6. Discussion

This paper developed techniques for analyzing the internalstructure of distributed measurements. We introducedentanglement, which quantifies the extent to which ameasurement is indecomposable. Entanglement can be shownto quantify context-dependence. Moreover, positive entanglementis necessary for a system to generate more information than thesum of its subsystems. Along the way, we constructed the quale,which geometrically represents the compositional structure of adistributed measurement. The information-theoretic approachdeveloped here is dual, in a precise sense, to the algorithmicperspective on computation. Studying duals ${𝔪}^{\mathrm{♮}}$instead of mechanisms $𝔪$ shifts the focus from whatthe algorithm does to how it does it: instead ofanalyzing rules we analyze functional dependencies.

The intuition driving the paper is that the structure presheaf$ℱ$ is an information-theoretic analogue of a tangentspace. A particle moving in a manifold $X$ defines a vectorfield – a section of the tangent space to $X$, which is asheaf. The tangent vector at a point depends on the particle’slocation at “nearby time-points”: it is computed by takingthe limit of difference in positions at $t$ and $t+h$ as$h\to 0$. Similarly, a system performing a measurementgenerates a quale, a section of the structure presheafconsisting of “nearby counterfactuals”. The quale is computedby applying Bayes’ rule to determine which inputs could have ledto the output. How far thisanalogy can be developed remains to be seen.

Entanglement can be loosely considered as aninformation-theoretic analogue of curvature: the extent towhich interactions within a system “warp” sections of$ℱ$ away from a product structure. A related approachto geometrically analyzing the complexity of interactions wasproposed in [1]. In fact, this project began as anattempt to reformulate [2] in terms of sheafcohomology using ideas from [1]. We failed at thefirst step since the structure presheaf is not a sheaf.However, the failure was instructive since it is preciselythe obstruction to forming a sheaf that is of interestsince it is the obstruction (entanglement) that quantifiesindecomposability and context-dependence, and only systemswhose measurements are entangled are able to generate moreinformation than the sum of their subsystems.