Agents, subsystems, and the conservation of information
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Dividing the world into subsystems is an important component of the scientific method. The choice of subsystems, however, is not defined a priori. Typically, it is dictated by our experimental capabilities, and, in general, different agents may have different capabilities. Here we propose a construction that associates every agent with a subsystem, equipped with its set of states and its set of transformations. In quantum theory, this construction accommodates the traditional notion of subsystems as factors of a tensor product, as well as the notion of classical subsystems of quantum systems. We then restrict our attention to systems where all physical transformations act invertibly. For such systems, the future states are a faithful encoding of the past states, in agreement with a requirement known as the Conservation of Information. For systems satisfying the Conservation of Information, we propose a dynamical definition of pure states, and show that all the states of all subsystems admit a canonical purification. This result extends the purification principle to a broader setting, in which coherent superpositions can be interpreted as purifications of incoherent mixtures. As an example, we illustrate the general construction for subsystems associated with group representations.
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