩,|10⟩,|11⟩$, are constructed using the Kronecker product.

Let's watch this process unfold.

Let's write out the calculation for one of these, $|10⟩=|1⟩\otimes |0⟩$, to reinforce the concept:

This resulting 4D vector is the third standard basis vector in the two-qubit space.

4. Two Sides of the Same Coin: Kronecker vs. Outer Product

For someone with your background in computer science, it's useful to see an alternative but equivalent way to think about this construction. The tensor product of two vectors can also be represented as a matrix via the outer product, $\mathbf{v}{\mathbf{w}}^T$.

Let's use our previous example: $\mathbf{v}=\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{c}{w}_1\\ w_2\end{array}\right)$.

The outer product is the matrix:

Notice the entries of this matrix. If you read them out row by row, you get $(v_1{w}_1,v_1{w}_2,v_2{w}_1,v_2{w}_2)$. This is exactly the sequence of components in the vector we calculated using the Kronecker product!

This reveals a deep connection: the Kronecker product vector can be seen as a "reshaping" of the outer product matrix. This idea of "matrix-vector duality" is powerful. The matrix represents a linear map (a process), while the vector represents a state. In the context of tensor products, they are just two different ways of arranging the same set of numbers.

This short reading elaborates on this very connection.

One final clarification on terminology: physicists and engineers often use "tensor product" to refer to the Kronecker product operation on vectors and matrices. In pure mathematics, "tensor product" refers to the more abstract operation between vector spaces we discussed in the last lesson. For our practical purposes in this course, when we talk about the tensor product of vectors, we mean the computation performed by the Kronecker product.

Conclusion

In this lesson, we moved from the abstract theory to concrete computation, focusing on the Kronecker product as the workhorse for combining vectors.

Key Takeaways:

• The Kronecker product is the algorithm for computing the tensor product of two column vectors.
• The procedure involves scaling the second vector by each component of the first vector and stacking the results.
• If $\mathbf{v}\in {\mathbb{F}}^n$ and $\mathbf{w}\in {\mathbb{F}}^m$, their product $\mathbf{v}\otimes \mathbf{w}$ is a vector in ${\mathbb{F}}^{nm}$.
• This operation is not commutative: in general, $\mathbf{v}\otimes \mathbf{w}\ne \mathbf{w}\otimes \mathbf{v}$.
• The Kronecker product is the tool used to construct the basis states of composite quantum systems, like multi-qubit registers.
• The resulting vector can also be seen as a "vectorized" form of the outer product matrix $\mathbf{v}{\mathbf{w}}^T$.

Preview of the Next Lesson:

We've learned how to combine vectors, which represent states. The next logical step is to learn how to combine the operators that act on those states. In the next lesson, we will extend the Kronecker product to matrices, allowing us to construct the matrix representation of a tensor product of two linear operators. This will enable us to describe operations, such as quantum gates, that act on individual parts of a composite system.