simple tensor
The tensor product (http://planetmath.org/TensorProduct) \PMlinkescapephrasetensor product
of two vector spaces
and is
another vector space which is characterised by being universal
for bilinear maps on . As part of this package,
there is an operation
on vectors such that
for all
and , and the primary subject
of this article is the image of that operation.
Definition 1.
The element is said to be a simple tensor if there exist and such that .
More generally, the element is said to be a simple tensor (with respect to the decomposition of ) if there exist for such that .
For this definition to be interesting, there must also be tensors
which are not simple, and indeed most tensors arenโt. In order to
illustrate why, it is convenient to consider the tensor product of
two finite-dimensional vector spaces and
over some field . In this case one can let (the vector space of matrices), since
is isomorphic to any generic construction of
and the tensor product of two spaces is anyway only
defined up to isomorphism. Furthermore considering elements of
and as column vectors
, the tensor product of vectors can be
defined through
where denotes the product of two matrices (in this case an
matrix by a matrix). As a very concrete
example of this,
One reason the simple tensors in cannot exhaust this space (provded ) is that there are essentially only degrees of freedom in the choice of a simple tensor, but dimensions (http://planetmath.org/Dimension2) in the space as a whole. Hence
How can one to understand the non-simple tensors, then? In general, they are finite sums of simple tensors. One way to see this is from the theorem that has a basis consisting of products of pairs of basis vectors.
Theorem 2 (tensor product basis (http://planetmath.org/TensorProductBasis)).
Let and be vector spaces over with bases and respectively. Then is a basis for .
Expressing some arbitrary as a linear
combination
with respect to such a basis immediately produces the decomposition
as a sum of simple tensors, but this decomposition is often far from optimally short. Let and . The tensor is not simple, but as it happens the tensor is simple. In general it is not trivial to find the simplest way of expressing a tensor as a sum of simple tensors, so there is a name for the length of the shortest such sum.
Definition 3.
The rank of a tensor is the smallest natural
number such that
for some set of simple tensors , โฆ, .
In particular, the zero tensor has rank , and all other simple tensors have rank .
-
Warning.ย ย There is an entirely different concept which is also called โthe rank of a tensor (http://planetmath.org/Tensor)โ, namely the number of components
(factors) in the tensor product forming the space in which the tensor lives. This latter โrankโ concept does not generalise โrank of a matrix (http://planetmath.org/RankLinearMapping)โ. The โrankโ of Definitionย 3 does generalise โrank of a matrixโ. (It also generalises rank of a quadratic form (http://planetmath.org/Rank5).)
one way
One area where the distinction between simple and non-simple tensors
is particularly important is in Quantum Mechanics, because the state
space of a pair of quantum systems is in general the tensor product
of the state spaces of the component systems. When the combined state
is a simple tensor , then that
state can be understood as though one subsystem has state
and the other state , but when the combined state
is a non-simple tensor then the full system cannot be
understood by considering the two subsystems in isolation, even if
there is no apparent interaction between them. This situation is
often described by saying that the two subsystems are
entangled, or using phrases such as โeither is in state
and is in state , or else is in state
and is in state .โ
Entanglement is an important part of that which makes quantum systems
different from probabilistic classical systems. The physical
interpretations are often mind-boggling, but the mathematical meaning
is no more mysterious than โnon-simple tensorโ.
Entanglement can also be a useful concept for understanding pure mathematics. One reason that the comultiplication of a coalgebra cannot simply be replaced in the definition by two maps that compute the โleftโ and โrightโ parts of is that value of may be entangled, in which case one left part and one right part cannot fully encode .
Title | simple tensor |
---|---|
Canonical name | SimpleTensor |
Date of creation | 2013-03-22 15:26:07 |
Last modified on | 2013-03-22 15:26:07 |
Owner | lars_h (9802) |
Last modified by | lars_h (9802) |
Numerical id | 5 |
Author | lars_h (9802) |
Entry type | Definition |
Classification | msc 15A69 |
Synonym | tensor rank |
Synonym | entangled |
Related topic | TensorProduct |
Related topic | BasicTensor |
Defines | rank |