5. Entanglement

5. Entanglement

The proof of

Theorem

4 (

http://planetmath.org/3distributeddynamicalsystems#Thmthm4

)
showed the

structure

presheaf

has non-unique descent,
reflecting the fact that measuring devices do not necessarily
reduce to

products

of subdevices. Similarly, as we will see,
measurements do not in general decompose into

independent

submeasurements. Entanglement,

γ

, quantifies how far a
measurement

diverges

in bits from the product of its
submeasurements. It turns out that

γ

>

0

is necessary for
a system to generate more information than the sum of its
components: non-unique descent thus provides “room at the top”
to build systems that perform more precise measurements
collectively than the sum of their components.

Entanglement has no direct

relation

to quantum entanglement.
The name was chosen because of a formal resemblance between the
two quantities, see Supplementary Information of

[

1

]

.

Definition 10

.

Entanglement

over

partition

𝒫

=

{

M

1

⁢

…

⁢

M

m

}

of

s

⁢

r

⁢

c

⁢

(

𝔪

𝐃

)

is

γ

(

𝔪

𝐃

,

𝒫

,

d

o

⁢

u

⁢

t

)

=

H

[

𝔪

𝐃

♮

∘

d

o

⁢

u

⁢

t

∥

⊗

i

=

1

m

π

j

∘

𝔪

j

♮

∘

d

o

⁢

u

⁢

t

]

where

π

j

:

𝒱

⁢

S

𝐃

→

𝒱

⁢

S

M

j

and

𝔪

j

=

{

(

k

,

l

)

∈

𝔪

𝐃

|

k

∈

M

j

}

.

Projecting via

π

j

marginalizes onto the subspace

𝒱

⁢

S

M

j

. Entanglement thus compares the measurement
performed by the entire system with submeasurements over the
decomposition of the source occasions into partition

𝒫

.

Theorem 9

(effective information decomposes additively when
entanglement is zero).

γ

⁢

(

𝔪

𝐃

,

𝒫

,

d

o

⁢

u

⁢

t

)

=

0

⟹

e

⁢

i

⁢

(

𝔪

𝐃

,

d

o

⁢

u

⁢

t

)

=

∑

i

=

1

m

e

⁢

i

⁢

(

𝔪

j

,

d

o

⁢

u

⁢

t

)

.

Proof: Follows from the observations that (i)

H

[

p

∥

p

1

⊗

p

2

]

=

0

if and only if

p

=

p

1

⊗

p

2

; (ii)

H

[

p

1

⊗

p

2

∥

q

1

⊗

q

2

]

=

H

[

p

1

∥

q

1

]

+

H

[

p

2

∥

q

2

]

;
and (iii) the

uniform distribution

on

𝐃

is a tensor of
uniform distributions on subsystems of

𝐃

.

■

The theorem shows the relationship between effective information
and entanglement. If a system generates more information “than
it should” (meaning, more than the sum of its subsystems), then
the measurements it generates are entangled. Alternatively, only

indecomposable

measurements can be more precise than the sum of
their submeasurements.

We conclude with some detailed

computations

for

X

×

Y

→

𝑔

Z

,

Diagram

(11) (

http://planetmath.org/4measurement#id2

).
Let

𝒫

=

{

X

|

Y

}

.

Theorem 10

(entanglement and effective information for

g

:

X

×

Y

→

Z

).

γ

⁢

(

𝔪

X

⁢

Y

,

𝒫

,

δ

z

)

=

∑

(

x

,

y

)

∈

g

-

1

⁢

(

z

)

1

|

g

-

1

⁢

(

z

)

|

⁢

log

2

⁡

|

g

-

1

⁢

(

z

)

|

|

g

x

×

Y

-

1

⁢

(

z

)

|

⋅

|

g

X

×

Y

-

1

⁢

(

z

)

|

=

e

⁢

i

⁢

(

𝔪

X

⁢

Y

,

δ

z

)

-

e

⁢

i

⁢

(

𝔪

X

⁣

∙

,

δ

z

)

-

e

⁢

i

⁢

(

𝔪

∙

Y

,

δ

z

)

.

Proof:
The first

equality

follows from

Propositions

5 (

http://planetmath.org/4measurement#Thmthm5

)
and 6 (

http://planetmath.org/4measurement#Thmthm6

)

γ

⁢

(

𝔪

X

⁢

Y

,

𝒫

,

δ

z

)

=

∑

(

x

,

y

)

∈

g

-

1

⁢

(

z

)

=

∑

(

x

,

y

)

∈

g

-

1

⁢

(

z

)

1

|

g

-

1

⁢

(

z

)

|

⁢

log

2

⁡

[

1

|

g

-

1

⁢

(

z

)

|

⋅

|

g

-

1

⁢

(

z

)

|

|

g

x

×

Y

-

1

⁢

(

z

)

|

⁢

|

g

-

1

⁢

(

z

)

|

|

g

X

×

Y

-

1

⁢

(

z

)

|

]

.

