[Paper Review] Quaternionic quantum mechanics allows non-local boxes

Matthew McKague|ArXiv.org|Nov 9, 2009

Quantum Mechanics and Applications12 references3 citations

TL;DR

This paper demonstrates that quaternionic quantum mechanics supports non-local boxes—non-local correlations stronger than those allowed by standard quantum mechanics—by constructing a protocol using non-commutative quaternionic operations and time-ordered unitary transformations. The key result is that such non-locality implies trivial communication complexity for all boolean functions, ruling out quaternionic quantum mechanics under assumptions of non-trivial communication complexity or information causality.

ABSTRACT

We consider non-local properties of quanternionic quantum mechanics, in which the complex numbers are replaced by the quaternions as the underlying algebra. Specifically, we show that it is possible to construct a non-local box. This allows one to rule out quaternionic quantum mechanics using assumptions about communication complexity or information causality.

Motivation & Objective

• To investigate whether quaternionic quantum mechanics, which replaces complex amplitudes with quaternions, can support stronger non-local correlations than standard quantum mechanics.
• To determine if non-local boxes—systems exhibiting non-signaling but non-quantum correlations—can be constructed within the quaternionic framework.
• To assess whether the existence of non-local boxes in quaternionic quantum mechanics contradicts known physical principles such as communication complexity bounds or information causality.
• To clarify the physical viability of quaternionic quantum mechanics by comparing its non-local properties to those of complex quantum mechanics and classical theories.

Proposed method

• Construct a bipartite entangled state |ψ⟩ = (|00⟩ + |11⟩)/√2 in a quaternionic Hilbert space.
• Apply time-ordered unitary operations R_i and R_j—defined over quaternions—on the two subsystems based on local inputs a and b.
• Use the non-commutativity of quaternions (e.g., ij = k, ji = -k) to generate different relative phases depending on the order of operations.
• Define measurement bases |+⟩ and |−⟩ to detect the relative phase, yielding correlated or anti-correlated outcomes.
• Analyze the resulting output bit strings x and y to verify that x ⊕ y = ab, satisfying the non-local box condition.
• Leverage known results on communication complexity and information causality to argue that the existence of such a non-local box implies physical incompatibility with standard physical principles.

Experimental results

Research questions

• RQ1 Can non-local boxes be constructed within quaternionic quantum mechanics, given the non-commutativity of quaternions?
• RQ2 Does the existence of non-local boxes in quaternionic quantum mechanics imply trivial communication complexity for all boolean functions?
• RQ3 How does the non-commutative structure of quaternions enable stronger non-local correlations than those in standard complex quantum mechanics?
• RQ4 Can the non-locality in quaternionic quantum mechanics be ruled out using assumptions such as non-trivial communication complexity or information causality?
• RQ5 Is there a fundamental physical principle that distinguishes quaternionic quantum mechanics from standard quantum mechanics based on non-local correlations?

Key findings

• A non-local box can be explicitly constructed in quaternionic quantum mechanics using time-ordered quaternionic unitary operations R_i and R_j.
• The relative phase between |00⟩ and |11⟩ states changes to −k or +k depending on the order of operations, leading to correlated or anti-correlated measurement outcomes.
• The protocol achieves x ⊕ y = ab for all input pairs (a,b), satisfying the definition of a non-local box.
• The existence of such a non-local box implies that all boolean functions have trivial communication complexity in this framework.
• This result implies that quaternionic quantum mechanics is incompatible with physical principles such as non-trivial communication complexity or information causality, which hold in classical and quantum theories.
• Thus, the paper provides a physical principle—communication complexity or information causality—that rules out quaternionic quantum mechanics as a viable physical theory.