Twirled and swirled worlds: a compositional approach
This paper extends the concepts of twirling and swirling to a fully compositional framework for finite-dimensional generalised probabilistic theories, establishing conditions under which symmetry-restricted subtheories satisfy process tomography and demonstrating how specific physical theories like real quantum theory emerge from these constructions.
Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer
In the vast landscape of modern physics, symmetries act as the invisible architects of reality. They are the rules that say a system looks the same whether you rotate it, flip it, or shift it in time. When physicists study these symmetries, they often look for the parts of a system that remain unchanged, or "invariant," under these transformations. This search has led to the discovery of superselection rules, which are like invisible fences that prevent certain physical states from mixing, effectively dividing the world into separate, non-interacting sectors. For decades, scientists have used a mathematical technique called "twirling" to strip away the non-invariant parts of a theory, leaving behind a simplified version that respects these symmetry rules. This technique has been crucial in understanding quantum mechanics and the nature of reference frames, but it was largely limited to specific types of theories and simple scenarios involving just states and measurements.
A new study by Sina Soltani, Thomas Galley, Marco Erba, and John Selby expands this idea into a much broader and more flexible framework. They have developed a way to apply these symmetry restrictions to the entire structure of physical theories, treating states, measurements, and the transformations between them as a single, unified family of processes. By doing so, they have created a comprehensive map of what happens when you impose symmetry constraints on any physical theory, not just the familiar ones like quantum mechanics. Their work reveals that the resulting "invariant" worlds are not always as simple as previously thought. While earlier research showed that these restricted worlds still allow scientists to fully describe the state of a system using local measurements, this new analysis proves that the same is not automatically true for the processes that transform those states. In many cases, the symmetry restrictions hide information that can only be revealed by using extra systems or by creating specific types of entanglement, a phenomenon where parts of a system become deeply linked in ways that defy classical intuition.
The researchers approached this problem by treating physical theories as compositions of processes, much like building complex circuits from simple wires and components. They distinguished between two ways symmetries can be applied: either by acting directly on the individual systems (like rotating a particle) or by acting directly on the rules that govern how processes combine. They also separated these actions into two categories: physical actions, which could theoretically be performed in a lab, and non-physical actions, which are mathematical constructs that do not correspond to any realizable experiment. By averaging over these actions, they defined two types of resulting worlds: "twirled" worlds, which arise from physical symmetries, and "swirled" worlds, which arise from non-physical symmetries. This distinction allowed them to explore a wider range of theoretical possibilities, including the famous case of real quantum theory, which can be viewed as a "swirled" version of standard complex quantum theory.
One of the most significant findings of the paper is a precise condition for when these symmetry-restricted theories fail to be "locally tomographic." In a locally tomographic theory, you can fully describe a complex system by measuring its individual parts separately. The authors proved that this property breaks down in the new invariant worlds whenever different parts of the system share a specific type of mathematical connection known as a non-trivial irreducible representation. In simpler terms, if two parts of a system are linked by a symmetry in a way that cannot be broken down into independent pieces, then looking at them separately is no longer enough to understand the whole. The researchers showed exactly how this failure manifests and quantified the amount of information that becomes inaccessible to local measurements. They demonstrated that to recover this lost information, one must sometimes use entangled states or bring in extra "ancillary" systems to act as probes.
The paper also provides a complete recipe for determining when and how to perform these measurements. It turns out that if the input and output systems of a process share a non-trivial symmetry connection, you cannot distinguish between different processes without using an extra system. However, if they do not share such a connection, you can distinguish them using only the systems at hand, though you may still need to use entangled states rather than just simple, separate measurements. The authors applied these general results to specific examples, showing how a theory called "bilocal classical theory" naturally emerges as a twirled version of standard classical theory. They also showed how real quantum theory fits into their framework as a swirled theory, recovering its known properties from their general rules. This work unifies a wide variety of previously disconnected ideas, offering a powerful new tool for understanding the structural limits of physical theories and the role of symmetry in defining what is knowable about the universe.
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