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 HI and HII. Thus, using the bra-ket notation, the vectors ψI and ψII describe the states of system I and II with the state of the total system given by the tensor product ψI⟩ ⊗ ∣ψII.

The common way is to introduce tensor products for vector spaces VI and VII and their elements ψI and ψII. The tensor product of both vector spaces V = VI ⊗ VII is the vector space V of the overall system. If the dimensions of VI and VII are given by dim(VI) = nI and dim(VII) = nII, the dimension of V is given by the product dim(V) = nInII.

If the vectors ϕI, i form a base of VI and similar ϕII, j in VII, we get the base vectors of V wih the tensor product ϕij = ϕI, i ⊗ ϕII, j. Using the bra-ket notation, the abbreviation ij⟩ = ∣i⟩ ⊗ ∣j is quite common. The m-fold tensor product of a vector space is denoted by V ⊗ V ⊗ … ⊗ V = V ⊗ m. Each element of V can be written as a linear combination

ijcijϕI, i ⊗ ϕII, j = ψ ∈ 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 HI and HII instead of vector spaces, the inner product or scalar product of H = HI ⊗ HII is given by

(ϕI ⊗ ϕII, ψI ⊗ ψII) = (ϕI, ψI)(ϕII, ψII)
. 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 AI and AII acting on the Hilbert spaces HI and HII we can derive an operator acting on H = HI ⊗ HII. This operator A is defined by the tensor product A = AI ⊗ AII and acts on the elements of H as following:

Aψ⟩ = (AI ⊗ AII)(∣ψI⟩ ⊗ ∣ψII⟩) = (AIψI⟩) ⊗ (AIIψII⟩)

For linear operators AI and AII, 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 = ∑i, jti, jAI, i ⊗ AII, j

If an operator A is restricted to the subsystem I we can write A = AI ⊗ idHII, with idHII being the identity map on HII. Correspondingly the operator A restricted to subsystem II is A = idHI ⊗ AII.


An example of the tensor product of two vectors ϕ ∈ C2 and ψ ∈ C2 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 θij = ϕiψj, 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 ∈ Ck × l and B ∈ Cm × n is given by the matrix

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

$$A_{ij}B = \left(\begin{matrix}A_{ij}B_{11}&\ldots&A_{ij}B_{1n}\\\vdots&\ddots&\vdots\\A_{ij}B_{m1}&\ldots&A_{ij}B_{mn}\end{matrix}\right)$$

Category:Linear Algebra Category:Handbook of Quantum Information

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Tuesday, November 3, 2015 - 20:24