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 $II$ with their corresponding Hilbert spaces $\backslash mathcal\{H\}\_I$ and $\backslash mathcal\{H\}\_\{II\}$. Thus, using the [[bra-ket notation]], the vectors $|\backslash psi\_I\backslash rangle$ and $|\backslash psi\_\{II\}\backslash rangle$ describe the states of system $I$ and $II$ with the state of the total system given by the tensor product $|\backslash psi\_I\; \backslash rangle\; \backslash otimes\; |\backslash psi\_\{II\}\backslash rangle$. The common way is to introduce tensor products for vector spaces $\backslash mathcal\{V\}\_I$ and $\backslash mathcal\{V\}\_\{II\}$ and their elements $\backslash psi\_I$ and $\backslash psi\_\{II\}$. The tensor product of both vector spaces $\backslash mathcal\{V\}\; =\; \backslash mathcal\{V\}\_I\; \backslash otimes\; \backslash mathcal\{V\}\_\{II\}$ is the vector space $\backslash mathcal\{V\}$ of the overall system. If the dimensions of $\backslash mathcal\{V\}\_I$ and $\backslash mathcal\{V\}\_\{II\}$ are given by $\backslash operatorname\{dim\}(\backslash mathcal\{V\}\_I)=n\_I$ and $\backslash operatorname\{dim\}(\backslash mathcal\{V\}\_\{II\})=n\_\{II\}$, the dimension of $\backslash mathcal\{V\}$ is given by the product $\backslash operatorname\{dim\}(\backslash mathcal\{V\})\; =\; n\_In\_\{II\}$. If the vectors $\backslash phi\_\{I,i\}$ form a base of $\backslash mathcal\{V\}\_I$ and similar $\backslash phi\_\{II,j\}$ in $\backslash mathcal\{V\}\_\{II\}$, we get the base vectors of $\backslash mathcal\{V\}$ wih the tensor product $\backslash phi\_\{ij\}\; =\; \backslash phi\_\{I,i\}\; \backslash otimes\; \backslash phi\_\{II,j\}$. Using the [[bra-ket notation]], the abbreviation $|ij\backslash rangle\; =\; |i\backslash rangle\; \backslash otimes\; |j\backslash rangle$ is quite common. The $m$-fold tensor product of a vector space is denoted by $\backslash mathcal\{V\}\; \backslash otimes\; \backslash mathcal\{V\}\; \backslash otimes\; \backslash ldots\; \backslash otimes\; \backslash mathcal\{V\}\; =\; \backslash mathcal\{V\}^\{\backslash otimes\; m\}$. Each element of $\backslash mathcal\{V\}$ can be written as a linear combination :$\backslash sum\_\{ij\}\; c\_\{ij\}\; \backslash phi\_\{I,i\}\; \backslash otimes\; \backslash phi\_\{II,j\}\; =\; \backslash psi\; \backslash in\; \backslash mathcal\{V\}$. 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 $\backslash mathcal\{H\}\_I$ and $\backslash mathcal\{H\}\_\{II\}$ instead of vector spaces, the inner product or scalar product of $\backslash mathcal\{H\}\; =\; \backslash mathcal\{H\}\_I\; \backslash otimes\; \backslash mathcal\{H\}\_\{II\}$ is given by :$\backslash left(\backslash phi\_\{I\}\; \backslash otimes\; \backslash phi\_\{II\},\; \backslash psi\_\{I\}\; \backslash otimes\; \backslash psi\_\{II\}\backslash right)\; =\; \backslash left(\backslash phi\_I,\backslash psi\_I\backslash right)\backslash left(\backslash phi\_\{II\},\backslash psi\_\{II\}\backslash right)$. More general we can write :$\backslash left(\backslash sum\_\{i,k\}\; c\_\{i,k\}\; \backslash cdot\; \backslash phi\_\{I,i\}\; \backslash otimes\; \backslash phi\_\{II,k\},\; \backslash