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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.)

In this talk, I shall recall the nonlocality transitivity problem, which concerns the possibility of inferring the Bell-nonlocality of certain marginals in a multipartite scenario based on other given marginals. Then, I explain how considering this problem has led to a more general class of problems known as resource marginal problems (RMPs). More precisely, RMPs concern the possibility of having a resource-free target subsystem compatible with a given collection of marginal density matrices. We briefly discuss how a resource theory for a collection of marginal density matrices naturally arises from any given RMP and present some general features of such a theory. After that, we focus on a special case of RMPs known as the entanglement transitivity problems and explain how our progress on this problem has led to progress in the original nonlocality transitivity problem.

Entanglement in quantum many-body systems is typically fragile to interactions with the environment. Generic unital quantum channels, for example, have the maximally mixed state with no entanglement as their unique steady state. However, we find that for a unital quantum channel that is `strongly symmetric', i.e. it preserves a global on-site symmetry, the maximally mixed steady state in certain symmetry sectors can be highly entangled. For a given symmetry, we analyze the entanglement and correlations of the maximally mixed state in the invariant sector (MMIS), and show that the entanglement of formation and distillation are exactly computable and equal for any bipartition. For all Abelian symmetries, the MMIS is separable, and for all non-Abelian symmetries, the MMIS is entangled. Remarkably, for non-Abelian continuous symmetries described by compact semisimple Lie groups (e.g. SU(2)), the bipartite entanglement of formation for the MMIS scales logarithmically ∼logN with the number of qudits N.

While quantum correlations between two spacelike-separated systems are fully encoded by the bipartite density operator associated with the joint system, what operator encodes quantum correlations across space and time? I will describe the general theory of such "quantum states over time" as well as a canonical example that encodes the expectation values of certain observables measured sequentially in time. The latter extends the theory of pseudo-density matrices to arbitrary dimensions, not necessarily restricted to multi-qubit systems. In addition, quantum states over time admit a natural proposal for a general-purpose quantum Bayes' rule. Our results specialize to many well-studied examples, such as the state-update rule, the two-state vector formalism and weak values, and the Petz recovery map. This talk is based on joint work with James Fullwood and the two papers: arXiv: 2212.08088 [quant-ph] and 2405.17555 [quant-ph].