# 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 â€“ a section of the tangent space to $X$, which is a sheaf. The tangent vector at a point depends on the particleâ€™s location at â€śnearby time-pointsâ€ť: 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 â€śnearby counterfactualsâ€ť. The quale is computed by applying Bayesâ€™ rule to determine which inputs could have led to the output.11A counterfactual input is â€śnearbyâ€ť 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 â€śwarpâ€ť 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 NÂ Ay, EÂ Olbrich, NÂ Bertschinger & JÂ Jost (2006): A unifying framework for complexity measures of finite systems. In: Proceedings of ECCS06, European Complex Systems Society, Oxford, UK, pp. ECCS06â€“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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