# 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 ${\mathfrak{m}}^{\natural}$ instead of mechanisms ${\mathfrak{m}}$ 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 ${\mathcal{F}}$ 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 $h\rightarrow 0$. 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.11A 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 ${\mathcal{F}}$ 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.
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