[Paper Review] An Information-Theoretic Approach to Quantum Theory, I: The Abstract Quantum Formalism

Philip Goyal|ArXiv.org|Feb 13, 2007

Quantum Mechanics and Applications25 references4 citations

TL;DR

This paper proposes a derivation of the abstract quantum formalism for finite-dimensional systems using an information-theoretic invariance principle, showing that the structure of quantum mechanics—including canonical commutation relations and Dirac's Poisson bracket rule—arises naturally from assumptions about information gain in measurement processes. The key result is that uniform information gain over state space leads to a symmetric Gaussian posterior with standard deviation $1/(2\sqrt{n})$, enforcing the Hilbert space structure and non-classical features of quantum theory.

ABSTRACT

In this paper and a companion paper, we attempt to systematically investigate the possibility that the concept of information may enable a derivation of the quantum formalism from a set of physically comprehensible postulates. To do so, we formulate an abstract experimental set-up and a set of assumptions based on generalizations of experimental facts that can be reasonably taken to be representative of quantum phenomena, and on theoretical ideas and principles, and show that it is possible to deduce the quantum formalism. In particular, we show that it is possible to derive the abstract quantum formalism for finite-dimensional quantum systems and the formal relations, such as the canonical commutation relationships and Dirac's Poisson Bracket rule, that are needed to apply the abstract formalism to particular systems of interest. The concept of information, via an information-theoretic invariance principle, plays a key role in the derivation, and gives rise to some of the central structural features of the quantum formalism.

Motivation & Objective

• To derive the abstract quantum formalism for finite-dimensional systems from physically comprehensible postulates.
• To identify the role of information as a foundational principle in quantum theory, replacing or supplementing traditional axioms.
• To show that the structure of quantum mechanics—including commutation relations—can emerge from information-theoretic invariance.
• To provide a physical interpretation for the choice of complex numbers and Hilbert space structure in quantum theory.

Proposed method

• Formulates an abstract experimental setup involving state preparation and measurement, with outcomes described by probabilities.
• Applies Bayesian inference to estimate unknown state parameters from measurement frequencies.
• Uses the Shannon information measure to quantify information gain about the state from $n$ measurements.
• Imposes an invariance condition requiring information gain to be independent of the true state, leading to a uniform prior over the state space.
• Derives the posterior distribution over state parameters as a multivariate Gaussian with standard deviation $1/(2\sqrt{n})$, independent of the measurement basis.
• Demonstrates that the requirement of uniform information gain forces the state space to be a unit hypersphere, leading to the Hilbert space formalism.

Experimental results

Research questions

• RQ1 Can the abstract quantum formalism be derived from information-theoretic principles rather than postulates?
• RQ2 What physical constraints arise from requiring information gain to be invariant under changes in the true state?
• RQ3 How does the structure of quantum mechanics—such as Hilbert space and commutation relations—emerge from information constraints?
• RQ4 Can the choice of complex numbers in quantum theory be derived from information-theoretic principles?
• RQ5 What is the role of the uniform prior over state space in enforcing quantum structure?

Key findings

• Information gain is maximized when the prior distribution over the state space is uniform, leading to a symmetric Gaussian posterior with standard deviation $1/(2\sqrt{n})$.
• The requirement that information gain be independent of the true state forces the prior over the positive quadrant of the unit hypersphere to be uniform, which implies a symmetric, isotropic structure.
• The posterior distribution over state parameters becomes a product of independent Gaussians, each with variance $1/(4n)$, when orthogonal coordinates are used.
• The invariance of information gain under reparametrization leads to the conclusion that the state space must be a unit hypersphere, enforcing the Hilbert space structure.
• The derivation yields the canonical commutation relations and Dirac’s Poisson bracket rule as consequences of the information-theoretic invariance principle.
• The formalism naturally leads to the complex Hilbert space structure without assuming it a priori, by showing that only complex amplitudes satisfy the required invariance and information constraints.