In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and in particular
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, the tensor product of
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s is a way to extend the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
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 In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hi ...
. The tensor product allows Hilbert spaces to be collected into a
symmetric monoidal category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict s ...
.
[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
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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 and be two Hilbert spaces with inner products and respectively. Construct the tensor product of and as vector spaces as explained in the article on tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
s. We can turn this vector space tensor product into an inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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 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 and
Explicit construction
The tensor product can also be defined without appealing to the metric space completion. If and are two Hilbert spaces, one associates to every simple tensor product the rank one operator from to that maps a given as
This extends to a linear identification between and the space of finite rank operators from to The finite rank operators are embedded in the Hilbert space of Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm
\, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ...
s from to 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 and that is isometrically and linearly isomorphic to
Universal property
The Hilbert tensor product is characterized by the following universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
:
A weakly Hilbert-Schmidt mapping is defined as a bilinear map for which a real number exists, such that
for all and one (hence all) orthonormal basis of and of
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 In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict s ...
, and Hilbert spaces are a particular example thereof.
Infinite tensor products
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 [Bratteli, O. and Robinson, D: ''Operator Algebras and Quantum Statistical Mechanics v.1, 2nd ed.'', page 144. Springer-Verlag, 2002.]
Operator algebras
Let be the von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
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*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
s of operators, without defining reference states. This is one advantage of the "algebraic" method in quantum statistical mechanics.
Properties
If and have orthonormal bases
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
and respectively, then is an orthonormal basis for In particular, the Hilbert dimension of the tensor product is the product (as cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
s) of the Hilbert dimensions.
Examples and applications
The following examples show how tensor products arise naturally.
Given two measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
s 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 map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
ping 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. This shows that is isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. 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 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 space
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intr ...
s, which describe a variable number of particles.
References
Bibliography
* .
* .
{{Functional analysis
Functional analysis
Hilbert space
Linear algebra
Operator theory