Almost Commutative Kronecker Product
Introduction
Tensor product and Kronecker product are very important in quantum mechanics. It also have practical physical meanings for quantum processes. One of the interesting properties of Kronecker product is that it is “almost commutative”.
In this blog post, I would like to informally discuss the “almost commutative” property for Kronecker product.
Outer Product
Given two vectors
Using index notation,
If
The mapping of outer product could be described as
Tensor Product
Tensor product is essentially an general case of one dimensional array outer product. It is the outer product of two tensors, namely multidimensional tensors, that could have different dimensionality.
If
Kronecker Product
The outer product and Kronecker product are closely related. In fact the same symbol is commonly used to denote both operations.
If
Note that there is an implicit dimensional reduction process in the above expression.
The mapping of outer product could be described as
More concretely, let
Almost Commutative
Kronecker product is not commutative, i.e., usually
Suppose we have
We rearrange the columns of
We further rearrange the rows of
Therefore,
This means, there exist permutation matrices
Miscellaneous
In Numpy, the tensor product could be computed using numpy.ufunc.outer, and the Kronecker product could be computed using numpy.kron.
Almost Commutative Kronecker Product
https://leimao.github.io/blog/Almost-Commutative-Kronecker-Product/