# Tensor Product

- [[cartesian-product]]
- [[kronecker-product]]
- on two [[quantum-state]]s:
  - take the cartesian product of the set of component states, this is the new set of states
  - the scalar attached to each new state is the product of the two component scalars
  - $`\left(\alpha\ket{0}+\beta\ket{1}\right)\otimes\left(\gamma\ket{0}+\delta\ket{1}\right) = \alpha\gamma\ket{00} + \alpha\delta\ket{01} + \beta\gamma\ket{10} + \beta\delta\ket{11}`$
- on two [[unitary-operator]]s:
  - $`\mathbf{A} \otimes \mathbf{B} = \begin{bmatrix} a_{11}\mathbf{B} & \cdots & a_{1n}\mathbf{B} \\ \vdots & \ddots & \vdots \\ a_{m1} \mathbf{B} & \cdots & a_{mn} \mathbf{B} \end{bmatrix}`$
- $`\left(\mathbf{A} \otimes \mathbf{B}\right)\left(\mathbf{C} \otimes \mathbf{D}\right) = \mathbf{A}\mathbf{C} \otimes \mathbf{B}\mathbf{D}`$