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Physical Review A

Relation between wave-particle duality and quantum uncertainty

Hongyu Liu, Jie‐Hui Huang, Jiangrui Gao, Muhammad Suhail Zubairy, Shi‐Yao Zhu

Published Feb 6, 2012• 31 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.85.022106

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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.
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 theoretically investigate the influence of losses on the wave-particle duality in a Mach-Zehnder interferometer. Balanced losses (equal loss on the two paths) have no effect on the visibility and predictability (also the duality relation), while unbalanced losses do have influence on them. If the unbalanced losses occur inside the interferometer, the losses have an effect on both the visibility and predictability, while the losses have no effect on the duality relation. If the unbalanced losses occur after the interferometer, the losses have no effect on the visibility, but do have an effect on the predictability (also the wave-particle duality), which can lead to a result that the duality relation exceeds 1, P2+V2>1{P}^{2}+{V}^{2}>1P2+V2>1. The influence of losses can be eliminated by exchanging the two detectors or the two inputs (one photon and the vacuum) of the interferometer and then averaging the two results. Consequently, we have the visibility, predictability, and duality relation ${P}^{2}+{V}^{2}\ensuremath{\le}1$ independent of the unbalanced losses. The obtained PPP and VVV for the unbalanced losses in the two paths do not represent the original predictability and visibility, and the result of PD2+VD2>1{P}_{D}^{2}+{V}_{D}^{2}>1PD2​+VD2​>1 does not mean the ``violation'' of the original duality.
The notion of wave-particle duality may be quantified by the inequality V2+K2< 1,{V}^{2}{+K}^{2}<~1,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.
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.
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.