simple tensor

simple tensor

The

tensor product

(

http://planetmath.org/TensorProduct

)

\PMlinkescapephrase

tensor product

U

⊗

V

of two

vector spaces

U

and

V

is
another vector space which is characterised by being

universal

for

bilinear maps

on

U

×

V

. As part of this package,
there is an

operation

⊗

on vectors such that

𝐮

⊗

𝐯

∈

U

⊗

V

for all

𝐮

∈

U

and

𝐯

∈

V

, and the

primary

subject
of this article is the image of that operation.

Definition 1

.

The

element

𝐰

∈

U

⊗

V

is said to be a

simple tensor

if there exist

𝐮

∈

U

and

𝐯

∈

V

such that

𝐰

=

𝐮

⊗

𝐯

.

More generally, the element

𝐰

∈

W

=

U

1

⊗

⋯

⊗

U

k

is said to be a simple tensor (with respect to the

decomposition

U

1

⊗

⋯

⊗

U

k

of

W

) if there
exist

𝐮

i

∈

U

i

for

i

=

1

,

…

,

k

such that

𝐰

=

𝐮

1

⊗

⋯

⊗

𝐮

k

.

For this definition to be interesting, there must also be tensors
which are not simple, and indeed most tensors aren’t. In order to
illustrate why, it is convenient to consider the tensor product of
two

finite-dimensional

vector spaces

U

=

𝒦

m

and

V

=

𝒦

n

over some field

𝒦

. In this case one can let

U

⊗

V

=

𝒦

m

×

n

(the vector space of

m

×

n

matrices), since

𝒦

m

×

n

is

isomorphic

to any

generic

construction of

U

⊗

V

and the tensor product of two spaces is anyway only
defined up to

isomorphism

. Furthermore considering elements of

U

and

V

as

column vectors

, the tensor product of vectors can be
defined through

𝐮

⊗

𝐯

=

𝐮

⋅

𝐯

T

where

⋅

denotes the

product

of two matrices (in this case an

m

×

1

matrix by a

1

×

n

matrix). As a very concrete
example of this,

(

u

1

u

2

u

3

)

⊗

(

v

1

v

2

v

3

v

4

)

=

(

u

1

⁢

v

1

u

1

⁢

v

2

u

1

⁢

v

3

u

1

⁢

v

4

u

2

⁢

v

1

u

2

⁢

v

2

u

2

⁢

v

3

u

2

⁢

v

4

u

3

⁢

v

1

u

3

⁢

v

2

u

3

⁢

v

3

u

3

⁢

v

4

)

⁢

.

One reason the simple tensors in

U

⊗

V

cannot exhaust this
space (provded

m

,

n

⩾

2

) is that there are essentially
only

m

+

n

-

1

degrees of freedom in the choice of a simple tensor, but

m

⁢

n

dimensions (

http://planetmath.org/Dimension2

) in the space

U

⊗

V

as a whole. Hence

𝒦

m

⊗

𝒦

n

≠

{

𝐮

⊗

𝐯

𝐮

∈

𝒦

m

,

𝐯

∈

𝒦

n

}

when

m

,

n

⩾

2

.

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

U

⊗

V

has a basis consisting of products of
pairs of basis vectors.

Theorem 2

(tensor product basis (

http://planetmath.org/TensorProductBasis

)).

Let

U

and

V

be vector spaces over

K

with bases

{

e

i

}

i

∈

I

and

{

f

j

}

j

∈

J

respectively. Then

{

e

i

⊗

f

j

}

(

i

,

j

)

∈

I

×

J

is a basis for

U

⊗

V

.

Expressing some arbitrary

𝐰

∈

U

⊗

V

as a

linear
combination

𝐰

=

∑

r

=

1

n

λ

r

⁢

(

𝐞

i

r

⊗

𝐟

j

r

)

with respect to such a basis immediately produces the decomposition

𝐰

=

∑

r

=

1

n

(

λ

r

⁢

𝐞

i

r

)

⊗

𝐟

j

r

as a sum of simple tensors, but this decomposition is often far
from optimally short.
Let

𝐞

1

=

(

1

0

)

∈

𝒦

2

and

𝐞

2

=

(

0

1

)

∈

𝒦

2

. The tensor

𝐞

1

⊗

𝐞

1

+

𝐞

2

⊗

𝐞

2

=

(

1

0

0

1

)

is not
simple, but as it happens the tensor

𝐞

1

⊗

𝐞

1

+

𝐞

1

⊗

𝐞

2

+

𝐞

2

⊗

𝐞

1

+

𝐞

2

⊗

𝐞

2

=

(

1

1

1

1

)

=

(

1

1

)

⊗

(

1

1

)

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

𝐰

is the smallest

natural
number

n

such that

𝐰

=

𝐰

1

+

⋯

+

𝐰

n

for some set of

n

simple tensors

𝐰

1

, …,

𝐰

n

.

In particular, the zero tensor has rank

0

, and all other simple
tensors have rank

1

.

Warning.

There is an entirely different

concept

which is also called
‘the rank of a tensor (

http://planetmath.org/Tensor

)’,
namely the number of

components

(factors) in the tensor
product forming the space in which the tensor lives. This latter
‘rank’ concept does not generalise
‘rank of a matrix (

http://planetmath.org/RankLinearMapping

)’.
The ‘rank’ of Definition

3

does

generalise ‘rank of a matrix’. (It also generalises
rank of a

quadratic form

(

http://planetmath.org/Rank5

).)

\PMlinkescapephrase

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

𝐰

=

𝐮

⊗

𝐯

, then that
state can be understood as though one subsystem has state

𝐮

and the other state

𝐯

, but when the combined state

𝐰

is a non-simple tensor

𝐮

1

⊗

𝐯

1

+

𝐮

2

⊗

𝐯

2

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 “either

U

is in state

𝐮

1

and

V

is in state

𝐯

1

, or else

U

is in state

𝐮

2

and

V

is in state

𝐯

2

.”
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 ‘non-simple tensor’.

Entanglement can also be a useful concept for understanding pure mathematics.
One reason that the comultiplication

Δ

:

C

⟶

C

⊗

C

of a coalgebra

C

cannot simply be replaced in the definition by two
maps

Δ

L

,

Δ

R

:

C

⟶

C

that compute the ‘left’
and ‘right’ parts of

Δ

is that value of

Δ

may be
entangled, in which case one left part

Δ

L

⁢

(

c

)

and one right
part

Δ

R

⁢

(

c

)

cannot fully encode

Δ

⁢

(

c

)

.

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

Generated on Fri Feb 9 18:31:08 2018 by

LaTeXML