Complementarity Beyond Definite Causal Order

Abstract

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 linear additive complementarity relation exists that simultaneously captures path distinguishability, spatial coherence, and coherence between causal orders. This reveals a fundamental separation between spatial and causal resources, which reside on different subsystems and are therefore not jointly constrained by a single quantum state. While tracing out the order qubit recovers the standard duality relation at the level of the reduced quanton--detector state, coherence between causal orders is not accessible at the level of the reduced description. To capture this contribution, we introduce \emph{causal coherence}, defined as the coherence of the order qubit, which quantifies interference between alternative causal orders and is operationally measurable; we construct explicit processes in which spatial duality is saturated while causal coherence is maximal. We further show that complementarity admits a state-dependent entropic formulation based on incompatible measurements on the causal degree of freedom; unlike generic state-dependent relations, this formulation arises from a universal uncertainty principle and provides a canonical operationally meaningful description. These results establish that complementarity is fundamentally shaped by causal structure and cannot, in general, be fully captured at the level of reduced quantum states alone.

Figures (2)

• Figure 1: Schematic representation of a quantum switch controlling the temporal order between a which--path interaction $A$ and an interference operation $B$ in an interferometric setup. The order qubit places the two possible orders $A\prec B$ and $B\prec A$ in a superposition of alternative causal orders.
• Figure 2: Geometric representation of complementarity in the presence of indefinite causal order. The accessible region in the $(C_q + D_Q^{\mathrm{ICO}},\, C_{\mathrm{causal}})$ plane is the full unit square. In particular, the point $(1,1)$ is achievable in the commuting sector of the quantum switch, demonstrating the absence of any universal linear tradeoff relation between spatial and causal quantities.

Theorems & Definitions (6)

• Theorem 5.1: No universal linear additive complementarity
• proof
• Theorem 6.1: State--dependent entropic complementarity
• Lemma D.1: Detector correlations underlying causal-order discrimination
• proof
• Corollary D.2: Operational interpretation