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Physical Review A

Complementarity relations in multipath interferometers via decomposition of the uncertainty

Shuangshuang Fu, Xiaohui Li

Published May 29, 2025• 4 citations
Publisher

Abstract

In this paper, employing the metric-adjusted skew information, we bring up a family of decompositions of the total uncertainty as quantified by the variance into the quantum uncertainty and classical uncertainty. Based on the decompositions of uncertainty, we further establish a family of complete complementarity relations for predictability, interference, and correlation in multipath interferometers. Explicitly, we show that the predictability can be quantified by the (reverse) total uncertainty, the interference can be quantified by the quantum uncertainty, while the correlation can be quantified by the classical uncertainty. The family of complementarity relations established in this paper generalize previous characterizations of interference in multipath interferometers via the quantum Fisher information [Y. Sun et al., Phys. Rev. A 108, 062416 (2023)]. As an illustration, we calculate the relevant quantities for some extremal quantum states, and also give explicitly the complementarity relations in two-path interferometers, which demonstrate the validity of the quantifiers. We compare our results with two related complementarity relations established recently and highlight the main differences. The results obtained in this work reveal the close connections between uncertainty and complementarity.

DOI
10.1103/physreva.111.052451

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In this paper, employing the metric-adjusted skew information, we bring up a family of decompositions of the total uncertainty as quantified by the variance into the quantum uncertainty and classical uncertainty. Based on the decompositions of uncertainty, we further establish a family of complete complementarity relations for predictability, interference, and correlation in multipath interferometers. Explicitly, we show that the predictability can be quantified by the (reverse) total uncertainty, the interference can be quantified by the quantum uncertainty, while the correlation can be quantified by the classical uncertainty. The family of complementarity relations established in this paper generalize previous characterizations of interference in multipath interferometers via the quantum Fisher information [Y. Sun et al., Phys. Rev. A 108, 062416 (2023)]. As an illustration, we calculate the relevant quantities for some extremal quantum states, and also give explicitly the complementarity relations in two-path interferometers, which demonstrate the validity of the quantifiers. We compare our results with two related complementarity relations established recently and highlight the main differences. The results obtained in this work reveal the close connections between uncertainty and complementarity.
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.
In terms of operator, the two complementary quantities, the predictability and visibility, are reinvestigated in a two-way interferometer. One Hermitian operator and one non-Hermitian operator (composed of two Hermitian operators) are introduced for the predictability and visibility, respectively. The predictability and visibility can not be measured exactly simultaneously, due to the non-commutation between the two operators. The sum of the variances of the predictability and visibility (the total variance), is used to measure the uncertainty, which is linked to the complementarity relation through the equation, (δP)2+(δVf)2+P2+V2=2(δ_P)^2+(δ_Vf)^2+P^2+V^2=2(δP​)2+(δV​f)2+P2+V2=2 . This new description for the predictability and visibility connects the complementarity and the uncertainty relations, although neither of them can be derived directly from the other.
It is important to distinguish between classical information and quantum information in quantum information theory. In this paper, we first extend the concept of metric-adjusted correlation measure and some related measures to non-Hermitian operators, and establish several relations between the metric-adjusted skew information with different operator monotone functions. By employing operator monotone functions, we next introduce three uncertainty matrices generated by channels: the total uncertainty matrix, the classical uncertainty matrix, and the quantum uncertainty matrix. We establish a decomposition of the total uncertainty matrix into classical and quantum parts and further investigate their basic properties. As applications, we employ uncertainty matrices to quantify the decoherence caused by the action of quantum channels on quantum states, and calculate the uncertainty matrices of some typical channels to reveal certain intrinsic features of the corresponding channels. Moreover, we establish several uncertainty relations that improve the traditional Heisenberg uncertainty relations involving variance.
The complementarity relations impose the constraints on different aspects of quantum states. We study the complementarity relation within a multi-path interferometer that includes detectors and quantum memory. Here we consider the mixed states as the input states. We establish a duality relation between the visibility and the path distinguishability. Based on this duality, two triality relations, one is related with visibility, path distinguishability, and mixedness, the other is related with visibility, path distinguishability, and entanglement, were derived respectively. Therefore, the role of entanglement in multi-path interferometer is characterized quantitatively. These complementarity relations are all complete for the two-path interferometer.
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.
The complementarity relations in an interferometer reflect an important phenomenon in quantum mechanics: wave-particle duality. Here, we develop a method to quantify both wave and particle behaviors in a multi-path interferometer. In particular, we find that the trace distance is a good candidate for wave and particle measures. As a result, some duality relations and triality ralations are established respectively. This work not only extends the application of the trace distance to the interferometer, but also opens up new perspectives on the quantification of waveness and particleness.