What is a tensor product?

The state of a coin can be defined by a vector $\text{COIN} = \begin{pmatrix}\text{HEAD}\\\text{TAIL}\end{pmatrix}$. Let's consider the state of one coin to be$\text{COIN}_1 = \begin{pmatrix}0.7 \\ 0.3 \end{pmatrix}$; this means that the coin is $\text{HEAD}$ 70% of the time and $\text{TAIL}$ 30% of the time. Let's consider another coin with the state $\text{COIN}_2 = \begin{pmatrix} 0.4\\0.6 \end{pmatrix}$, i.e., it is $\text{HEAD}$ 50% of the time and $\text{TAIL}$ 50% of the time—it is an unbiased coin.

How can we define the combined state of a system comprising of $\text{COIN}_1$ and $\text{COIN}_2$?

Mathematically, composite systems are represented by the following relation:

Therefore, the state of the above system becomes $\begin{pmatrix} 0.28\\ 0.42 \\0.12\\0.18 \end{pmatrix}$.

Tensor product

In linear algebra, this can be done by computing the tensor product of two vectors. For two vectors $v$and $w$, their tensor product can be defined as follows:

Here, $v_1, v_2, \cdots, v_n$are the elements of $v.$

If $m$ is the dimension of $v$, and $n$ is the dimension of $w$, the dimension of $v\otimes w$is $m\times n.$

Note: Tensor products are defined on both vectors and matrices.

Code example

In NumPy, the tensor product of two arrays can be evaluated using the kron() function. Let's evaluate the tensor product of the two coin states under consideration:

Here, the kron() function is used for evaluating the tensor product between the two vectors we defined, coin_1 and coin_2.