Wave-particle duality is traditionally viewed as a binary complementarity between interference visibility V and path distinguishability D. In this work, we develop a unified continuous framework where "wave-like" and "particle-like" behaviors are represented as basis-dependent features of a single quantum state, smoothly interpolated by the fractional Fourier transform (FrFT) and broader metaplectic groups. We introduce two operational indices: a coherence index 𝐶(𝛼), defined as the ℓ₁-norm of off-diagonal density matrix elements in the 𝛼-representation, and a localization index 𝐿(𝛼), derived from the Shannon entropy or Fisher information of the 𝛼-domain distribution. We prove that for Gaussian superpositions, these indices reveal a continuous trade-off that generalizes the Englert-Greenberger-Yasin relation. Furthermore, we establish a state-independent entropic uncertainty bound for linear canonical transforms (LCTs) and propose experimental protocols using photonic and matter-wave platforms to map trajectories in the (C, L) plane.
Wave-particle duality is traditionally viewed as a binary complementarity between interference visibility V and path distinguishability D. In this work, we develop a unified continuous framework where "wave-like" and "particle-like" behaviors are represented as basis-dependent features of a single quantum state, smoothly interpolated by the fractional Fourier transform (FrFT) and broader metaplectic groups. We introduce two operational indices: a coherence index 𝐶(𝛼), defined as the ℓ₁-norm of off-diagonal density matrix elements in the 𝛼-representation, and a localization index 𝐿(𝛼), derived from the Shannon entropy or Fisher information of the 𝛼-domain distribution. We prove that for Gaussian superpositions, these indices reveal a continuous trade-off that generalizes the Englert-Greenberger-Yasin relation. Furthermore, we establish a state-independent entropic uncertainty bound for linear canonical transforms (LCTs) and propose experimental protocols using photonic and matter-wave platforms to map trajectories in the (C, L) plane.
An interferometer---no matter how clever the design---cannot reveal both the wave and the particle behavior of a quantum system. This fundamental idea has been captured by inequalities, so-called wave-particle duality relations (WPDRs), that upper bound the sum of the fringe visibility (wave behavior) and path distinguishability (particle behavior). Another fundamental idea is Heisenberg's uncertainty principle, stating that some pairs of observables cannot be known simultaneously. Recent work has unified these two principles for two-path interferometers. Here we extend this unification to n-path interferometers, showing that WPDRs correspond to a modern formulation of the uncertainty principle stated in terms of entropies. Furthermore, our unification provides a framework for solving an outstanding problem of how to formulate universally valid WPDRs for interferometers with more than two paths, and we employ this framework to derive some novel WPDRs.
An interferometer - no matter how clever the design - cannot allow the experimenter to see both the wave and particle behavior of a quantum system. This fundamental idea has been captured quantitatively by inequalities, so-called wave-particle duality relations (WPDRs), that upper bound the sum of the path distinguishability (particle behavior) and fringe visibility (wave behavior). Another fundamental quantum concept is Heisenberg's uncertainty principle, stating that some pairs of observables cannot be known simultaneously. The distinction between these two concepts has been debated in the literature. Here we provide closure to the debate by showing that WPDRs correspond to a modern formulation of the uncertainty principle, namely, the uncertainty relation for the min- and max-entropies, which is used in quantum cryptography. At the technical level, our unification relies on a novel operational meaning for the max-entropy in terms of the guessing probability. Furthermore, our unification provides a framework for solving an outstanding problem of how to rigorously prove WPDRs for interferometers with more than two paths, and we employ this framework to derive some novel WPDRs.
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.
Abstract The wave-particle duality, as a manifestation of Bohr’s complementarity, is usually quantified in terms of path predictability and interference visibility. Various characterizations of the wave-particle duality have been proposed from an operational perspective, most of them are in forms of inequalities, and some of them are expressed in forms of equalities by incorporating entanglement or coherence. In this work, we shed different insights into the nature of the wave-particle duality by casting it into a form of information conservation in a multi-path interferometer, with uncertainty as a unified theme. More specifically, by employing the simple yet fundamental concept of variance, we establish a resolution of unity, which can be interpreted as a complementarity relation among wave feature, particle feature, and mixedness of a quantum state. This refines or reinterprets some conventional approaches to wave-particle duality, and highlights informational aspects of the issue. The key idea of our approach lies in that a quantum state, as a Hermitian operator, can also be naturally regarded as an observable, with measurement uncertainty (in a state) and state uncertainty (in a measurement) being exploited to quantify particle feature and wave feature of a quantum state, respectively. These two kinds of uncertainties, although both are defined via variance, have fundamentally different properties and capture different features of a state. Together with the mixedness, which is a kind of uncertainty intrinsic to a quantum state, they add up to unity, and thus lead to a characterization of the wave-particle-mixedness complementarity. This triality relation is further illustrated by examples and compared with some popular wave-particle duality or triality relations.
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.
The Englert–Greenberger wave–particle duality relation V 2 + D2 ≤ 1, where V is fringevisibility and D is which-path distinguishability, is algebraically identical to the Thales con-straint L2 + h2 = R2 under the identification V = 2h (visibility equals twice the Thalesaltitude) and D = 2L (distinguishability equals twice the lateral displacement). We showthat this identification unifies three previously separate frameworks: (i) the hypotenuse rota-tion diagnostic for binary black hole inspiral, where the depth angle θh = arctan(h/L) tracksprocess progress; (ii) the torus vantage formula, where the observer angle α = arccos √aclassifies the measurement; and (iii) the Englert–Greenberger complementarity, where thewhich-path angle θ = arctan(D/V ) = arctan(L/h) quantifies wave–particle balance.The three frameworks are related by the exact identity 2α−θ = 90, which states that theobserver angle α always exceeds half the which-path angle by 45. This encodes a fixed 2:1angular relation: the two angles are linked so that each degree of which-path angle increasecorresponds to two degrees of observer-angle decrease toward equipartition at 45.Applied to laboratory data from entangled nuclear spins in isotopically engineered sili-con carbide (Bourassa et al., 2020), all identities hold to machine precision: the Englert–Greenberger constraint V 2 + D2 = 1 is conserved to 10−16, and 2α − θ = 90 holds to10−14 degrees. Through the structural mapping to effective Englert coordinates, the densitymatrix gives V = 0.990 (near-perfect fringe contrast in the interferometric analogue) andD = 0.139 (minimal which-path information), placing the system near the wave pole atθ = 8.0—consistent with the near-maximal entanglement (S = 0.986 bits) found indepen-dently from the ER = EPR diagnostic. Decoherence maps to which-path information leakingto the spin bath, with CPMG dynamical decoupling extending fringe persistence by 12×.The Born rule P = cos2(θBorn/2), the observer equation a = cos2 α, and the fringe-minimum intensity a = (1 − D)/2 are shown to be the same number: the partition fractionat the observer’s angle on the constraint circle.