Complementarity Beyond Definite Causal Order
Wave--particle duality is a cornerstone of quantum mechanics, traditionally formulated under definite causal order. We investigate how complementarity is modified when the temporal order of operations is coherently superposed, as in the quantum switch. We show that no universal, state-independent, linear additive complementarity relation exists that combines spatial coherence, path distinguishability, and causal coherence within a common unit-normalized bound. This result follows from explicit extremal constructions in the quantum-switch family: although the three quantities are derived from the same global switch state, the standard spatial duality relation can be saturated while the reduced control coherence is simultaneously maximal. Tracing out the order qubit recovers the standard duality relation at the level of the reduced quanton--detector state, while coherence associated with alternative causal orders remains inaccessible in this reduced description. To characterize this contribution, we introduce \emph{causal coherence}, defined as the coherence of the reduced order qubit in the causal-order basis. We further show that complementarity admits a state-dependent entropic formulation based on incompatible measurements on the causal degree of freedom, which arises from a universal uncertainty principle and provides a canonical operationally meaningful description. These results show that operational formulations of complementarity can depend on causal structure and cannot, in general, be fully characterized at the level of reduced quantum states alone.