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.
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.
Wave-particle duality, an important and fundamental concept established upon pure quantum systems, is central to the complementarity principle in quantum mechanics. However, due to the environment effects or even the entanglement between the quanton and the which-way detector (WWD), the quanton should be described by a mixed quantum state but not a pure quantum state. Although there are some attempts to clarify the complementarity principle for mixed quantum systems, it is still unclear how the mixedness affects the complementary relation. Here, we give a ternary complementary relation (TCR) among wave, particle and mixedness aspects for arbitrary multi-state systems, which are respectively quantified by the l1 measure for quantum coherence, the which-path predictability, and the linear entropy. In particular, we show how a WWD can transform entropy into predictability and coherence. Through modifying the POVM (positive-operator valued measure) measurement on WWD, our TCR can be simplified as the wave-mixedness and particle-mixedness duality relations. Beyond enclosing the wave-particle duality relation [PRL 116, 160406 (2016)], our TCR relates to the wave-particle-entanglement complementarity relation [PRL 98, 250501 (2008); Opt. Commun. 283, 827 (2010)].
Quantum mechanics allows events to happen with no definite causal order: this can be verified by measuring a causal witness, in the same way that an entanglement witness verifies entanglement. Here, we realize a photonic quantum switch, where two operations A[over ^] and B[over ^] act in a quantum superposition of their two possible orders. The operations are on the transverse spatial mode of the photons; polarization coherently controls their order. Our implementation ensures that the operations cannot be distinguished by spatial or temporal position-further it allows qudit encoding in the target. We confirm our quantum switch has no definite causal order by constructing a causal witness and measuring its value to be 18 standard deviations beyond the definite-order bound.
We derive a generalized wave-particle duality relation for arbitrary multipath quantum interference phenomena. Beyond the conventional notion of the wave nature of a quantum system, i.e., the interference fringe visibility, we introduce a quantifier as the normalized quantum coherence, recently defined in the framework of quantum information theory. To witness the particle nature, we quantify the path distinguishability or the which-path information based on unambiguous quantum state discrimination. Then, the Bohr complementarity principle for multipath quantum interference can be stated as a duality relation between the quantum coherence and the path distinguishability. For two-path interference, the quantum coherence is identical to the interference fringe visibility, and the relation reduces to the well-known complementarity relation. The duality relation continues to hold in the case where mixedness is introduced due to possible decoherence effects.
Delayed-choice experiments challenge classical intuitions by allowing the configuration of an interferometer to be chosen after a quantum system has already entered it. While these experiments support Bohr’s complementarity principle, the introduction of quantum control reveals that the manifestation of complementarity depends critically on the full physical context — including quantum correlations and the causal order of operations. In this work, we propose and analyze a modified entanglement-assisted delayed-choice experiment that maintains the same final visibility as the standard setup but reverses its causal structure. Here, we go beyond output statistics to examine how wave-particle realism evolves throughout the interferometer. Using a contextual realism quantifier, we show that wave and particle realism can be meaningfully assigned at intermediate stages with a clear connection with the final interferometric visibility. Specifically, our results show that modifying the causal order of operations and information structure, whether through entanglement tuning or post-selection, leads to significant variations in the wave-particle complementarity relation.
The complementarity principle is a cornerstone of quantum mechanics. In this work, we investigate a complementarity relation between the predictability and quantum coherence, respectively, representing the particle-like and wave-like behaviour in wave--particle duality, in a multi-path setup. We introduce a basis-dependent predictability defined by the Bures distance between the state dephased in the chosen basis and the maximally mixed state. The predictability depends only on the observed basis statistics and admits a closed form in terms of the Bhattacharyya overlap. We derive a trade-off relation between the predictability and a coherence measure defined based on the nonclassicality of the Kirkwood--Dirac quasiprobability. For pure states, the trade-off relation is an exact equality. Remarkably, this wave--particle duality relation endows the coherence measure with an operational interpretation as the classically irreducible part of measurement randomness, yielding a tight worst-case bound on the guessing probability in source-independent QRNG.
This paper formulates a hypothesis concerning the logical structure of observational, temporal, causal, and geometrical propositions in timelike quantum-causal domains. In classical relativity, timelike order is invariant once a Lorentzian geometry is fixed. In quantum theory, however, propositions associated with incompatible operational contexts need not belong to a single Boolean algebra. This paper investigates the possibility that, when observers, clocks, matter sources, or effective geometrical degrees of freedom are quantum systems, the set of observer-indexed propositions describing records, clock readings, simultaneity, temporal order, causal accessibility, and effective spacetime geometry may be locally Boolean without admitting a single observer-independent global Boolean valuation. This hypothesis shows to be compatible with the postulates of quantum mechanics, allowing context-dependent propositional structures, and with relativistic causality, causal consistency is preserved within each effective Lorentzian context. The resulting framework states that temporal and causal propositions may remain well defined within compatible operational contexts, while the total structure of facts across observers, clocks, and quantum geometrical degrees of freedom need not possess a single global Boolean representation.