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.
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.
We propose a theoretical framework demonstrating that the traditional interpretation of Bohr’s complementarity principle, asserting mutual exclusivity of wave and particle signatures, arises not from a fundamental physical prohibition but from the historical restriction to projective (von Neumann) measurements. Using positive operator valued measures (POVMs), the Lindblad master equation, and quantum Fisher information, we establish that simultaneous access to interference visibility V and path distinguishability D is mathematically permitted within the Englert–Greenberger bound D² + V² ≤ 1. Five experimental architectures are proposed: a photovoltaic detection screen (baseline), an asymmetric photovoltaic slit apparatus, an optomechanical mirror configuration, a quantum memory delayed emission scheme, and a room temperature telecom fiber duality disc mapper. Three closed form metrological identities are derived: flat Fubini–Study geometry of the Englert circle (F_Q(θ) = 1), reciprocity bounds √N σ(V̂) ≥ D and √N σ(D̂) ≥ V with the estimation cost sum rule, and autonomous duality flow equations with Gudermannian (tanh/sech) solutions. We term the resulting reframing the “duality illusion”: complementarity is a constraint on joint measurement precision, not a prohibition against joint measurement. This is a purely theoretical proposal. No experiments have been performed; all numerical values are predictions. Keywords: wave particle duality, complementarity, quantum measurement, POVM, Lindblad equation, quantum Fisher information, Englert–Greenberger inequality, optomechanics, quantum dots
Updating the wave-particle duality Maria C. Bosca e-mail: bosca@ugr.es Departamento de Fisica Atomica, Molecular y Nuclear Universidad de Granada E-18071. Granada Spain The wave-particle-duality, the fundamental component of the new quantum formalism in Bohr’s opinion, must be reformulated by incorporating the results of some experiments accomplished in the last decades of twentieth century. The Bohr´s complementarity principle stated the mutual exclusiveness and joint full completeness of the two (classical) descriptions of quantum systems; after Einstein-Podolsky-Rosen’paper, the wave-particle duality, or wave-particle complementarity, could be expressed by stating that it is impossible to build up an experimental arrangement in which we observe at the same time both corpuscular and wave aspects. In a two-slit experiment, they would correspond, respectively, to the which-way knowledge and the observation of interference pattern. Bohr showed this mutual exclusivity in numerous examples , and linked it to the unavoidable disturbance inherent in any measurement event. In quantum mechanical formalism, the complementarity principle has a clear mathematical expression: two observables are complementary if precise knowledge of one of them implies that all possible outcomes of the other are equally probable; their extension to classical concepts (as wave and particle) is not concerned. In 1991 Scully et al published a variant of the two-slit experiment that incorporates two micromasers cavities and a laser beam to provide which-path information without net momentum transferred during the interaction ; the impossibility of know which slit an atom went through and still observe the interference fringes is preserved by the establishing of quantum correlations between the measuring apparatus and the system being observed. They claimed that complementarity, of which wave-particle duality would be to them a manifestation, is more fundamental than the uncertainty principle . In 1996 B-G. Englert, following an approach originally due to Wooters and Zurek , derived , without making use of Heisenberg’s uncertainty relation, an inequality that quantifies the mutual compatibility relation between fringe visibility and which-way information. The inequality, that they denominated as “interferometric duality”, has the expression D^2 + V^2 ≤ 1, where D stands for the distinguishability of the ways and V for the fringe visibility; both of them are mathematical expressions that can be measured to check experimentally the inequality . Today, it is clear that intermediate particle-wave behaviours exist and, in addition to that, there are single experiments in which both classical wave-like and particle-like behaviours are showed total and simultaneously on an individual system. For instance, in the Bose’s double-prism experiment , tunnelling and perfect anticoincidence were observed in single photon states. Consequently, the meaning of the wave-particle duality must incorporate the simultaneous use of the two classical descriptions in the interpretation of experiments, loosing their original mutual exclusivity , which is incorporated as an extreme case in the new interferometric duality, a continuous quantum concept . The wave-particle duality, therefore, must be formulated as an interpretative addition to quantum mechanics, to which is possible to renounce if any pretension of visualize quantum phenomena in terms of classical concepts and intuitions is abandoned.
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.
We present a geometric reparameterization of quantum complementarity data using the arc-length of a Thales semicircle as a natural coordinate on the constraint manifold defined by the Englert relation V^2 + D^2 \leq 1. Applying this transformation to atom-interferometry measurements from Schwindt et al. (1999), which span the full range of wave–particle complementarity, reveals a previously hidden geometric structure: the corridor boundaries at amplitude ratios \lambda = 1/e and \lambda = e are conjugate, occurring at identical visibility, distinguishability, and equal arc-length displacement from the symmetric apex. While the standard V(D) representation saturates the Englert bound and shows no distinguished interior structure, the arc-length representation V(\Delta s) exposes a symmetric transition region corresponding to an intermediate compatibility class previously identified through principal component analysis (PCA) diagnostics in two-channel systems. The observed correspondence does not imply deviations from quantum mechanics or the introduction of new constants; rather, it provides a geometric organization of existing experimental results and clarifies why specific partition ratios require additional physical constraints beyond generic complementarity. This work positions Thales arc-length as a diagnostic coordinate that unifies experimental complementarity data with constraint-geometry analyses, suggesting a systematic framework for comparing diverse interferometric systems without modifying established theory.
In a recent paper, D\"urr [Phys. Rev. A 64, 042113 (2001)] proposed an interesting multibeam generalization of the quantitative formulation of interferometric wave-particle duality, discovered by Englert for two-beam interferometers. The proposed generalization is an inequality that relates a generalized measure of the fringe visibility to certain measures of the maximum amount of which-way knowledge that can be stored in a which-way detector. We construct an explicit example where, with three beams in a pure state, the scheme proposed by D\"urr leads to the possibility of an ideal which-way detector that can achieve a better path discrimination at the same time as a better fringe visibility. In our opinion, this seems to be in contrast with the intuitive idea of complementarity, as it is implemented in the two-beams case, where an increase in path discrimination always implies a decrease of fringe visibility, if the beams and the detector are in pure states.
The wave–particle duality relation derived by Englert sets an upper bound of the extractable information from wave and particle properties in a two-path interferometer. Surprisingly, previous studies demonstrated that the introduction of a quantum beamsplitter in the interferometer could break the limitation of this upper bound, due to interference between wave and particle states. Along the other line, a lot of efforts have been made to generalize this relation from the two-path setup to the N-path case. Thus, it is an interesting question that whether a quantum N-path beamsplitter can break the limitation as well. This paper systemically studies the model of a quantum N-path beamsplitter, and finds that the generalized wave–particle duality relation between interference visibility and path distinguishability is also broken in certain situations. We further study the maximal extractable information’s reliance on the interference between wave and particle properties, and derive a quantitative description. We then propose an experimental methodology to verify the break of the limitation. Our work reflects the effect of quantum superposition on wave–particle duality, and exhibits a new aspect of the relation between visibility and path distinguishability in N-path interference. Moreover, it implies the observer’s influence on wave–particle duality.