An Updated Quantum Complementarity Principle

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.}

In this arXiv version, we have corrected a minor error present in the published version. Specifically, in the 5th line after Equation 3.7, where it originally states ${\mathcal{V}}_0={\mathcal{V}}_1=1$, it should instead read ${\mathcal{V}}_0=1$. This error does not affect the results or conclusions of the paper. We are grateful to Stephen Walborn for bringing this to our attention

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• Quantum observable's reality erasure with spacelike-separated operations
• Correlating Local Quantum Reality with Causally Disconnected Choices
• Entanglement swapping for partially entangled qudits and the role of quantum complementarity
• Digital quantum simulation of bosonic systems and quantum complementarity