Tensor product of Hilbert spaces

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 ${\displaystyle \langle \cdot ,\cdot \rangle _{1}}$ and ${\displaystyle \langle \cdot ,\cdot \rangle _{2}}$, respectively. Construct the tensor product of H₁ and H₂ as vector spaces as explained in the article on tensor products. We can turn this vector space tensor product into an inner product space by defining

${\displaystyle \langle \phi _{1}\otimes \phi _{2},\psi _{1}\otimes \psi _{2}\rangle =\langle \phi _{1},\psi _{1}\rangle _{1}\,\langle \phi _{2},\psi _{2}\rangle _{2}\quad {\mbox{for all }}\phi _{1},\psi _{1}\in H_{1}{\mbox{ and }}\phi _{2},\psi _{2}\in H_{2}}$

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.

If ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ have orthonormal bases ${\displaystyle \{\phi _{k}\}}$ and ${\displaystyle \{\psi _{l}\},}$ respectively, then ${\displaystyle \{\phi _{k}\otimes \psi _{l}\}}$ is an orthonormal basis for ${\displaystyle H_{1}\otimes H_{2}.}$ In particular, the Hilbert dimension of the tensor product is the product (as cardinal numbers) of the Hilbert dimensions.

The following examples show how tensor products arise naturally.

Given two measure spaces ${\displaystyle X}$ and ${\displaystyle Y}$, with measures ${\displaystyle \mu }$ and ${\displaystyle \nu }$ respectively, one may look at ${\displaystyle L^{2}(X\times Y)}$, the space of functions on ${\displaystyle X\times Y}$ that are square integrable with respect to the product measure ${\displaystyle \mu \times \nu .}$ If ${\displaystyle f}$ is a square integrable function on ${\displaystyle X,}$ and ${\displaystyle g}$ is a square integrable function on ${\displaystyle Y,}$ then we can define a function ${\displaystyle h}$ on ${\displaystyle X\times Y}$ by ${\displaystyle h(x,y)=f(x)g(y).}$ The definition of the product measure ensures that all functions of this form are square integrable, so this defines a bilinear mapping ${\displaystyle L^{2}(X)\times L^{2}(Y)\to L^{2}(X\times Y).}$ Linear combinations of functions of the form ${\displaystyle f(x)g(y)}$ are also in ${\displaystyle L^{2}(X\times Y)}$. It turns out that the set of linear combinations is in fact dense in ${\displaystyle L^{2}(X\times Y),}$ if ${\displaystyle L^{2}(X)}$ and ${\displaystyle L^{2}(Y)}$ are separable. This shows that ${\displaystyle L^{2}(X)\otimes L^{2}(Y)}$ is isomorphic to ${\displaystyle L^{2}(X\times Y),}$ 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 ${\displaystyle L^{2}(X;H)}$, denoting the space of square integrable functions ${\displaystyle X\to H}$, is isomorphic to ${\displaystyle L^{2}(X)\otimes H}$ if this space is separable. The isomorphism maps ${\displaystyle f(x)\otimes \phi \in L^{2}(X)\otimes H}$ to ${\displaystyle f(x)\phi \in L^{2}(X;H)}$ We can combine this with the previous example and conclude that ${\displaystyle L^{2}(X)\otimes L^{2}(Y)}$ and ${\displaystyle L^{2}(X\times Y)}$ are both isomorphic to ${\displaystyle L^{2}(X;L^{2}(Y)).}$

Tensor products of Hilbert spaces arise often in quantum mechanics. If some particle is described by the Hilbert space ${\displaystyle H_{1},}$ and another particle is described by ${\displaystyle H_{2},}$ then the system consisting of both particles is described by the tensor product of ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}.}$ For example, the state space of a quantum harmonic oscillator is ${\displaystyle L^{2}(\mathbb {R} ),}$ so the state space of two oscillators is ${\displaystyle L^{2}(\mathbb {R} )\otimes L^{2}(\mathbb {R} ),}$ which is isomorphic to ${\displaystyle L^{2}(\mathbb {R} ^{2})}$. Therefore, the two-particle system is described by wave functions of the form ${\displaystyle \psi (x_{1},x_{2}).}$ A more intricate example is provided by the Fock spaces, which describe a variable number of particles.

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