New answers tagged matrix-representation
1
In general, given two matrices $A$ and $B$ of dimensions $n_1\times n_2$ and $m_1\times m_2$, respectively, their tensor product $A\otimes B$ can be represented using the Kronecker product as
$$(A\otimes B)_{n_1 m_1,n_2m_2}=A_{n_1,n_2}B_{m_1, m_2}.$$
The indices on the left hand side are a standard way to enumerate the integers from $1$ to $n_1 m_1$ and from ...
1
What you are looking for is https://en.wikipedia.org/wiki/Kronecker_product
Note that a column-vector can be considered as a matrix with the size $n \times 1$, so the Kronecker product rule also applies.
5
$\newcommand{\bs}[1]{{\boldsymbol #1}}
\newcommand{\tildebssigma}{\tilde{\bs\sigma}}
\newcommand{\bssigma}{{\bs\sigma}}$Yes, products of Pauli matrices form a basis for the set of Hermitian matrices (of dimensions that are powers of $2$).
More specifically, fix an integer $n$ and let $N\equiv 2^n$, define $\bssigma\equiv(\sigma_x,\sigma_y,\sigma_y)$, and $\...
6
For any matrix $A$ we can write
$$
A =\sum_{i,j,k,l}h_{ijkl}\cdot \frac{1}{4}\sigma_i\otimes\sigma_j\otimes\sigma_k\otimes\sigma_l,
$$
where
$$
h_{ijkl} = \frac{1}{4}\text{Tr}\big((\sigma_i\otimes\sigma_j\otimes\sigma_k\otimes\sigma_l)^\dagger \cdot A\big) = \frac{1}{4}\text{Tr}\big((\sigma_i\otimes\sigma_j\otimes\sigma_k\otimes\sigma_l) \cdot A\big)
$$
...
Top 50 recent answers are included
