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.
A quantum system (quanton) traverses an interferometer with
N equally probable paths and interacts with another quantum system (detector) that stores path information in a set of symmetric states. In this framework, we present a systematic quantitative study of wave-particle duality based on entropic trade-off relations between quantum coherence, characterized by the relative entropy of coherence of the quanton state, and which-path knowledge, quantified by the mutual information obtained through detector-state discrimination. Unlike prior upper-bound-based formulations, we obtain an exact quantification of which-path knowledge for the discrimination strategies considered here, encompassing previous strategies and extending beyond them. A general optimal measurement is implemented through a state-separation map governed by a continuous parameter
ξ∈[0,1], followed by minimum-error (ME) discrimination of either the successful outputs only (standard approach) or both successful and failure outputs (concatenated approach). Our framework identifies a strategy-dependent hierarchy that generally emerges in the tightness of the resulting duality relations. At
ξ=0, both approaches reduce to the ME measurement, which typically yields the tightest relation. For
ξ>0, each approach yields a distinct relation that generally becomes less tight as
ξ increases, with the concatenated strategy always yielding a relation at least as tight as its standard counterpart. We further characterize saturation of these relations by identifying the detector-state sets that lead to it and show that nontrivial saturation occurs only for interferometers with a nonprime number of paths. From these sets, we elucidate how the underlying structure of the quanton-detector correlations gives rise to this behavior and explains the lack of universal saturation in multipath duality relations for pure states.
Wave-particle duality demonstrates a competition relation between wave and particle behavior for a particle going through an interferometer. This duality can be formulated as an inequality, which upper bounds the sum of interference visibility and path information. However, if the particle is entangled with a quantum memory, then the bound may decrease. Here, we find the duality relation between coherence and path information for a particle going through a multipath interferometer in the presence of a quantum memory, offering an upper bound on the duality relation which is directly connected with the amount of entanglement between the particle and the quantum memory.
Abstract Wave-particle duality demonstrates a competition relation between wave and particle behavior for a particle going through an interferometer. This duality can be formulated as an inequality, which upper bounds the sum of interference visibility and path information. However, if the particle is entangled with a quantum memory, then the bound may decrease. Here, we find the duality relation between coherence and path information for a particle going through a multipath interferometer in the presence of a quantum memory, offering an upper bound on the duality relation which is directly connected with the amount of entanglement between the particle and the quantum memory.
Within the established theoretical framework of quantum mechanics, interference always occurs between pairs of paths through an interferometer.Higher order interferences with multiple constituents are excluded by Born's rule and can only exist in generalized probabilistic theories.Thus, high-precision experiments searching for such higher order interferences are a powerful method to distinguish between quantum mechanics and more general theories.Here, we perform such a test in an optical multi-path interferometer, which avoids crucial systematic errors, has access to the entire phase space and is more stable than previous experiments.Our results are in accordance with quantum mechanics and rule out the existence of higher order interference terms in optical interferometry to an extent that is more than four orders of magnitude smaller than the expected pairwise interference, refining previous bounds by two orders of magnitude.
Within the established theoretical framework of quantum mechanics, interference always occurs between pairs of trajectories. Higher order interferences with multiple constituents are, however, excluded by Born's rule and can only exist in generalized probabilistic theories. Thus, high-precision experiments searching for such higher order interferences are a powerful method to test the validity of the Born rule and distinguish between quantum mechanics and more general theories. Here, we perform such a test in optical multi-path interferometers. Our results permit to rule out the existence of higher order interference terms to an extent which is more than four orders of magnitude smaller than the expected pairwise interference, refining previous bounds by two orders of magnitudes. This establishes the hitherto tightest constraints on generalized interference theories.