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arXiv:2111.06068 · Physical Review A

Coherence, path predictability, and I concurrence: A triality

Abhinash Kumar Roy, Neha Pathania, Nitish Kumar Chandra, Prasanta Kumar Panigrahi, Tabish Qureshi

Published Mar 16, 2022• 25 citations
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Abstract

It is well known that fringe contrast is not a good quantifier of the wave nature of a quanton in multipath interference. An alternative interference visibility, based on the Hilbert-Schmidt coherence is introduced. It is demonstrated that this visibility is a good quantifier of wave nature and can be experimentally measured. A generalized path predictability is introduced, which reduces to the predictability of Greenberger and Yasin for the case of two paths. In a multipath, which-way interference experiment, the new visibility, the predictability, and the $I$ concurrence (quantifying the entanglement between the quanton and the path detector) are shown to follow a tight triality relation. It quantifies the essential role that entanglement plays in multipath quantum complementarity.

DOI
10.1103/physreva.105.032209

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It is well known that fringe contrast is not a good quantifier of the wave nature of a quanton in multipath interference. An alternative interference visibility, based on the Hilbert-Schmidt coherence is introduced. It is demonstrated that this visibility is a good quantifier of wave nature and can be experimentally measured. A generalized path predictability is introduced, which reduces to the predictability of Greenberger and Yasin for the case of two paths. In a multipath, which-way interference experiment, the new visibility, the predictability, and the III concurrence (quantifying the entanglement between the quanton and the path detector) are shown to follow a tight triality relation. It quantifies the essential role that entanglement plays in multipath quantum complementarity.
The interference observed for a quanton, traversing more than one path, is believed to characterize its wave nature. Conventionally, the sharpness of interference has been quantified by its visibility or contrast, as defined in optics. Based on this visibility, wave-particle duality relations have been formulated for two-path interference. However, as one generalizes the situation to multi-path interference, it is found that conventional interference visibility is not a good quantifier. A recently introduced measure of quantum coherence has been shown to be a good quantifier of the wave nature. The subject of quantum coherence, in relation to the wave nature of quantons and to interference visibility, is reviewed here. It is argued that coherence can be construed as a more general form of interference visibility, if the visibility is measured in a different manner, and not as contrast. Quanta 2019; 8: 24–35.
We theoretically analyze the observations reported in a four-path quantum interference experiment via multiple-beam Ramsey interference [Phys. Rev. Lett. 86, 559 (2001)]. In this experiment, a selective scattering of photons from just one interfering path causes decoherence. However, contrary to expectations, there is an increase in the contrast of the interference pattern, demonstrating that path selective decoherence can not only lead to a decrease but, under certain conditions, lead to an increase of the fringe contrast. Here we explain this seemingly counterintuitive effect based on a model for a multipath interference, with four to six slits, in the presence of decoherence. The effect of the environment is modeled via a coupling to a bath of harmonic oscillators. When decoherence is introduced in one of the multiple paths, an enhancement in fringe contrast is seen under certain conditions. A similar effect is shown to appear if instead of path-selective decoherence, a selective path detector is introduced. Our analysis points to the fact that while traditional fringe visibility captures the wave nature in the two-path case, it can fail in multipath situations. We explain the enhancement of fringe visibility and also show that quantum coherence based on the l1{l}_{1}l1​ norm of coherence, in contrast to traditional visibility, remains a good quantifier of wave nature, even in such situations. The enhancement of fringe contrast in the presence of environmental decoherence underscores the limitations of traditional visibility as a good measure for wave nature in quantifying complementarity and also makes it an unlikely candidate for quantifying decoherence. Our analysis could lead to better insight in ways to quantify decoherence in multipath interference and in studies that seek to exploit quantum superpositions and quantum coherence for quantum information applications.
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.
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.
Abstract In an asymmetric multislit interference experiment, a quanton is more likely to pass through certain slits than some others. In such a situation one may be able to predict which slit a quanton is more likely to go through, even without using any path-detecting device. This allows one to talk of path predictability . It has been shown earlier that for a two-slit interference, the predictability and fringe visibility are constrained by the inequality  2 +  2 ≤ 1 . Generalizing this relation to the case of more than two slits is still an unsolved problem. A new definition for predictability for multi-slit interference is introduced. It is shown that this predictability and quantum coherence follow a duality relation  2 +  2 ≤ 1 , which saturates for all pure states. For the case of two slits, this relation reduces to the previously known one.
Recent times have seen a spurt of research activity focused on "completing" certain wave-particle duality relations using entanglement or polarization. These studies use a duality relation involving path-predictability, and not path-distinguishability. Quantum origins of these results are explored here, in the more general framework of multipath quantum interference. Multipath interference with a path-detector is theoretically analyzed to find the connection between predictability and distinguishability. It is shown that entanglement is what quantitatively connects distinguishability with predictability. Thus, a duality relation between distinguishability and coherence can also be viewed as a triality between predictability, entanglement, and coherence. There exist two different kinds of duality relations in the literature, which pertain to two different kinds of interference experiments, with or without a path-detector. Results of this study show that the two duality relations are quantitatively connected via entanglement. The roots of the new, to the best of our knowledge, results in the classical optical domain, including the polarization coherence theorem, can be understood in the light of this work.