Complementarity is the principle that certain pairs of quantum properties cannot be simultaneously accessed within a single experimental arrangement. Position and momentum. Path and interference. Spin along different axes. The trade-off is quantitative — measuring one property more precisely necessarily increases the uncertainty in the complementary property — and the degree of incompatibility between different pairs of observables can be quantified. Some pairs are more incompatible than others. Position and momentum are maximally complementary; certain spin measurements are partially complementary. The incompatibility is a property of the observables themselves, determined by the algebraic structure of the operators.
This establishes a hierarchy: observable pair A is more incompatible than pair B, as an intrinsic fact about those observables. The hierarchy should be absolute — it shouldn't depend on how you choose to probe the system, only on the mathematical structure of the observables.
Agarwal, Naik, Chakraborty, Kar, Patra, Roy Chowdhury, and Banik (arXiv 2602.22792, February 2026) prove that this hierarchy is not absolute. They establish a No-Comparison Theorem: no global ordering of incompatible observable sets is preserved across all finite-copy resource configurations.
The mechanism operates in the multi-copy regime, where the experimenter has access to multiple copies of the quantum state. With a single copy, the complementarity ordering between observables is fixed — pair A is more incompatible than pair B, full stop. But with multiple copies, the experimenter can arrange the copies in different configurations — all identical, or in parallel-antiparallel pairs, or in other collective arrangements — and the complementarity ordering between the same observable pairs can reverse depending on the configuration.
Two sets of observables that exhibit ordering A > B when the available copies are arranged as identical states can exhibit the reversed ordering B > A when the same copies are arranged as parallel-antiparallel pairs. The incompatibility of the observables did not change. The observables are the same operators with the same algebraic properties. What changed is the resource structure — how the quantum probes are organized relative to each other. The hierarchy depends on the measurement context, not just the observables.
The No-Comparison Theorem is not a statement about experimental limitations or finite precision. It is a structural result about the theory: quantum mechanics does not admit a universal ordering of observable incompatibility that is independent of the resource configuration. The ordering is contextual in a precise sense — it depends on how many copies you have and how they are arranged.
This matters for quantum information applications where incompatibility is a resource. Quantum key distribution, quantum random number generation, and certain quantum computing protocols exploit the impossibility of simultaneously knowing complementary properties. If you need to choose which observables to measure to maximize your information advantage, the optimal choice depends on the resource structure of your probes. The “most incompatible” pair of observables — the pair that gives you the strongest quantum advantage — is not fixed. It changes with your experimental arrangement.
Complementarity is not a ranking of observables. It is a relation between observables and the configuration of the observer.