In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.
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Definition
Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arise naturally from those of the factors. Let H1 and H2 be two Hilbert spaces with inner products
and extending by linearity. That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on H1 × H2 and linear functionals on their vector space tensor product. Finally, take the completion under this inner product. The resulting Hilbert space is the tensor product of H1 and H2.
Explicit construction
The tensor product can also be defined without appealing to the metric space completion. If H1 and H2 are two Hilbert spaces, one associates to every simple tensor product
This extends to a linear identification between
where
Under the preceding identification, one can define the Hilbertian tensor product of H1 and H2, that is isometrically and linearly isomorphic to HS(H1∗, H2).
Universal property
The Hilbert tensor product
A weakly Hilbert-Schmidt mapping L : H1 × H2 → K is defined as a bilinear map for which a real number d exists, such that
As with any universal property, this characterizes the tensor product H uniquely, up to isomorphism. The same universal property, with obvious modifications, also applies for the tensor product of any finite number of Hilbert spaces. It is essentially the same universal property shared by all definitions of tensor products, irrespective of the spaces being tensored: this implies that any space with a tensor product is a symmetric monoidal category, and Hilbert spaces are a particular example thereof.
Infinite tensor products
If
Operator algebras
Let
Properties
If H1 and H2 have orthonormal bases {φk} and {ψl}, respectively, then {φk ⊗ ψl} is an orthonormal basis for H1 ⊗ H2. In particular, the Hilbert dimension of the tensor product is the product (as cardinal numbers) of the Hilbert dimensions.
Examples and applications
The following examples show how tensor products arise naturally.
Given two measure spaces X and Y, with measures μ and ν respectively, one may look at L2(X × Y), the space of functions on X × Y that are square integrable with respect to the product measure μ × ν. If f is a square integrable function on X, and g is a square integrable function on Y, then we can define a function h on X × Y by h(x,y) = f(x) g(y). The definition of the product measure ensures that all functions of this form are square integrable, so this defines a bilinear mapping L2(X) × L2(Y) → L2(X × Y). Linear combinations of functions of the form f(x) g(y) are also in L2(X × Y). It turns out that the set of linear combinations is in fact dense in L2(X × Y), if L2(X) and L2(Y) are separable. This shows that L2(X) ⊗ L2(Y) is isomorphic to L2(X × Y), and it also explains why we need to take the completion in the construction of the Hilbert space tensor product.
Similarly, we can show that L2(X; H), denoting the space of square integrable functions X → H, is isomorphic to L2(X) ⊗ H if this space is separable. The isomorphism maps f(x) ⊗ φ ∈ L2(X) ⊗ H to f(x)φ ∈ L2(X; H). We can combine this with the previous example and conclude that L2(X) ⊗ L2(Y) and L2(X × Y) are both isomorphic to L2(X; L2(Y)).
Tensor products of Hilbert spaces arise often in quantum mechanics. If some particle is described by the Hilbert space H1, and another particle is described by H2, then the system consisting of both particles is described by the tensor product of H1 and H2. For example, the state space of a quantum harmonic oscillator is L2(R), so the state space of two oscillators is L2(R) ⊗ L2(R), which is isomorphic to L2(R2). Therefore, the two-particle system is described by wave functions of the form φ(x1, x2). A more intricate example is provided by the Fock spaces, which describe a variable number of particles.