[Paper Review] The Ambiguity of Simplicity

Cina Aghamohammadi, John R. Mahoney+1 more|arXiv (Cornell University)|Feb 27, 2016

Quantum Mechanics and Applications39 references3 citations

TL;DR

This paper demonstrates that the simplicity of physical systems is ambiguous between classical and quantum descriptions: a system deemed simpler classically may be more complex quantum mechanically, and vice versa. Using statistical complexity measures on stochastic processes like the Ising spin chain, the authors show that relative simplicity can reverse between frameworks, challenging the notion of absolute physical simplicity and undermining principles like Ockham’s Razor in foundational physics.

ABSTRACT

A system's apparent simplicity depends on whether it is represented classically or quantally. This is not so surprising, as classical and quantum physics are descriptive frameworks built on different assumptions that capture, emphasize, and express different properties and mechanisms. What is surprising is that, as we demonstrate, simplicity is ambiguous: the relative simplicity between two systems can change sign when moving between classical and quantum descriptions. Thus, notions of absolute physical simplicity---minimal structure or memory---at best form a partial, not a total, order. This suggests that appeals to principles of physical simplicity, via Ockham's Razor or to the "elegance" of competing theories, may be fundamentally subjective, perhaps even beyond the purview of physics itself. It also raises challenging questions in model selection between classical and quantum descriptions. Fortunately, experiments are now beginning to probe measures of simplicity, creating the potential to directly test for ambiguity.

Motivation & Objective

• To investigate whether the concept of physical simplicity is absolute or relative across classical and quantum descriptions of stochastic processes.
• To examine whether quantum mechanics can genuinely simplify the description of classical systems, as suggested by quantum advantage in modeling.
• To determine whether the relative simplicity of two systems can reverse when switching between classical and quantum frameworks.
• To assess the implications of this ambiguity for model selection in physics and statistical inference.
• To propose experimental tests for measuring simplicity in quantum and classical regimes.

Proposed method

• The authors use statistical complexity, defined as the minimal memory required to model a stochastic process via causal states, to quantify classical simplicity.
• They apply a quantum extension of the ε-machine formalism to compute quantum statistical complexity, using quantum causal states and density matrices.
• The method relies on closed-form expressions for quantum complexity derived from the overlap of quantum states, avoiding approximation errors.
• The analysis is applied to the 1D and 2D Ising spin chains, where classical and quantum complexity are computed across temperature regimes.
• The authors derive general conditions under which two complexity measures (classical and quantum) must yield ambiguous ordering, proving ambiguity is necessary for quantum simplification.
• They analyze parameter regimes where the ambiguity persists under alternative quantum representations, demonstrating robustness.

Experimental results

Research questions

• RQ1 Can a system be simpler in a classical description than in a quantum one, and vice versa, for the same physical process?
• RQ2 Is there a regime in which the relative simplicity between two systems reverses when switching between classical and quantum modeling frameworks?
• RQ3 Does the existence of quantum simplification necessarily imply ambiguity in the ordering of simplicity?
• RQ4 How does the ambiguity of simplicity affect model selection in statistical inference when classical and quantum descriptions are both viable?
• RQ5 Can experimental measurements of complexity in quantum systems test the predicted ambiguity in simplicity?

Key findings

• For the 1D and 2D Ising spin chains, classical and quantum complexity measures produce opposite orderings of simplicity at low and high temperatures, demonstrating ambiguity.
• At zero temperature, the classical ε-machine complexity is low, while quantum complexity remains high; at high temperature, the reverse occurs, with quantum complexity decreasing faster.
• The ambiguity is robust: it persists across different quantum representations and is not an artifact of specific modeling choices.
• The authors prove that if quantum mechanics provides a genuine simplification, then ambiguity in simplicity ordering must exist—making ambiguity necessary for quantum advantage.
• The reversal of simplicity orderings implies that principles like Ockham’s Razor cannot be applied unambiguously across classical and quantum frameworks.
• The findings suggest that appeals to simplicity in theory selection may be fundamentally subjective and context-dependent, challenging the universality of simplicity as a physical criterion.