Photons in a two-path interferometer best embody wave-particle duality (WPD), which is a core concept of quantum theory. So far, the WPD relation is commonly written as
V2+D2≤1, where
V is the interference fringe visibility and
D is path distinguishability, i.e., the distinguishability of which path a photon passed. This inequality is saturated only when the which-way marker (WWM), which embodies which-path information (WPI) via an internal degree of freedom of photons, such as polarization, is in a pure state. For mixed-state WWM, conventionally defined distinguishability underestimates the amount of WPI and thus does not saturate the WPD relation. Here, we introduce a generalized measure of distinguishability
D that properly quantifies the WPI and saturates the WPD relation for all pure- and mixed-state WWM within a purification-based framework. To this end, mixed-state WWM is treated as a result of entanglement formation between the WWM and an external degree of freedom, e.g., environment, and
D is defined so that it incorporates the total WPI shared between the WWM and the environment. We show that
D thus defined is experimentally quantifiable, independently of
V, without access to the environment. We experimentally evaluate
V and
D using true single photons generated in the completely mixed (unpolarized) state, and thus verify the saturated WPD relation.
The first- and second-order wave-particle dualities for two-photon states are investigated in a Mach-Zehnder interferometer. These duality relations are experimentally tested through two-photon pure states and two-photon mixed states. The experimental results are in full agreement with the theoretical predictions. The optical fields exhibiting the same first-order particlelike and wavelike behaviors may carry entirely different second-order duality information, which implies that the higher-order duality relations are a useful tool for analyzing the wave-particle duality for multiphoton states.
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.
One of the most puzzling consequences of interpreting quantum mechanics in terms of concepts borrowed from classical physics, is the so-called wave-particle duality. Usually, wave-particle duality is illustrated in terms of complementarity between path distinguishability and fringe visibility in interference experiments. In this work, we instead propose a new type of complementarity, that between the continuous nature of waves and the discrete character of particles. Using the probabilistic methods of quantum field theory, we show that the simultaneous measurement of the wave amplitude and the number of photons in the same beam of light is, under certain circumstances, prohibited by the laws of quantum mechanics. Our results suggest that the concept of “interferometric duality'' could be eventually replaced by the more general one of “continuous-discrete duality''.
The notion of wave-particle duality may be quantified by the inequality
V2+K2< 1, relating interference fringe visibility V, and path knowledge K. With a single-photon interferometer in which polarization is used to label the paths, we have investigated the relation for various situations, including pure, mixed, and partially mixed input states. A quantum-eraser scheme has been realized that recovers interference fringes even when no which way information is available to erase.
The concept of wave-particle duality holds significant importance in the field of quantum mechanics, as it elucidates the dual nature encompassing both wavelike and particlelike properties exhibited by microscopic particles. In this paper, we construct generalized measures for the predictability and visibility of
n-path interference fringes to quantify the wave and particle properties in quantum high-dimensional systems. By employing the Morozova-Chentsov function, we ascertain that the wave-particle relationship can be delineated by the average coherence. This function exhibits a close correlation with the metric-adjusted skew information, thereby we establish complementary relations between visibility, predictability, and quantum
f entropy, which reveal deep connections between wave-particle duality and other physical quantities. Through our methodology, diverse functions can be selected to yield corresponding complementary relationships.