6. Discussion

6. Discussion

This paper developed techniques for analyzing the internal
structure of distributed measurements. We introduced
entanglement, which quantifies the extent to which a
measurement is

indecomposable

. Entanglement can be shown
to quantify context-dependence. Moreover,

positive

entanglement
is necessary for a system to generate more information than the
sum of its subsystems. Along the way, we constructed the quale,
which geometrically

represents

the compositional structure of a
distributed measurement. The information-theoretic approach
developed here is dual, in a precise sense, to the algorithmic

perspective

on computation. Studying duals

𝔪

♮

instead of mechanisms

𝔪

shifts the focus from

what

the algorithm does to

how

it does it: instead of
analyzing rules we analyze functional dependencies.

The intuition driving the paper is that the structure

presheaf

ℱ

is an information-theoretic analogue of a tangent
space. A particle moving in a manifold

X

defines a

vector
field

– a

section

of the tangent space to

X

, which is a
sheaf. The tangent vector at a point depends on the particle’s
location at “nearby time-points”: it is computed by taking
the limit of

difference

in positions at

t

and

t

+

h

as

h

→

0

. Similarly, a system performing a measurement
generates a quale, a section of the structure presheaf
consisting of “nearby counterfactuals”. The quale is computed
by applying Bayes’ rule to determine which inputs could have led
to the output.

1

1

A counterfactual input is “nearby” to
an output if it causes (leads to) that output.

How far this

analogy

can be developed remains to be seen.

Entanglement can be loosely considered as an
information-theoretic analogue of curvature: the extent to
which interactions within a system “warp” sections of

ℱ

away from a

product structure

. A related approach
to geometrically analyzing the complexity of interactions was
proposed in

[

1

]

. In fact, this project began as an
attempt to reformulate

[

2

]

in terms of

sheaf
cohomology

using ideas from

[

1

]

. We failed at the
first step since the structure presheaf is not a sheaf.
However, the failure was instructive since it is precisely
the

obstruction

to forming a sheaf that is of

interest

since it is the obstruction (entanglement) that quantifies
indecomposability and context-dependence, and only systems
whose measurements are

entangled

are able to generate more
information than the sum of their subsystems.

References

1

N Ay, E Olbrich,
N Bertschinger & J Jost
(2006):

A unifying framework for
complexity measures of finite systems

.

In:

Proceedings of ECCS06, European

Complex
Systems

Society

, Oxford, UK, pp.
ECCS06–174.

2

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

6. Discussion

Canonical

name

6Discussion

Date of creation

2014-04-22 15:13:32

Last modified on

2014-04-22 15:13:32

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

2

Author

rspuzio (6075)

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Feature

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