Decompositional theories describe the ways in which a global physical system can be split into subsystems, facilitating the study of how different possible partitions of a same system interplay, e.g. in terms of inclusions or signalling. In quantum theory, subsystems are usually framed as sub-C* algebras of the algebra of operators on the global system. However, most decompositional approaches have so far restricted their scope to the case of systems corresponding to factor algebras. We argue that this is a mistake: one should cater for the possibility for non-factor subsystems, arising for instance from symmetry considerations. Building on simple examples, we motivate and present a definition of partitions into an arbitrary number of parts, each of which is a possibly non-factor sub-C* algebra. We discuss its physical interpretation and study its properties, in particular with regards to the structure of algebras' centres. We prove that partitions, defined at the C*-algebraic level, can be represented in terms of a splitting of Hilbert spaces, using the framework of routed quantum circuits. For some partitions, however, such a representation necessarily retains a residual pseudo-nonlocality. We provide an example of this behaviour, given by the partition of a fermionic system into local modes.
In systems with infinitely many degrees of freedom, fundamental results from quantum information theory can fail. An important example is the uniqueness of purifications: Even when two subsystems, described by commuting von Neumann algebras
A and
B, are tomographically complete, purifications of a state on
A need not be related by unitaries in
B. It was recently shown that this occurs precisely when Haag duality fails, i.e., when the commutant
B′ is strictly larger than
A. This raises the question of which fundamental entanglement properties survive in such a setting. We show that, for a pure global state, the ability to steer any ensemble decomposition of the marginal state on
A by measurements on
B is equivalent to the existence of a state-preserving conditional expectation from
B′ onto
A. This establishes a direct connection between quantum steering and subfactor theory. The key observation is that steering is equivalent to the existence of extensions of ensemble decompositions from
A to
B′. Working with general Jordan algebras, we prove that unital positive maps have state-preserving left inverses if and only if ensemble decompositions can be lifted. For the inclusion
A↪B′, a left inverse is precisely a conditional expectation, yielding the characterization above.
Both classical and quantum mechanics assume that physical laws are invariant under changes in the way that the world is labeled. This Principle of Decompositional Equivalence is formalized, and shown to forbid finite experimental demonstrations that given pointer states |p1> and |p2> refer to the same physical system S. It is then shown that any extra-theoretical assumption that given pointer states |p1> and |p2> with indistinguishable coefficients in a Schmidt basis for the universe refer to the same physical system S is stronger than the assumption of classicality. Standard interpretations of quantum mechanics make such as assumption in analyzing measurement; hence they are logically inconsistent.
Introduction Quantum measurement problem is a technical euphemism for a much deeper and less well-defined question: How do we, “the observers,” fit within the physical universe? The problem is especially apparent in quantum physics because, for the first time in the history of science a majority of (but not all) physicists seriously entertains the possibility that the framework for the ultimate universal physical theory provided by quantum mechanics is here to stay. Quantum physics relevant for this discussion is (contrary to the common prejudice) relatively simple. By this I mean that some of the key features of its predictions can be arrived at on the basis of overarching principles of quantum theory and without reference to the minutiae of other specific ingredients (such as the details of the forces). Quantum superposition principle is such an overarching principle of quantum theory. It leads to predictions that seem difficult to reconcile with our perception of the familiar classical universe of everyday experience. The aim of this paper is to show that appearance of the classical reality can be viewed as a result of the emergence of the preferred states from within the quantum substrate through the Darwinian paradigm, once the survival of the fittest quantum states and selective proliferation of the information about them throughout the universe are properly taken into account.
A recent concept in theoretical physics, motivated in string duality and M-theory, is the notion that not all quantum theories arise from quantising a classical system. Also, a given quantum model may possess more than just one classical limit. In view of these developments, we analyse some general properties that quantum mechanics must satisfy, if it is not to be formulated as a quantisation of a given classical mechanics. Instead, our approach to quantum mechanics is modelled on a statement that is close in spirit to the equivalence principle of general relativity, thus bearing a strong resemblance with the equivalence principle of quantum mechanics formulated by Faraggi-Matone.