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Chinese Physics Letters

Wave-Particle Duality via Quantum Fisher Information

Chang Niu, Sixia Yu

Published Sep 25, 2023• 6 citations
Publisher

Abstract

Quantum Fisher information (QFI) plays an important role in quantum metrology, placing the ultimate limit to how precise we can estimate some unknown parameter and thus quantifying how much information we can extract. We observe that both the wave and particle properties within a Mach–Zehnder interferometer can naturally be quantified by QFI. Firstly, the particle property can be quantified by how well one can estimate the a priori probability of the path taken by the particle within the interferometer. Secondly, as the interference pattern is always related to some phase difference, the wave property can be quantified by how well one can estimate the phase parameter of the original state. With QFI as the unified figure of merit for both properties, we propose a more general and stronger wave-particle duality relation than the original one derived by Englert.

DOI
10.1088/0256-307x/40/11/110301

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Quantum Fisher information (QFI) plays an important role in quantum metrology, placing the ultimate limit to how precise we can estimate some unknown parameter and thus quantifying how much information we can extract. We observe that both the wave and particle properties within a Mach–Zehnder interferometer can naturally be quantified by QFI. Firstly, the particle property can be quantified by how well one can estimate the a priori probability of the path taken by the particle within the interferometer. Secondly, as the interference pattern is always related to some phase difference, the wave property can be quantified by how well one can estimate the phase parameter of the original state. With QFI as the unified figure of merit for both properties, we propose a more general and stronger wave-particle duality relation than the original one derived by Englert.
Wave-particle duality, as complementarity displayed in interferometry, has been investigated extensively from both theoretical and experimental perspectives. In this work, we quantify coherence in multipath interference via quantum Fisher information and illuminate its basic properties. We prove that any rational quantifier of coherence satisfying monotonicity under incoherent operations is nonincreasing in the presence of path detectors. This quantitatively implies that increase on the path information will lead to the loss of coherence or interference (wave feature), which is consistent with wave-particle duality. Furthermore, we provide an operational illustration of wave feature as quantum Fisher information of phase shift parameters encoded in each interference path. By dividing the variance of a quantum state relative to a von Neumann measurement into quantum uncertainty and classical uncertainty, we establish a coherence-predictability-correlations triality relation, with coherence (wave feature) quantified by quantum Fisher information, predictability quantified by purity of a classical probability distribution, and correlations quantified by classical uncertainty relative to the von Neumann measurement determined by the interference paths. Finally, we make a comparison between this triality relation and the wave-particle-mixedness triality relation in the literature.
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.
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.
Abstract Quantum interference is known to become extinct with distinguishing information, as illustrated by the ubiquitous double-slit experiment or the two-photon Hong–Ou–Mandel effect. In the former case single particle interference is destroyed with which-path information while in the latter bunching interference tails-off as photons become distinguishable. It has been observed that when more than two particles are involved, these interference patterns are in general a non-monotonic function of the distinguishability. Here we perform a comprehensive characterization, both theoretically and experimentally, of four-photon interference by analyzing the corresponding correlation functions, contemplating several degrees of distinguishability across different parameters. This study provides all the necessary tools to quantify the impact of multi-photon interference on precision measurements of parameters such as phase, frequency, and time difference. We apply these insights to quantify the precision in the estimation of an interferometric phase in a two-port interferometer using a four-photon state. Our results reveal that, for certain phase values, partially distinguishable multi-photon states can achieve higher Fisher information values compared to the two-photon experiment. These findings highlight the potential of distinguishable multi-photon states for enhanced precision in quantum metrology and related applications.
Abstract The Mach–Zehnder interferometer is a fundamental tool for measuring phase shifts between two light paths, serving as a crucial prototype for achieving high-precision measurements in various scientific and technological applications. In this study, we analyze different models for estimating the relative phase shift in a general two-arm Mach–Zehnder interferometer. We demonstrated that single-parameter estimation models can be reduced from the two-parameter estimation model by imposing appropriate constraints on the parameter space. To make quantum Fisher information of the single-parameter estimation models meaningful, the corresponding constraints must be guaranteed in the experiment implementation. Furthermore, we apply the quantum Fisher information approach to analyze the Mach–Zehnder interferometer with the an input state composed of a displaced squeezed vacuum state and a coherent state, providing insights into the precision limits of such configurations.