matrix_ops.tensor

Tensor product operation.

Module Contents

Functions

tensor(*args)

Compute the Kronecker tensor product [1].

matrix_ops.tensor.tensor(*args)

Compute the Kronecker tensor product [1].

Tensor two matrices or vectors together using the standard Kronecker operation provided from numpy.

Given two matrices \(A\) and \(B\), computes \(A \otimes B\). The same concept also applies to two vectors \(v\) and \(w\) which computes \(v \otimes w\).

One may also compute the tensor product one matrix \(n\) times with itself.

For a matrix, \(A\) and an integer \(n\), the result of this function computes \(A^{\otimes n}\).

Similarly for a vector \(v\) and an integer \(n\), the result of of this function computes \(v^{\otimes n}\).

One may also perform the tensor product on a list of matrices.

Given a list of \(n\) matrices \(A_1, A_2, \ldots, A_n\) the result of this function computes

\[A_1 \otimes A_2 \otimes \cdots \otimes A_n.\]

Similarly, for a list of \(n\) vectors \(v_1, v_2, \ldots, v_n\), the result of this function computes

\[v_1 \otimes v_2 \otimes \cdots \otimes v_n.\]

Examples

Tensor product two matrices or vectors

Consider the following ket vector

\[e_0 = \left[1, 0 \right]^{\text{T}}.\]

Computing the following tensor product

This can be accomplished in toqito as follows.

>>> from toqito.states import basis
>>> from toqito.matrix_ops import tensor
>>> e_0 = basis(2, 0)
>>> tensor(e_0, e_0)
array([[1],
       [0],
       [0],
       [0]])

Tensor product one matrix \(n\) times with itself.

We may also tensor some element with itself some integer number of times. For instance we can compute

\[e_0^{\otimes 3} = \left[1, 0, 0, 0, 0, 0, 0, 0 \right]^{\text{T}}\]

in toqito as follows.

>>> from toqito.states import basis
>>> from toqito.matrix_ops import tensor
>>> e_0 = basis(2, 0)
>>> tensor(e_0, 3)
array([[1],
       [0],
       [0],
       [0],
       [0],
       [0],
       [0],
       [0]])

Perform the tensor product on a list of vectors or matrices.

If we wish to compute the tensor product against more than two matrices or vectors, we can feed them in as a list. For instance, if we wish to compute \(e_0 \otimes e_1 \otimes e_0\), we can do so as follows.

>>> from toqito.states import basis
>>> from toqito.matrix_ops import tensor
>>> e_0, e_1 = basis(2, 0), basis(2, 1)
>>> tensor([e_0, e_1, e_0])
array([[0],
       [0],
       [1],
       [0],
       [0],
       [0],
       [0],
       [0]])

References

[1] (1,2)

Wikipedia. Tensor product. https://en.wikipedia.org/wiki/Tensor_product.

Raises:

ValueError – Input must be a vector or matrix.

Parameters:

args – Input to the tensor function is expected to be either: - list[np.ndarray]: List of numpy matrices, - np.ndarray, … , np.ndarray: An arbitrary number of numpy arrays, - np.ndarray, int: A numpy array and an integer.

Returns:

The computed tensor product.

Return type:

numpy.ndarray