Tensor product

Take two vector spaces $V$ and $W$ , and two states in those spaces $\ket v$ and $\ket w$ . To represent the state corresponding to both $\ket v$ and $\ket w$ simultaneously (such as a multi particle system), we use a tensor product

\begin{align*} \ket v \otimes \ket w. \end{align*}

This tensor product is in the tensor product space $V \otimes W$ , which is itself a Hilbert space. We define the usual vector operations.

Scalar multiplication: $\alpha (\ket v \otimes \ket w) = (\alpha \ket v) \otimes \ket w = \ket v \otimes (\alpha \ket w)$ .

Addition: $(\ket{v_1} \otimes \ket {w}) + (\ket{v_2} \otimes \ket w) = (\ket{v_1} + \ket{v_2}) \otimes \ket w$ .

The zero vector has either of the tensor product states $\ket 0$ . The dimension of the tensor product space $\dim(V \otimes W) = \dim V \dim W$ .

The basis of $V \otimes W$ can be written in terms of the basis of the original vector spaces. Let $\{ \ket{e_i} \}, \{ \ket{f_i} \}$ be bases of $V,W$ . Then

\begin{align*} \{ \ket{e_i} \otimes \ket{f_j} \forall \,i,j \}. \end{align*}

Is a basis for $V \otimes W$ .

Within this space there are two important classes of states. A product state is the outer product of two states, and can be written in a way that the coefficients depend independently on the index $i$ and $j$

\begin{align*} \ket v \otimes \ket w = \sum_{ij} v_i w_j \ket{e_i} \otimes \ket{f_j}. \end{align*}

By contrast, an entangled state is the sum of several outer products. The coefficients cannot be written independently like they can for a product state

\begin{align*} \ket a \otimes \ket b + \ket c \otimes \ket d = \sum_{ij} q_{ij} \ket{e_i} \otimes \ket{f_j}. \end{align*}

Entangled states are one of the very cool things about quantum mechanics! Learning something about one particle can tell you some information about another particle without interacting with it.