An Updated Quantum Complementarity Principle

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: $\mathit{\text{For}}\mathit{\text{a}}\mathit{\text{given}}\mathit{\text{quantum}}\mathit{\text{state}}\mathit{\text{preparation}}{\rho }_t\mathit{\text{at}}\mathit{\text{a}}\mathit{\text{specific}}\mathit{\text{instant}}\mathit{\text{of}}\mathit{\text{time}}t\mathit{\text{,}}\mathit{\text{the}}\mathit{\text{wave}}\mathit{\text{and}}\mathit{\text{particle}}\mathit{\text{behaviors}}\mathit{\text{of}}\mathit{\text{a}}\mathit{\text{quanton}}\mathit{\text{are}}\mathit{\text{constrained}}\mathit{\text{by}}\mathit{\text{a}}\mathit{\text{complementarity}}\mathit{\text{relation}}\mathfrak{W}({\rho }_t)+\mathfrak{P}({\rho }_t)\le \alpha (d)\mathit{\text{,}}\mathit{\text{which}}\mathit{\text{is}}\mathit{\text{derived}}\mathit{\text{directly}}\mathit{\text{from}}\mathit{\text{the}}\mathit{\text{axioms}}\mathit{\text{of}}\mathit{\text{quantum}}\mathit{\text{mechanics.}}$