Intuitively, tensor products (also known as outer products) is a way of capturing all possible combinations of basis vectors.
A basis provides a way to write down quantum states in a defined matter. For e.g. a quantum coin has two outcomes (heads and tails) so the state of the quantum coin can be represented in the basis , where the state is represented as the linear combination of the basis .
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A quantum die (with 6 basis vectors) and a quantum coin (with two basis vectors) can be considered to be 2 different vector spaces with its own set of basis vectors, so that the tensor product of the two would capture all the possible combinations
Formal definition = The tensor product of two vector spaces V and W is a vector space, containing a bilinear map () from the cartesian product to
any two vector spaces V and W can be combined with the tensor product to generate a new vector space
Consider two vectors:
a_{1} \\ a_{2} \end{bmatrix} , b = \begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \end{bmatrix}$$ $$a \otimes b = \begin{bmatrix} a_{1} (b) \\ a_{2} (b) \end{bmatrix} = \begin{bmatrix} a_{1} (\begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \end{bmatrix}) \\ a_{2} (\begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \end{bmatrix}) \end{bmatrix} = \begin{bmatrix} a_{1} b_{1} \\ a_{1} b_{2} \\ a_{1} b_{3} \\ a_{2} b_{1} \\ a_{2} b_{2} \\ a_{3} b_{3} \end{bmatrix} $$ ##### applications in quantum theory In [[Quantum Physics]], the combined system for n systems with individual quantum states = $$\ket{\Phi} = \ket{\psi_{0}} \otimes \ket{\psi_{1}} \otimes \ket{\psi_{2}} . . . $$