From the same propositions it follows that

e

⁢

i

⁢

(

𝔪

X

⁢

Y

,

δ

z

)

-

e

⁢

i

⁢

(

𝔪

X

⁣

∙

,

δ

z

)

-

e

⁢

i

⁢

(

𝔪

∙

Y

,

δ

z

)

equals

log

2

⁡

|

X

|

⋅

|

Y

|

|

g

-

1

⁢

(

x

)

|

-

∑

x

|

g

x

×

Y

-

1

⁢

(

z

)

|

|

g

-

1

⁢

(

z

)

|

⁢

log

2

⁡

|

X

|

⋅

|

g

x

×

Y

-

1

⁢

(

z

)

|

|

g

-

1

⁢

(

z

)

|

-

∑

y

|

g

X

×

y

-

1

⁢

(

z

)

|

|

g

-

1

⁢

(

z

)

|

⁢

log

2

⁡

|

Y

|

⋅

|

g

X

×

y

-

1

⁢

(

z

)

|

|

g

-

1

⁢

(

z

)

|

=

log

2

⁡

1

g

-

1

⁢

(

z

)

-

∑

(

x

,

y

)

∈

g

-

1

⁢

(

z

)

1

|

g

-

1

⁢

(

z

)

|

⋅

log

2

⁡

|

g

X

×

y

-

1

⁢

(

z

)

|

|

g

-

1

⁢

(

z

)

|

⋅

|

g

x

×

Y

-

1

⁢

(

z

)

|

|

g

-

1

⁢

(

z

)

|

.

Entanglement quantifies how far the size of the pre-image of

g

-

1

⁢

(

z

)

deviates from the sizes of its

X

×

y

and

x

×

Y

slices as

x

and

y

are varied.

■

By Corollary 8 (

http://planetmath.org/4measurement#Thmthm8

)
entanglement also equals

e

i

(

𝔪

X

⁣

∙

→

𝔪

X

⁢

Y

,

δ

z

)

-

e

i

(

𝔪

∙

Y

,

δ

z

)

. In
Diagram (11) (

http://planetmath.org/4measurement#id2

)
entanglement is the vertical arrow minus both arrows at the
bottom, or the

difference

between opposing diagonal arrows.
Note that the diagonal arrows from left to right are
constructed by adding edge

v

Y

→

v

Z

to the null
system and the subsystem

𝔪

X

⁣

∙

=

{

v

X

→

v

Z

}

respectively. Entanglement is the
difference between the information generated by the diagonal
arrows. It quantifies the difference between the information

{

v

Y

→

v

Z

}

generates in two different

contexts

.

Corollary 11

(characterization of disentangled set-valued functions).

Function

X

×

Y

→

𝑔

Z

performs a
disentangled measurement when outputting

z

iff

g

-

1

⁢

(

z

)

=

g

x

×

Y

-

1

⁢

(

z

)

×

g

X

×

y

-

1

⁢

(

z

)

for any

x

,

y

such that

g

⁢

(

x

,

y

)

=

z

.

Proof:
By Theorem

10

entanglement is zero iff

|

g

-

1

⁢

(

z

)

|

=

|

g

x

×

Y

-

1

⁢

(

z

)

|

⋅

|

g

X

×

y

-

1

⁢

(

z

)

|

for any

x

,

y

such that

g

⁢

(

x

,

y

)

=

z

. This implies the desired
result since

g

-

1

⁢

(

z

)

↪

g

x

×

Y

-

1

⁢

(

z

)

×

g

X

×

y

-

1

⁢

(

z

)

.

■

Thus, the measurement generated by

g

is disentangled iff its
pre-image

g

-

1

⁢

(

z

)

satisfies

a strong geometric
“rectangularity” constraint: that the pre-image decomposes
into the product of its

x

×

Y

and

X

×

y

slices for
all pairs of slices intersecting

g

-

1

⁢

(

z

)

. The
categorizations performed within a disentangled measuring
device have nothing to do with each other, so that the device
is best considered as two (or more) distinct devices that
happen to have been grouped together for the purposes of
performing a computation.

Example 4

.

An XOR-gate

g

:

X

×

Y

→

Z

outputting 0
generates an entangled measurement. The pre-image is

g

-

1

⁢

(

0

)

=

{

00

,

11

}

so the XOR-gate generates 1 bit
of information about occasions

v

X

and

v

Y

. However,
the bit is

not localizable

. The measurement
generates no information about occasion

v

X

taken singly,
since its output could have been 0 or 1 with equal

probability

; and similarly for

v

Y

.

Finally, and unsurprisingly, a function is completely
disentangled across all its measurements iff it is a
product of two simpler functions:

Corollary 12

(completely disentangled functions are products).

If

X

×

Y

→

𝑔

Z

is

surjective

, then

γ

⁢

(

m

X

⁢

Y

,

P

,

δ

z

)

=

0

for all

z

∈

Z

iff

g

decomposes into

X

×

Y

→

g

1

×

g

2

Z

1

×

Z

2

=

Z

for

X

→

g

1

Z

1

and

Y

→

g

2

Z

2

.

Proof:
The reverse

implication

is

trivial

.
In the forward direction, note that

Z

=

{

g

-

1

⁢

(

z

)

|

z

∈

Z

}

and, by Corollary

11

, each pre-image has

product
structure

g

-

1

⁢

(

z

)

=

g

x

×

Y

-

1

⁢

(

Z

)

×

g

X

×

Y

-

1

⁢

(

z

)

. Let

Z

1

=

{

g

X

×

y

-

1

|

y

∈

Y

⁢

and

⁢

z

∈

Z

}

and similarly for

Z

2

. Define

g

1

:

X

→

Z

1

:

x

↦

the unique element of
form

⁢

g

X

×

y

-

1

⁢

(

z

)

⁢

containing it,

and similarly for

g

2

.

References

1

David Balduzzi & Giulio Tononi
(2009):

Qualia: the geometry of
integrated information

.

PLoS Comput Biol

5(8), p. e1000462,
doi:

10.1371/journal.pcbi.1000462

.

Title

5. Entanglement

Canonical name

5Entanglement

Date of creation

2014-04-22 23:06:40

Last modified on

2014-04-22 23:06:40

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Feature

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