matrix_ops.outer_product

Outer product operation.

Module Contents

Functions

outer_product(v1, v2)

Compute the outer product \(|v_1\rangle\langle v_2|\) of two vectors.

matrix_ops.outer_product.outer_product(v1, v2)

Compute the outer product \(|v_1\rangle\langle v_2|\) of two vectors.

The outer product is calculated as follows [1] :

\[\begin{split}\left|\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}\right\rangle\left\langle\begin{pmatrix}b_1\\\vdots\\b_n\end{pmatrix}\right|=\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}\begin{pmatrix}b_1&\cdots&b_n\end{pmatrix}=\begin{pmatrix}a_1b_1&\cdots&a_1b_n\\\vdots&\ddots&\vdots\\a_1b_n&\cdots&a_nb_n\end{pmatrix}\end{split}\]

Example

The outer product of the vectors \(v1 = \begin{pmatrix}1 \\ 2 \\ 3 \end{pmatrix}\) and \(v2 = \begin{pmatrix}4 \\ 5 \\ 6 \ \end{pmatrix}\) looks as follows:

\[\begin{split}\left|\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\right\rangle \left\langle \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}\right|= \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \begin{pmatrix} 4 & 5 & 6 \end{pmatrix}= \begin{pmatrix} 1 \times 4 & 1 \times 5 & 1 \times 6 \\ 2 \times 4 & 2 \times 5 & 2 \times 6 \\ 3 \times 4 & 3 \times 5 & 3 \times 6 \end{pmatrix}= \begin{pmatrix} 4 & 5 & 6 \\ 8 & 10 & 12 \\ 12 & 15 & 18 \end{pmatrix}\end{split}\]

In toqito, this looks like this:

>>> import numpy as np
>>> from toqito.matrix_ops import outer_product
>>> v1, v2 = np.array([1,2,3]), np.array([4,5,6])
>>> outer_product(v1,v2)
array([[ 4,  5,  6],
       [ 8, 10, 12],
       [12, 15, 18]])

References

[1]

Wikipedia. Outer product. https://en.wikipedia.org/wiki/Outer_product.

Raises:

ValueError – Vector dimensions are mismatched.

Parameters:

• v1 (numpy.ndarray) – v1 and v2, both vectors of dimensions \((n,1)\) where \(n>1\).

• v2 (numpy.ndarray) – v1 and v2, both vectors of dimensions \((n,1)\) where \(n>1\).

Returns:

The computed outer product.

Return type:

numpy.ndarray