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The standard perspective on subsystems in quantum theory is a bottom-up, compositional one: one starts with individual "small" systems, viewed as primary, and composes them together to form larger systems. The top-down, decompositional perspective goes the other way, starting with a "large" system and asking what it means to partition it into smaller parts. In this talk, I will 1/ argue that the adoption of the top-down perspective is the key to progress in several current areas of foundational research; and 2/ present an integrated mathematical framework for partitions into three or more subsystems, using sub-C* algebras. Concerning the first item, I will explain how the top-down perspective becomes crucial whenever the way in which a quantum system is partitioned into smaller subsystems is not unique, but might depend on the physical situation at hand. I will display how that precise feature lies at the heart of a flurry of current hot foundational topics, such as quantum causal models, Wigner's friend scenarios, superselection rules, quantum reference frames, and debates over the implementability of the quantum switch. Concerning the second item, I will argue that partitions in (finite-dimensional) quantum theory can be naturally pinned down using sub-C* algebras. Building on simple illustrative examples, I will discuss the often-overlooked existence of non-factor C*-algebras, and how it leads to numerous subtleties -- in particular a generic failure of local tomography. I will introduce a sound framework for quantum partitions that overcomes these challenges; it is the first top-down framework that allows to consider three or more subsystems. Finally, as a display of this framework's technical power, I will briefly present how its application to quantum causal modelling unlocked the proof that all 1D quantum cellular automata admit causal decompositions.

(This is joint work with Octave Mestoudjian and Pablo Arrighi. This talk is complementary to my Causalworlds 2024 presentation, which will focus on the issue of causal decompositions.)