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arXiv:2104.04230

Quantitative Complementarity of Wave-Particle Duality

Tai Hyun Yoon, Minhaeng Cho

Published Apr 9, 2021• 0 citations
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Abstract

To test the principle of complementarity and wave-particle duality quantitatively, we need a quantum composite system that can be controlled by experimental parameters. Here, we demonstrate that a double-path interferometer consisting of two parametric downconversion crystals seeded by coherent idler fields, where the generated coherent signal photons are used for quantum interference and the conjugate idler fields are used for which-path detectors with controllable fidelity, is useful for elucidating the quantitative complementarity. We show that the source purity $μ_s$ is tightly bounded by the entanglement measure $E$ by the relation $μ_s=\sqrt{1-E^2 }$ and the visibility $V$ and detector fidelity $F$ determine the coherence of the quantons, i.e., $C = V|F|$. The quantitative complementarity of the double-path interferometer we developed recently is explained in terms of the quanton-detector entanglement or quanton source purity that are expressed as functions of injected seed photon numbers. We further suggest that the experimental scheme utilizing two stimulated parametric downconversion processes is an ideal tool for investigating and understanding wave-particle duality and complementarity quantitatively.

DOI
10.48550/arxiv.2104.04230

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To test the principle of complementarity and wave-particle duality quantitatively, we need a quantum composite system that can be controlled by experimental parameters. Here, we demonstrate that a double-path interferometer consisting of two parametric downconversion crystals seeded by coherent idler fields, where the generated coherent signal photons are used for quantum interference and the conjugate idler fields are used for which-path detectors with controllable fidelity, is useful for elucidating the quantitative complementarity. We show that the source purity μsμ_sμs​ is tightly bounded by the entanglement measure EEE by the relation μs=1−E2μ_s=\sqrt{1-E^2 }μs​=1−E2​ and the visibility VVV and detector fidelity FFF determine the coherence of the quantons, i.e., C=V∣F∣C = V|F|C=V∣F∣. The quantitative complementarity of the double-path interferometer we developed recently is explained in terms of the quanton-detector entanglement or quanton source purity that are expressed as functions of injected seed photon numbers. We further suggest that the experimental scheme utilizing two stimulated parametric downconversion processes is an ideal tool for investigating and understanding wave-particle duality and complementarity quantitatively.
We propose a novel quantum interferometer based on two stimulated Parametric Down-Conversion (PDC) crystals seeded by weak coherent idler fields to investigate the relationship between the nonclassicality of the signal photon state and the fidelity of the conjugate idler photon state. We show that the purity of the signal photon state is tightly bounded by the entanglement between the signal photon and its conjugate idler photon, i.e., the fidelity of the idler state in the bipartite system, resulting in a novel complementarity relation for wave-particle duality. Moreover, we find that the nonclassical measure of the signal photon state is identical to the fidelity of the conjugate idler photon states prepared in each crystal. In our novel interferometer, the input beam splitter of the conventional SU(2) interferometer is replaced with two seeded SU(1,1) PDC systems that emit only a single photon at once from two crystals, thus producing an entangled superposition state where the conjugate idler photon states act as path detectors with controllable fidelity. We demonstrate that the presented quantum interferometer consisting of two stimulated PDC crystals positioned in parallel is an ideal setup for investigating and quantitatively understanding the complementarity relation of wave-particle duality.
According to the relevant theories on duality relation, the summation of the extractable information of a quanton’s wave and particle properties, which are characterized by interference visibility V and path distinguishability D, respectively, is limited. However, this relation is violated upon quantum superposition between the wave-state and particle-state of the quanton, which is caused by the quantum beamsplitter (QBS). Along another line, recent studies have considered quantum coherence C in the l1-norm measure as a candidate for the wave property. In this study, we propose an interferometer with a quantum which-path detector (QWPD) and examine the generalized duality relation based on C. We find that this relationship still holds under such a circumstance, but the interference between these two properties causes the full-particle property to be observed when the QWPD system is partially present. Using a pair of polarization-entangled photons, we experimentally verify our analysis in the two-path case. This study extends the duality relation between coherence and path information to the quantum case and reveals the effect of quantum superposition on the duality relation.
The wave-particle duality, one of the expressions of Bohr complementarity, is usually quantified by path predictability and the visibility of interference fringes. With the development of quantum resource theory, quantitative analysis of wave-particle duality is increasing, most of which are expressed in the form of specific functions. In this paper, we obtain the path-information measure for pure states by converting the coherence measure for pure states into a symmetric concave function. Then we prove the function as a path-information measure is also valid for mixed states. Furthermore, we also establish a generalized wave-particle-mixedness triality. Although the mixedness proposed in the text is not a complete mixedness measure, it also satisfies some conditions of the mixedness measure. From the perspective of resource theory, the path information we establish can be used as the measure of the resource of predictability, and the triality relationship we establish reveals the relationship among coherence, predictability, purity, and mixedness degree to a certain extent. According to our method, given either the coherence measure or path information, a particular form of wave-particle-mixedness triality can be established. This will play an important role in establishing connections between wave, particle, and other physical quantifiers.
The wave-particle duality, as one of the expressions of Bohr complementarity, is usually quantified by path predictability and the visibility of interference fringes. With the development of quantum resource theory, the quantitative analysis of wave-particle duality is increasing, most of which are expressed in the form of specific functions. In this paper, we obtain the path information measure for pure states by converting the coherence measure for pure state into a symmetric concave function. Then we prove the function as a path information measure is also valid for mixed states. Furthermore, we also establish a generalized wave-particle-mixedness traility. Although the mixedness proposed in the text is not a complete mixedness measure, it also satisfies some conditions of mixdness measure. From the perspective of resource theory, the path information we establish can be used as the measure of the resource of predictability, and the triaility relationship we establish reveals the relationship among coherence, predictability, purity and mixdness degree to a certain extent. According to our method, given either coherence measure or path information, a particular form of wave-particle-mixedness traility can be established. This will play an important role in establishing connections between wave, particle and other physical quantifiers.
A quantum system (quanton) traverses an interferometer with NNN 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]ξ\in[0,1]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=0ξ=0ξ=0, both approaches reduce to the ME measurement, which typically yields the tightest relation. For ξ>0ξ>0ξ>0, each approach yields a distinct relation that generally becomes less tight as ξξξ 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.