An Updated Quantum Complementarity Principle

TL;DR

This paper proposes an updated formulation of the quantum complementarity principle, deriving a rigorous relation between wave and particle behaviors directly from quantum mechanics axioms to resolve previous ambiguities and potential violations.

Contribution

It introduces a new complementarity relation based on quantum axioms, clarifying the coexistence of wave and particle aspects in quantum systems.

Findings

Derived a rigorous complementarity relation from quantum axioms.

Reconciled previous claims of violations with a new formal framework.

Provided a consistent description of wave-particle duality in quantum states.

Abstract

Bohr's complementarity principle has long been a fundamental concept in quantum mechanics, positing that, within a given experimental setup, a quantum system (or quanton) can exhibit either its wave-like character, denoted as $W$, or its particle-like character, denoted as $P$, but not both simultaneously. Modern interpretations of Bohr's complementarity principle acknowledge the coexistence of these aspects in the same experiment while introducing the constraint $W + P \le \alpha$. Notably, estimations of $W$ or $P$ frequently rely on indirect retrodiction methods, a practice that has led to the claim of the violation of Bohr's complementarity principle. By taking a different route, recent advancements demonstrate that quantum complementarity relations can be rigorously derived from the axioms of quantum mechanics. To reconcile these observations and eliminate potential paradoxes or violations, we propose an updated formulation for the quantum complementarity principle, which is stated as follows: \textit{For a given quantum state preparation $\rho_t$ at a specific instant of time $t$, the wave and particle behaviors of a quanton are constrained by a complementarity relation $\mathfrak{W}(\rho_t) + \mathfrak{P}(\rho_t) \le \alpha(d)$, which is derived directly from the axioms of quantum mechanics.}