The wave–particle duality of a single particle with an n-dimensional internal degree of freedom is re-examined theoretically in a Mach–Zehnder interferometer. The famous duality relation D2+V2≤ 1 is always valid in this situation, where D is the distinguishability and V is the visibility. However, the sum of the particle information and the wave information,D2+V2, can be smaller than one for the input of a pure state if this initial pure state includes the internal degree of freedom of the particle, while the quantity D2+V2is always equal to one when the internal degree of freedom of the particle is excluded.
We present a study of the relation between wave-particle duality and quantum uncertainty in a two-path interferometer and derive equalities and inequalities involving the visibility (representing wave-like behavior), the predictability (representing particle-like behavior), and their variances. We experimentally demonstrate that, for a single photon in a Mach-Zehnder interferometer, these quantities are related via an equation that connects both duality and uncertainty. This relation holds for the single-photon source prepared either in a pure state or a mixed state.
For a particle travelling through an interferometer, the trade-off between the available which-way information and the interference visibility provides a lucid manifestation of the quantum mechanical wave–particle duality. Here we analyse this relation for a particle possessing an internal degree of freedom such as spin. We quantify the trade-off with a general inequality that paints an unexpectedly intricate picture of wave–particle duality when internal states are involved. Strikingly, in some instances which-way information becomes erased by introducing classical uncertainty in the internal degree of freedom. Furthermore, even imperfect interference visibility measured for a suitable set of spin preparations can be sufficient to infer absence of which-way information. General results are illustrated with a proof-of-principle single-photon experiment. Quantum mechanics dictates that the interference pattern cast by particles after passing through a double slit depends on how much information it is possible to know about which slit they went through. Banaszek et al. show how this behaviour extends to a system’s internal degrees of freedom.
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.
Classical wisdom of wave–particle duality regulates that a quantum object shows either the particle or wave nature but never both. Consequently, it would be impossible to observe simultaneously the complete wave and particle nature of the quantum object. Mathematically the principle requests that the interference visibility V and which-path distinguishability D satisfy an orthodox limit of V 2 + D 2 ≤ 1 . The present work reports a new wave–particle duality test experiment using single photons in a modified Mach–Zehnder interferometer to demonstrate the possibility of breaking the limit. The key element of the interferometer is a weakly scattering total internal reflection prism surface, which exhibits a pronounced single-photon interference with a visibility of up to 0.97 and simultaneously provides a path distinguishability of 0.83. Apparently, the result of V 2 + D 2 ≈ 1.63 exceeds the orthodox limit set by the classical principle of wave–particle duality for single photons. We expect that more delicate experiments in the future should be able to demonstrate the ultimate limit of V 2 + D 2 ≈ 2 and shed new light on the foundations of contemporary quantum mechanics.