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 I and II. 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 = 𝒱m. 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 I and II 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

(i,kci,kϕI,iϕII,k,j,ldj,lϕI,jϕII,l)=i,j,k,l¯ci,kdj,l(ϕI,i,ϕI,j)(ϕII,k,ϕII,l).

Tensor products of operators

If we assume operators AI and AII acting on the Hilbert spaces I and II 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 as following:

A|ψ⟩ = (AI ⊗ AII)(|ψI⟩⊗|ψII⟩) = (AI|ψI⟩) ⊗ (AII|ψII⟩).

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 ⊗ idII, with idII being the identity map on II. Correspondingly the operator A restricted to subsystem II is A = idI ⊗ AII.

Examples

An example of the tensor product of two vectors ϕ ∈ ℂ2 and ψ ∈ ℂ2 is

(ϕ1ϕ2)(ψ1ψ2)=(ϕ1ψ1ϕ1ψ2ϕ2ψ1ϕ2ψ2). By rearranging this result we get the dyadic product of two vectors θij = ϕiψj, or

θ=(ϕ1ϕ2)(ψ1ψ2)=(ϕ1ϕ2)(ψ1,ψ2)=(ϕ1ψ1amp;ϕ1ψ2ϕ2ψ1amp;ϕ2ψ2)

Correspondingly, the tensor product of matrices A ∈ ℂk × l and B ∈ ℂm × n is given by the matrix

AB=(A11amp;amp;A1lamp;amp;Ak1amp;amp;Akl)(B11amp;amp;B1namp;amp;Bm1amp;amp;Bmn)=(A11Bamp;amp;A1lBamp;amp;Ak1Bamp;amp;AklB), with the entries

AijB=(AijB11amp;amp;AijB1namp;amp;AijBm1amp;amp;AijBmn).

Category:Linear Algebra Category:Handbook of Quantum Information

Last modified: 

Tuesday, November 3, 2015 - 20:24