sum\_\{j,l\}\; d\_\{j,l\}\; \backslash cdot\; \backslash phi\_\{I,j\}\; \backslash otimes\; \backslash phi\_\{II,l\}\backslash right)\; =\; \backslash sum\_\{i,j,k,l\}\; \backslash overline\{c\_\{i,k\}\}\; d\_\{j,l\}\; \backslash cdot\; \backslash left(\backslash phi\_\{I,i\},\backslash phi\_\{I,j\}\backslash right)\backslash left(\backslash phi\_\{II,k\},\backslash phi\_\{II,l\}\backslash right)$. == Tensor products of operators == If we assume operators $A\_I$ and $A\_\{II\}$ acting on the Hilbert spaces $\backslash mathcal\{H\}\_I$ and $\backslash mathcal\{H\}\_\{II\}$ we can derive an operator acting on $\backslash mathcal\{H\}=\backslash mathcal\{H\}\_\{I\}\; \backslash otimes\; \backslash mathcal\{H\}\_\{II\}$. This operator $A$ is defined by the tensor product $A\; =\; A\_I\; \backslash otimes\; A\_\{II\}$ and acts on the elements of $\backslash mathcal\{H\}$ as following: :$A\; |\backslash psi\backslash rangle\; =\; (A\_I\; \backslash otimes\; A\_\{II\})\backslash left(|\backslash psi\_I\backslash rangle\; \backslash otimes\; |\backslash psi\_\{II\}\backslash rangle\backslash right)\; =\; \backslash left(A\_I|\backslash psi\_I\backslash rangle\backslash right)\; \backslash otimes\; \backslash left(A\_\{II\}|\backslash psi\_\{II\}\backslash rangle\backslash right)$. For linear operators $A\_I$ and $A\_\{II\}$, $A$ 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 :$T=\backslash sum\_\{i,j\}t\_\{i,j\}A\_\{I,i\}\; \backslash otimes\; A\_\{II,j\}$. If an operator $A$ is restricted to the subsystem $I$ we can write $A\; =\; A\_I\; \backslash otimes\; \backslash operatorname\{id\}\_\{\backslash mathcal\{H\}\_\{II\}\}$, with $\backslash operatorname\{id\}\_\{\backslash mathcal\{H\}\_\{II\}\}$ being the identity map on $\backslash mathcal\{H\}\_\{II\}$. Correspondingly the operator $A$ restricted to subsystem $II$ is $A=\backslash operatorname\{id\}\_\{\backslash mathcal\{H\}\_I\}\; \backslash otimes\; A\_\{II\}$. == Examples == An example of the tensor product of two vectors $\backslash phi\; \backslash in\; \backslash mathbb\{C\}^2$ and $\backslash psi\; \backslash in\; \backslash mathbb\{C\}^2$ is :$\backslash left(\backslash begin\{matrix\}\backslash phi\_1\backslash \backslash \backslash phi\_2\backslash end\{matrix\}\backslash right)\; \backslash otimes\; \backslash left(\backslash begin\{matrix\}\backslash psi\_1\backslash \backslash \backslash psi\_2\backslash end\{matrix\}\backslash right)\; =\; \backslash left(\backslash begin\{matrix\}\backslash phi\_1\backslash psi\_1\backslash \backslash \backslash phi\_1\backslash psi\_2\backslash \backslash \backslash phi\_2\backslash psi\_1\backslash \backslash \backslash phi\_2\backslash psi\_2\backslash end\{matrix\}\backslash right)$. By rearranging this result we get the dyadic product of two vectors $\backslash theta\_\{ij\}\; =\; \backslash phi\_i\backslash psi\_j$, or : Correspondingly, the tensor product of matrices $A\; \backslash in\; \backslash mathbb\{C\}^\{k\backslash times\; l\}$ and $B\; \backslash in\; \backslash mathbb\{C\}^\{m\backslash times\; n\}$ is given by the matrix :, with the entries :. [[Category:Linear Algebra]] [[Category:Handbook of Quantum Information]]