We propose a reformulation of quantum mechanics as a theory of unresolved uncertainty. This theory of potentiality is formulated in the language of complex-valued measure theory, regarded as a pre-probabilistic counterpart of ordinary probability theory. In this formulation, additivity, conditioning, independence, mixtures, transition kernels, and temporal divisibility retain natural linear forms at the potentiality level, while non-classical probability-level features such as interference arise from the nonlinear Born map. Measurement is described as Bayesian-type conditioning of potentialities on actualized information, and non-selective measurement as the replacement of coherent potentiality by statistical mixtures of conditional potentiality branches. Mixed states, decoherence, composite systems, entanglement, and Bell-type correlations are also given a unified potentiality-level interpretation. The density matrix is interpreted as a coherence kernel whose off-diagonal blocks encode retained phase relations. For pure bipartite states, potentiality independence is shown to be equivalent to factorization of the Born distribution in every pair of local contexts. The resulting formulation is empirically equivalent to standard quantum mechanics, but it makes explicit a pre-probabilistic description of physical reality that is usually implicit in the Hilbert-space formalism.
Textbook quantum superposition refers to the feature that certain linear combinations of Hilbert space rays, each representing a valid quantum state, are themselves valid states. This notion is not operational, and it relies on the underlying Hilbert space formalism. Recent proposals for experimental tests of indefinite causal order, as well as tests probing the non-classicality of gravity, pivot on superposition, thereby calling for a theory-independent, operational formalisation of the concept. Here, we define superposition within the framework of Generalised Probabilistic Theories, based on observed statistics in prepare-and-measure experiments. Using this, we formulate three superposition principles to investigate which structural features of quantum theory carry over to other theories. We study conditions under which these principles carry over from subsystems to their compositions; to this end, we show that the quantum tensor product emerges as the largest composition rule for quantum systems respecting all three principles. Furthermore, we show how non-classical features such as entanglement and preparational uncertainty can be viewed as special forms of superposition.
QBism understands quantum mechanics to be probability theory supplemented by additional nonclassical coherence conditions. In this dissertation, we develop these nonclassical coherence conditions from first principles, emphasizing the role of a well chosen reference measurement. After treating standard probability on subjective Bayesian lines, we demonstrate an equivalence between the QBist approach and the existing framework of generalized probabilistic theories. We show that the fundamental nonclassical coherence relation may almost always be taken to be a gentle modification of the law of total probability, and give a coherentist account of when an experimental scenario has a classical explanation. Finally, we show that when the reference measurement is chosen to correspond to a complex projective 3-design, the shape of quantum state space is implicit in the probabilities which characterize the reference measurement itself. Thus coherence with this single reference measurement, properly understood, implies coherence with all of finite dimensional quantum mechanics. We then attempt a modest reconstruction of quantum theory along these lines, one practical consequence of which is a method for self-testing complex projective $t$-designs for $t\ge 3$ in a theory-agnostic way.
Reverse Physics is a methodology that breaks physical theories into separate mathematical and physical conditions to establish their logical relationships. To showcase the power of the methodology, we present several results for quantum mechanics and their related insights. The standard Hilbert-space formulation conflicts with basic physical requirements, while a minimal topological modification can solve these problems. The ensemble space, rather than the pure-state space, distinguishes classical from quantum systems. The Born rule is an additional assumption linking orthogonality, mutual exclusivity and information entropy. Under explicit background conditions, unitary evolution is equivalent to deterministic and reversible evolution. Nonselective projective measurements can be characterized as Lindblad equilibration processes, while unitary evolution can be characterized as a limit of infinitesimal projective processes. Classical mechanics is recovered as the high-entropy limit of quantum mechanics, and every quantum state, pure or mixed, is a dynamical, spectral and thermodynamic equilibrium. These results are self-contained, use the standard vector-space representation and can thus be used as common tools and constraints for teaching, interpretations, reconstructions and future theories.
G. Carcassi, Tobias Thrien, C. Aidala· 0 citations
This doctoral dissertation on the foundations of quantum theory isolates and then formalizes a physically relevant concept that I have called"Epistemic Constraint."Here, epistemic constraints are the definite, intersubjectively agreeable, ordinary-language conditions under which experiments are described. The usual formulation of the quantum measurement problem, which I call the Schrodingerian measurement problem, has the structure of an anomaly: if we take quantum theory at face value, we expect no definite values, and yet we see definite values in experiments. The responses to this problem have been either to solve it or to dissolve it. These responses, which have taken the form of interpretation, modification, or reconstruction of quantum mechanics, seek either to derive (conceptually or mathematically) epistemic constraints from within quantum mechanics or suitable modifications of it, as is the case with certain interpretations and modifications, or to posit the epistemic constraint, or parts of it, as a primitive assumption with the goal of deriving quantum mechanics, as is the case in some reconstruction programs. In contrast to the Schrodingerian measurement problem, which had the structure of an anomaly, this dissertation develops the Bohrian Program, which (for lack of a better comparison) has a structure similar to the problem historically associated with Euclid's fifth postulate. It seeks to keep epistemic constraints as primitive in an onto-epistemic sense. It then seeks new physical conclusions from the joint consideration of quantum mechanics and epistemic constraints, without seeking to derive one from the other. Among other results, this leads to a notion of the probability of instantiability of the Born Rule that specifies when to apply the Born Rule and when to apply a unitary transformation to a quantum state.
The physical foundation of the mathematical formalism of quantum theory is still an iffy mystery. Here it is presumed that a physically reasonable mathematical model needs only three basic features. The first one are the transition probabilities, which are so typical of quantum theory. The other two constitute a variation of the postulate that continuous reversible dynamical processes exist and act transitively on the underlying space. One class of mathematical models with these features arises from the atomic JBW factors, which include the atomic von Neumann factors and become identical with the Jordan matrix algebras, when the dimension is finite. A further model is known, on which the exceptional Lie group E6 acts transitively. Interestingly, E6 is sometimes considered a candidate for internal symmetries in particle physics, but many familiar features of quantum theory get lost in this case (particularly the general existence of post-measurement states). The paper concludes with some open issues, concerning this problem and the classification of the mathematical structures with the three features.
We draw on ’t Hooft’s seminal formulation of the holographic principle to analyse the methodological and conceptual role of information in quantum gravity. We argue that, in 't Hooft's work and in later developments, information functions as a substantive guiding principle. We distinguish three aspects of this role: First, holographic bounds on the amount of information that can be stored in a region function as theory selection criteria that constrain viable quantum gravity theories. Second, holography functions as a principle of theoretical equivalence: the bulk and boundary theories must describe the same physical content, even though ’t Hooft privileges a more fundamental, lower-dimensional, and potentially deterministic boundary description. Third, the distribution and encoding of information link ’t Hooft’s proposals to contemporary work on bulk reconstruction, holographic quantum error correction, and ER=EPR, where emergent spacetime structure is tied to patterns of entanglement and redundancy. On the basis of these three roles, we argue that purely epistemic or Shannon-style conceptions of information are inadequate in this context: we instead outline a distinction between what we call maximal and intermediate conceptions that aim to capture the methodological and interpretative roles of information in holographic quantum gravity. We therefore suggest that, in the context of holography and quantum gravity, a more systematic philosophical treatment of the role of information as an interpretation-guiding principle would be desirable.
S. De Haro, Enrico Cinti, Aude Corbeel· International Journal of Mod...· 0 citations
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