Quantum Computing Explained

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TENSOR PRODUCTS

In quantum mechanics we don’t always work with single particles in isolation. In many cases, some of which are seen in the context of quantum information processing, it is necessary to work with multiparticle states. Mathematically, to understand multiparticle systems in quantum mechanics, it is necessary to be able to construct a Hilbert space H that is a composite of the independent Hilbert spaces that are associated with each individual particle. The machinery required to do this goes by the name of the Kronecker or tensor product. We consider the two-particle case.

Suppose that H₁ and H₂ are two Hilbert spaces of dimension N₁ and N₂. We can put these two Hilbert spaces together to construct a larger Hilbert space. We denote this larger space by H and use the tensor product operation symbol ⊗. So we write

The dimension of the larger Hilbert space is the product of the dimensions of H₁ and H₂. Once again, we assume that dim(H₁) = N₁ and dim(H₂) = N₂. Then

Next we start getting down to business and learn how to represent state vectors in the composite Hilbert space.

REPRESENTING COMPOSITE STATES IN QUANTUM MECHANICS

A state vector belonging to H is the tensor product of state vectors belonging to H1 and H2. We will show how to represent such vectors explicitly ...