No Absolute Hierarchy of Quantum Complementarity

TL;DR

This paper demonstrates that the traditional hierarchy of quantum complementarity is not absolute but depends on the configuration of multiple quantum resources, challenging longstanding assumptions in quantum mechanics.

Contribution

It introduces a No-Comparison Theorem showing that no universal ordering of incompatible observables exists across different multi-copy configurations.

Findings

Hierarchy of complementarity varies with resource arrangement

Reversed complementarity ordering depends on resource configuration

Quantum incompatibility is context-dependent rather than intrinsic

Abstract

Bohr's principle of complementarity, prohibiting simultaneous access to certain physical properties within a single experimental arrangement, is considered to be a defining feature of quantum mechanics. It is commonly viewed as inducing an intrinsic hierarchy among incompatible observables: some sets of quantum properties are fundamentally more incompatible than others, as quantified by the maximal sharpness permitting their joint measurement. We show that this hierarchy ceases to be absolute in the multi-copy regime. Analyzing qubit spin observables, we prove a No-Comparison Theorem establishing that no global ordering of incompatible observable sets is preserved across all finite-copy configurations. In particular, two sets of observables can exhibit reversed complementarity ordering depending solely on whether the available resources are arranged as identical copies or as parallel-antiparallel pairs. Thus, the degree of quantum incompatibility is not an intrinsic property of observables alone but depends on the global configuration of the prepared quantum probes. Our results uncover a configuration-dependent structure of complementarity, reveal a subtle role of entanglement in shaping the structure of measurement limitations, and call for a reassessment of quantum information protocols under finite resources.