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Tensor product

Tensor Products are used to describe systems consisting of multiple subsystems. Each subsystem is described by a vector in a vector space (Hilbert space). For example, let us have two systems $I$ and $I I$ with their corresponding Hilbert spaces $\mathcal{H}_I$ and $\mathcal{H}_{II}$ . Thus, using the bra-ket notation, the vectors $|\psi_I\rang$ and $|\psi_{II}\rang$ describe the states of system $I$ and $I I$ with the state of the total system given by the tensor product $|\psi_I \rang \otimes |\psi_{II}\rang$ .

The common way is to introduce tensor products for vector spaces $\mathcal{V}_I$ and $\mathcal{V}_{II}$ and their elements $ψ I$ and $ψ I I$ . The tensor product of both vector spaces $\mathcal{V} = \mathcal{V}_I \otimes \mathcal{V}_{II}$ is the vector space $\mathcal{V}$ of the overall system. If the dimensions of $\mathcal{V}_I$ and $\mathcal{V}_{II}$ are given by $\operatorname{dim}(\mathcal{V}_I)=n_I$ and $\operatorname{dim}(\mathcal{V}_{II})=n_{II}$ , the dimension of $\mathcal{V}$ is given by the product $\operatorname{dim}(\mathcal{V}) = n_In_{II}$ .

If the vectors $φ I, i$ form a base of $\mathcal{V}_I$ and similar $φ I I, j$ in $\mathcal{V}_{II}$ , we get the base vectors of $\mathcal{V}$ wih the tensor product $\phi_{ij} = \phi_{I,i} \otimes \phi_{II,j}$ . Using the bra-ket notation, the abbreviation $|ij\rang = |i\rang \otimes |j\rang$ is quite common. The $m$ -fold tensor product of a vector space is denoted by $\mathcal{V} \otimes \mathcal{V} \otimes \ldots \otimes \mathcal{V} = \mathcal{V}^{\otimes m}$ . Each element of $\mathcal{V}$ can be written as a linear combination

$\sum_{ij} c_{ij} \phi_{I,i} \otimes \phi_{II,j} = \psi \in \mathcal{V}$ .

The tensor product is linear in both factors. Contrary to the common multiplication it is not necessarily commutative as each factor corresponds to an element of different vector spaces.

If we have Hilbert spaces $\mathcal{H}_I$ and $\mathcal{H}_{II}$ instead of vector spaces, the inner product or scalar product of $\mathcal{H} = \mathcal{H}_I \otimes \mathcal{H}_{II}$ is given by

$\left(\phi_{I} \otimes \phi_{II}, \psi_{I} \otimes \psi_{II}\right) = \left(\phi_I,\psi_I\right)\left(\phi_{II},\psi_{II}\right)$ .

More general we can write

$\left(\sum_{i,k} c_{i,k} \cdot \phi_{I,i} \otimes \phi_{II,k}, \sum_{j,l} d_{j,l} \cdot \phi_{I,j} \otimes \phi_{II,l}\right) = \sum_{i,j,k,l} \overline{c_{i,k}} d_{j,l} \cdot \left(\phi_{I,i},\phi_{I,j}\right)\left(\phi_{II,k},\phi_{II,l}\right)$ .

Tensor products of operators

If we assume operators $A I$ and $A I I$ acting on the Hilbert spaces $\mathcal{H}_I$ and $\mathcal{H}_{II}$ we can derive an operator acting on $\mathcal{H}=\mathcal{H}_{I} \otimes \mathcal{H}_{II}$ . This operator $A$ is defined by the tensor product $A = A_I \otimes A_{II}$ and acts on the elements of $\mathcal{H}$ as following:

$A |\psi\rang = (A_I \otimes A_{II})\left(|\psi_I\rang \otimes |\psi_{II}\rang\right) = \left(A_I|\psi_I\rang\right) \otimes \left(A_{II}|\psi_{II}\rang\right)$ .

For linear operators $A I$ and $A I I$ , $A$ is a linear operator, too. This property of the tensor product is valid for some more important operator properties, that are unitarity, positivity, normality, Hermiticity and the adjoint. Similar to the elements of the vector space of the overall system, every operator $T$ can be written as a linear combination

$T=\sum_{i,j}t_{i,j}A_{I,i} \otimes A_{II,j}$ .

If an operator $A$ is restricted to the subsystem $I$ we can write $A = A_I \otimes \operatorname{id}_{\mathcal{H}_{II}}$ , with $\operatorname{id}_{\mathcal{H}_{II}}$ being the identity map on $\mathcal{H}_{II}$ . Correspondingly the operator $A$ restricted to subsystem $I I$ is $A=\operatorname{id}_{\mathcal{H}_I} \otimes A_{II}$ .

Examples

An example of the tensor product of two vectors $\phi \in \mathbb{C}^2$ and $\psi \in \mathbb{C}^2$ is

$\left(\begin{matrix}\phi_1\\\phi_2\end{matrix}\right) \otimes \left(\begin{matrix}\psi_1\\\psi_2\end{matrix}\right) = \left(\begin{matrix}\phi_1\psi_1\\\phi_1\psi_2\\\phi_2\psi_1\\\phi_2\psi_2\end{matrix}\right)$ .

By rearranging this result we get the dyadic product of two vectors $θ i j = φ i ψ j$ , or

$\theta = \left(\begin{matrix}\phi_1\\\phi_2\end{matrix}\right) \otimes \left(\begin{matrix}\psi_1\\\psi_2\end{matrix}\right) = \left(\begin{matrix}\phi_1\\\phi_2\end{matrix}\right) \cdot \left(\psi_1, \psi_2\right) = \left(\begin{matrix}\phi_1\psi_1&\phi_1\psi_2\\\phi_2\psi_1&\phi_2\psi_2\end{matrix}\right)$

Correspondingly, the tensor product of matrices $A \in \mathbb{C}^{k\times l}$ and $B \in \mathbb{C}^{m\times n}$ is given by the matrix

$A \otimes B = \left(\begin{matrix}A_{11}&\ldots&A_{1l}\\\vdots&\ddots&\vdots\\A_{k1}&\ldots&A_{kl}\end{matrix}\right) \otimes \left(\begin{matrix}B_{11}&\ldots&B_{1n}\\\vdots&\ddots&\vdots\\B_{m1}&\ldots&B_{mn}\end{matrix}\right) = \left(\begin{matrix}A_{11}B&\ldots&A_{1l}B\\\vdots&\ddots&\vdots\\A_{k1}B&\ldots&A_{kl}B\end{matrix}\right)$ ,