Informational derivation of Quantum Theory

Abstract: Quantum theory can be derived from purely informational principles. Five elementary axioms-causality, perfect distinguishability, ideal compression, local distinguishability, and pure conditioning-define a broad class of theories of information-processing that can be regarded as a standard. One postulate-purification-singles out quantum theory within this class. The main structures of quantum theory, such as the representation of mixed states as convex combinations of perfectly distinguishable pure states, are derived directly from the principles without using the Hilbert space framework.

• The paper derives quantum theory from operational information principles, replacing traditional Hilbert space foundations with an axiomatic framework.
• It introduces five key axioms and the crucial Purification Postulate to uniquely characterize quantum mechanics through observable operational implications.
• By demonstrating derivations for qubits and state transformations, the work paves the way for theoretical extensions in quantum gravity and non-causal physical frameworks.

Informational Derivation of Quantum Theory: A Review

The paper "Informational Derivation of Quantum Theory" presents a foundational exploration into the axiomatic underpinnings of quantum theory using information-theoretic principles. Authored by Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti, the research aims to characterize quantum theory uniquely through operational principles without relying on the mathematical framework traditionally associated with quantum mechanics, such as Hilbert spaces.

Overview and Structure

The authors propose that quantum theory arises naturally from a set of informational axioms rather than abstract mathematical assumptions. The core idea is to derive the quantum formalism as a physical theory based on principles expressing conditions for information processing. The paper's structure is methodical, beginning with an operational framework that underlies all information-processing theories. From this, it introduces several axioms and a critical postulate that distinguishes quantum theory within this framework.

The paper extensively details the operational-probabilistic framework, emphasizing circuits and transformations, drawing on the effects and states that build the foundation for reasoning about physical theories. It systematically outlines and proves the consequences of each axiom, leading to the derivation of quantum mechanics.

Axioms and Principles

Five informational axioms are deployed:

1. Causality ensures the independence of preparation probabilities from future measurement choices.
2. Perfect distinguishability posits that every non-completely mixed state can be distinguished from some other state.
3. Ideal compression allows for optimal encoding of state information without loss.
4. Local distinguishability requires the differentiation of bipartite states through local measurements.
5. Pure conditioning stipulates that measurement outcomes on one part of a pure state induce pure states on another.

Crucially, the paper introduces the Purification Postulate, which asserts that every mixed state can be seen as part of a pure state of a larger system. This postulate is pivotal in characterizing quantum theory uniquely.

Insights and Results

The paper achieves a significant result by demonstrating the informational derivation of the qubit, showing that systems with two perfectly distinguishable states adhere to the properties of quantum bits in conventional theory. An important numerical aspect of this paper is the establishment that the dimension $D_A$ of a system is equal to $d_A^2$, aligning with the dimension of density matrices for quantum systems with a $d_A$-dimensional Hilbert space.

Further, the derivation traces quantum operations and transformations, detailing how these emerge naturally from the axioms, with the superposition principle for general systems being a noteworthy advancement from this work.

Implications and Future Directions

This derivation method not only enriches the conceptual understanding of quantum theory but also establishes a path for exploring theoretical extensions and constraints of quantum mechanics through informational lenses. It provides a framework for interpreting quantum phenomena without presupposing mathematical edifice, opening the door to potential theoretical expansions in domains where traditional quantum mechanics falters, such as gravitation and spacetime theories that lack a fixed causal structure.

The work hints at prospective investigations into relaxing some of the operational axioms, leading to theories that might incorporate non-causal interactions or deviations from local distinguishability, thereby contributing to quantum gravity or other speculative physics areas.

In conclusion, the paper sets a robust foundation for the informational derivation of quantum mechanics, with the purification principle firmly demarcating its boundary within a class of information-processing theories. This work is crucial for theoretical physicists and computer scientists interested in the foundational aspects of quantum theory, providing compelling evidence that informational paradigms can fundamentally define the quantum world.