Constructor Theory of Information

Abstract: We present a theory of information expressed solely in terms of which transformations of physical systems are possible and which are impossible - i.e. in constructor-theoretic terms. Although it includes conjectured laws of physics that are directly about information, independently of the details of particular physical instantiations, it does not regard information as an a priori mathematical or logical concept, but as something whose nature and properties are determined by the laws of physics alone. It does not suffer from the circularity at the foundations of existing information theory (namely that information and distinguishability are each defined in terms of the other). It explains the relationship between classical and quantum information, and reveals the single, constructor-theoretic property underlying the most distinctive phenomena associated with the latter, including the lack of in-principle distinguishability of some states, the impossibility of cloning, the existence of pairs of variables that cannot simultaneously have sharp values, the fact that measurement processes can be both deterministic and unpredictable, the irreducible perturbation caused by measurement, and entanglement (locally inaccessible information).

Constructor Theory of Information: A Bridge Between Classical and Quantum Realms

Deutsch and Marletto's paper on the constructor theory of information proposes a novel approach to understanding information that diverges from traditional frameworks like Shannon's classical information theory. The authors suggest a reevaluation of information in distinctly physical terms, asserting that information's properties are derived fundamentally from the laws of physics, without relegating it to abstract mathematical or logical concepts.

Key Insights and Novel Contributions

The paper underscores the non-circular definition of information, addressing the limitations inherent in classical and quantum theories of information. One of the critical contributions of this work is resolving the oft-debated problem of defining distinguishability without relying on previously assumed, yet undefined, notions. They assert that information, predominantly seen as an abstract entity, is primarily physical, evident in how information can be moved between distinct physical media while retaining its informational qualities.

A major highlight of this work is its emphasis on the "substrate-independence" of information — drawing parallels with conserved quantities in physics like energy, which are conserved across transformations. They challenge traditional theories like Shannon's by demonstrating the inadequacy in dealing with quantum phenomena such as the cloning impossibility and entanglement.

Constructor Theory Framework

The authors advance the constructor theory as a framework that shifts focus from states of systems to the possible and impossible transformations these systems can undergo. The basic premise is that all relevant laws of physics are expressible in terms of these transformations. This starkly contrasts with the conventional state-space and probabilistic approach prevalent in both classical and quantum mechanics.

Key to this framework is the distinction between two kinds of systems: constructors, which effect changes repeatedly without changing, and substrates, the entities being transformed. The possible and impossible tasks define the theory’s landscape, encompassing tasks that can forever approach arbitrarily high precision and those prohibited by laws of physics.

Bridging Classical and Quantum Information

In integrating classical and quantum information concepts, Deutsch and Marletto propose defining computations and information in terms of constructor theory. Here, an information medium is conceptualized to allow the reversible computation over a set of its possible states. A significant insight is the emergent coherence and entanglement which they relate to the clearer notion of "superinformation," bridging traditional information concepts with quantum mechanics' counter-intuitive phenomena like entanglement and measurement perturbation.

Implications and Future Developments

The implications of constructor theory extend beyond academic elegance, suggesting potential refactoring of how we handle information processing at a foundational level. As classical constructs fail to govern quantum phenomena adequately, or to address the fundamental uncertainties of state determinism in quantum mechanics, constructor theory provides a vivid framework for re-evaluating these with clear demarcation between tasks designated as possible or impossible.

Future developments in AI and information physics could benefit from this theory, offering robust axiomatic foundations for machine reasoning and information transfer, especially as we edge towards integrating quantum computation and classical data processing.

Conclusion

Deutsch and Marletto's exploration of the constructor theory of information presents a critical shift in understanding information as fundamentally physical, with axioms grounded in transformations rather than states. This refashioning aligns classical theories with quantum phenomena, fostering a potentially richer, integrated framework that could invigorate further inquiry into the fundamental constructs of information science and quantum theory.