Information Processing in Convex Operational Theories

Electronic Proceedings in Theoretical Computer Science, 2018

This paper charts a very direct path between the categorical approach to quantum mechanics, due to Abramsky and Coecke, and the older convex-operational approach based on ordered vector spaces (recently reincarnated as "generalized probabilistic theories"). In the former, the objects of a symmetric monoidal category C are understood to represent physical systems and morphisms, physical processes. Elements of the monoid C (I, I) are interpreted somewhat metaphorically as probabilities. Any monoid homomorphism from the scalars of a symmetric monoidal category C gives rise to a covariant functor V o from C to a category of dual-pairs of ordered vector spaces. Specifying a natural transformation u : V o → 1 (where 1 is the trivial such functor) allows us to identify normalized states, and, thus, to regard the image category V o (C) as consisting of concrete operational models. In this case, if A and B are objects in C , then V o (A ⊗ B) defines a non-signaling composite of V o (A) and V o (B). Provided either that C satisfies a "local tomography" condition, or that C is compact closed, this defines a symmetric monoidal structure on the image category, and makes V o a (strict) monoidal functor.

Arxiv preprint arxiv:0908.2354, 2009

We review some of our recent results (with collaborators) on information processing in an ordered linear spaces framework for probabilistic theories. These include demonstrations that many "inherently quantum" phenomena are in reality quite general characteristics of non-classical theories, quantum or otherwise. As an example, a set of states in such a theory is broadcastable if, and only if, it is

2010

We consider symmetric monoidal categories of convex operational models, and adduce necessary and sufficient conditions for these to be compact-closed or daggercompact. Compact closure amounts to the condition that all processes be implementable by means of a "remote evaluation" protocol (generalizing standard conclusive quantum teleportation protocols), which amounts to a form of classical conditioning. This requires the existence, for each system, of a bipartite state involving a further system, whose corresponding conditioning map is an isomorphism, and an an effect whose corresponding map is the inverse of this isomorphism. Degenerate compact closure, in which systems act as their own duals in the compact structure, means that one may take this extension to be the system itself, so the isomorphism implies that systems are weakly self-dual as ordered vector spaces. Degenerate dagger compact categories emerge from a further restriction, namely, that the bipartite "isomorphism" state and effect be symmetric.

2012

Abstract: A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in this context are operational theories, and categorical quantum mechanics. The aim of the present paper is to establish strong connections between these two formalisms.

2018

Since its inception, many physicists have seen in quantum mechanics the possibility, if not the necessity, of bringing cognitive aspects into the play, which were instead absent, or unnoticed, in the previous classical theories. In this article, we outline the path that led us to support the hypothesis that our physical reality is fundamentally conceptual-like and cognitivisticlike. However, contrary to the ‘abstract ego hypothesis’ introduced by John von Neumann and further explored, in more recent times, by Henry Stapp, our approach does not rely on the measurement problem as expressing a possible ‘gap in physical causation’, which would point to a reality lying beyond the mind-matter distinction. On the contrary, in our approach the measurement problem is considered to be essentially solved, at least for what concerns the origin of quantum probabilities, which we have reasons to believe they would be epistemic. Our conclusion that conceptuality and cognition would be an integral ...

2021

Operational frameworks are very useful to study the foundations of quantum mechanics, and are sometimes used to promote antirealist attitudes towards the theory. The aim of this paper is to review three arguments aiming at defending an antirealist reading of quantum physics based on various developments of standard quantum mechanics appealing to notions such as quantum information, non-causal correlations and indefinite causal orders. Those arguments will be discussed in order to show that they are not convincing. Instead, it is argued that there is conceptually no argument that could favour realist or antirealist attitudes towards quantum mechanics based solely on some features of some formalism. In particular, both realist and antirealist views are well accomodable within operational formulations of the theory. The reason for this is that the realist/antirealist debate is located at a purely epistemic level, which is not engaged by formal aspects of theories. As such, operational ...

Entropy, 2024

The new logic of partitions is dual to the usual Boolean logic of subsets (usually presented only in the special case of the logic of propositions) in the sense that partitions and subsets are category-theoretic duals. The new information measure of logical entropy is the normalized quantitative version of partitions. The new approach to interpreting quantum mechanics (QM) is showing that the mathematics (not the physics) of QM is the linearized Hilbert space version of the mathematics of partitions. Or, putting it the other way around, the math of partitions is a skeletal version of the math of QM. The key concepts throughout this progression from logic, to logical information, to quantum theory are distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. The distinctions of a partition are the ordered pairs of elements from the underlying set that are in different blocks of the partition and logical entropy is defined (initially) as the normalized number of distinctions. The cognate notions of definiteness and distinguishability run throughout the math of QM, e.g., in the key non-classical notion of superposition (= ontic indefiniteness) and in the Feynman rules for adding amplitudes (indistinguishable alternatives) versus adding probabilities (distinguishable alternatives).

The Frontiers Collection, 2015

We believe that the hypothesis 'it from bit' originates from the assumption that probabilities have a fundamental, irremovable status in quantum theory. We argue against this assumption and highlight four well-known reformulations / modifications of the theory in which probabilities and the measuring apparatus do not play a fundamental role. These are: Bohmian Mechanics, Dynamical Collapse Models, Trace Dynamics, and Quantum Theory without Classical Time. Here the 'it' is primary and the 'bit' is derived from the 'it'.

Annalen der Physik

The reconstruction of quantum physics has been connected with the interpretation of the quantum formalism, and has continued to be so with the recent deeper consideration of the relation of information to quantum states and processes. This recent form of reconstruction has mainly involved conceiving quantum theory on the basis of informational principles, providing new perspectives on physical correlations and entanglement that can be used to encode information. By contrast to the traditional, interpretational approach to the foundations of quantum mechanics, which attempts directly to establish the meaning of the elements of the theory and often touches on metaphysical issues, the newer, more purely reconstructive approach sometimes defers this task, focusing instead on the mathematical derivation of the theoretical apparatus from simple principles or axioms. In its most pure form, this sort of theory reconstruction is fundamentally the mathematical derivation of the elements of theory from explicitly presented, often operational principles involving a minimum of extra-mathematical content. Here, a representative series of specifically information-based treatments-from partial reconstructions that make connections with information to rigorous axiomatizations, including those involving the theories of generalized probability and abstract systems-is reviewed.