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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
/ref>


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 H_1 and H_2 be two Hilbert spaces with inner products \langle\cdot, \cdot\rangle_1 and \langle\cdot, \cdot\rangle_2, respectively. Construct the tensor product of H_1 and H_2 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 \left\langle\phi_1 \otimes \phi_2, \psi_1 \otimes \psi_2\right\rangle = \left\langle\phi_1, \psi_1\right\rangle_1 \, \left\langle\phi_2, \psi_2\right\rangle_2 \quad \mbox \phi_1,\psi_1 \in H_1 \mbox \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 H_1 \times H_2 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 H_1 and H_2.


Explicit construction

The tensor product can also be defined without appealing to the metric space completion. If H_1 and H_2 are two Hilbert spaces, one associates to every simple tensor product x_1 \otimes x_2 the rank one operator from H_1^* to H_2 that maps a given x^*\in H^*_1 as x^* \mapsto x^*(x_1) \, x_2. This extends to a linear identification between H_1 \otimes H_2 and the space of finite rank operators from H_1^* to H_2. The finite rank operators are embedded in the Hilbert space HS(H_1^*, H_2) 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 H_1^* to H_2. The scalar product in HS(H_1^*, H_2) is given by \langle T_1, T_2 \rangle = \sum_n \left \langle T_1 e_n^*, T_2 e_n^* \right \rangle, where \left(e_n^*\right) is an arbitrary orthonormal basis of H_1^*. Under the preceding identification, one can define the Hilbertian tensor product of H_1 and H_2, that is isometrically and linearly isomorphic to HS(H_1^*, H_2).


Universal property

The Hilbert tensor product H = H_1 \otimes H_2 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 L : H_1 \times H_2 \to K is defined as a bilinear map for which a real number d exists, such that \sum_^\infty \bigl, \left\langle L(e_i, f_j), u \right \rangle\bigr, ^2 \leq d^2\,\, u\, ^2 for all u \in K and one (hence all) orthonormal basis e_1, e_2, \ldots of H_1 and f_1, f_2, \ldots of H_2. 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 H_n is a collection of Hilbert spaces and \xi_n is a collection of unit vectors in these Hilbert spaces then the incomplete tensor product (or Guichardet tensor product) is the L^2 completion of the set of all finite linear combinations of simple tensor vectors \bigotimes_^ \psi_n where all but finitely many of the \psi_n's equal the corresponding \xi_n.Bratteli, O. and Robinson, D: ''Operator Algebras and Quantum Statistical Mechanics v.1, 2nd ed.'', page 144. Springer-Verlag, 2002.


Operator algebras

Let \mathfrak_i 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 H_i for i=1,2. 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 A_1\otimes A_2 where A_i \in \mathfrak_i for i = 1, 2. This is exactly equal to the von Neumann algebra of bounded operators of H_1\otimes H_2. 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 H_1 and H_2 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, ...
\left\ and \left\, respectively, then \left\ is an orthonormal basis for H_1\otimes H_2. 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 X and Y, with measures \mu and \nu respectively, one may look at L^2(X \times Y), the space of functions on X \times Y that are square integrable with respect to the product measure \mu\times\nu. 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\times 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 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 L^2(X) \times L^2(Y) \to L^2(X\times Y). Linear combinations of functions of the form f(x) g(y) are also in L^2(X \times Y). It turns out that the set of linear combinations is in fact dense in L^2(X \times Y), if L^2(X) and L^2(Y) are separable. This shows that L^2(X) \otimes L^2(Y) 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 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 L^2(X;H), denoting the space of square integrable functions X \to H, is isomorphic to L^2(X) \otimes H if this space is separable. The isomorphism maps f(x) \otimes \phi \in L^2(X)\otimes H to f(x) \phi \in L^2(X;H) We can combine this with the previous example and conclude that L^2(X) \otimes L^2(Y) and L^2(X \times Y) are both isomorphic to L^2\left(X; L^2(Y)\right). 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 H_1, and another particle is described by H_2, then the system consisting of both particles is described by the tensor product of H_1 and H_2. For example, the state space of a
quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最� ...
is L^2(\R), so the state space of two oscillators is L^2(\R) \otimes L^2(\R), which is isomorphic to L^2\left(\R^2\right). Therefore, the two-particle system is described by wave functions of the form \psi\left(x_1, x_2\right). 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