Tensor product of Hilbert spaces

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.[1]

DefinitionEdit

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  , 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

 

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 constructionEdit

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   the rank one operator from   to H2 that maps a given   as

 

This extends to a linear identification between   and the space of finite rank operators from   to H2. The finite rank operators are embedded in the Hilbert space   of Hilbert–Schmidt operators from   to H2. The scalar product in   is given by

 

where   is an arbitrary orthonormal basis of  

Under the preceding identification, one can define the Hilbertian tensor product of H1 and H2, that is isometrically and linearly isomorphic to  

Universal propertyEdit

The Hilbert tensor product   is characterized by the following universal property (Kadison & Ringrose 1997, Theorem 2.6.4):

There is a weakly Hilbert–Schmidt mapping p : H1 × H2 → H such that, given any weakly Hilbert–Schmidt mapping L : H1 × H2 → K to a Hilbert space K, there is a unique bounded operator T : H → K such that L = Tp.

A weakly Hilbert-Schmidt mapping L : H1 × H2 → K is defined as a bilinear map for which a real number d exists, such that

 

for all   and one (hence all) orthonormal basis e1, e2, ... of H1 and f1, f2, ... of H2.

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 productsEdit

If   is a collection of Hilbert spaces and   is a collection of unit vectors in these Hilbert spaces then the incomplete tensor product (or Guichardet tensor product) is the   completion of the set of all finite linear combinations of simple tensor vectors   where all but finitely many of the  's equal the corresponding  .[2]

Operator algebrasEdit

Let   be the von Neumann algebra of bounded operators on   for   Then the von Neumann tensor product of the von Neumann algebras is the strong completion of the set of all finite linear combinations of simple tensor products   where   for   This is exactly equal to the von Neumann algebra of bounded operators of   Unlike for Hilbert spaces, one may take infinite tensor products of von Neumann algebras, and for that matter C*-algebras of operators, without defining reference states.[2] This is one advantage of the "algebraic" method in quantum statistical mechanics.

PropertiesEdit

If   and   have orthonormal bases   and   respectively, then   is an orthonormal basis for   In particular, the Hilbert dimension of the tensor product is the product (as cardinal numbers) of the Hilbert dimensions.

Examples and applicationsEdit

The following examples show how tensor products arise naturally.

Given two measure spaces   and  , with measures   and   respectively, one may look at  , the space of functions on   that are square integrable with respect to the product measure   If   is a square integrable function on   and   is a square integrable function on   then we can define a function   on   by   The definition of the product measure ensures that all functions of this form are square integrable, so this defines a bilinear mapping   Linear combinations of functions of the form   are also in  . It turns out that the set of linear combinations is in fact dense in   if   and   are separable.[citation needed] This shows that   is isomorphic to   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  , denoting the space of square integrable functions  , is isomorphic to   if this space is separable. The isomorphism maps   to   We can combine this with the previous example and conclude that   and   are both isomorphic to  

Tensor products of Hilbert spaces arise often in quantum mechanics. If some particle is described by the Hilbert space   and another particle is described by   then the system consisting of both particles is described by the tensor product of   and   For example, the state space of a quantum harmonic oscillator is   so the state space of two oscillators is   which is isomorphic to  . Therefore, the two-particle system is described by wave functions of the form   A more intricate example is provided by the Fock spaces, which describe a variable number of particles.

ReferencesEdit

  1. ^ B. Coecke and E. O. Paquette, Categories for the practising physicist, in: New Structures for Physics, B. Coecke (ed.), Springer Lecture Notes in Physics, 2009. arXiv:0905.3010
  2. ^ a b Bratteli, O. and Robinson, D: Operator Algebras and Quantum Statistical Mechanics v.1, 2nd ed., page 144. Springer-Verlag, 2002.

BibliographyEdit

  • Kadison, Richard V.; Ringrose, John R. (1997). Fundamentals of the theory of operator algebras. Vol. I. Graduate Studies in Mathematics. Vol. 15. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-0819-1. MR 1468229..
  • Weidmann, Joachim (1980). Linear operators in Hilbert spaces. Graduate Texts in Mathematics. Vol. 68. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90427-6. MR 0566954..