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.
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.
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.
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 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.
Abstract Wave-particle duality as one of the expression of Bohr complementarity is a significant concept in the field of quantum mechanics. Quantitative analysis of wave-particle duality aims to establish a complementary relation between the particle and wave properties. Beyond the conventional quantitative analysis depending on special choice of quantum information measures, we are aimed to provide a measure-independent complementary relation via coherence. By employing maximally coherent states in the set of all states with fixed diagonal elements, we give a measure-independent description of wave-particle-mixedness triality in d -path interferometers. Our complementary relations reveal the relationship between wave-particle duality and quantum coherence, and also give a justification to coherence as it truly brings out the wave nature of quantum systems at its heart.
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.
Complementarity, or more specifically, the wave-particle duality, and quantum coherence are two fundamental concepts in quantum mechanics. Recently, motivated by the progress of the quantification of quantum coherence, the complementary relation between coherence and path information is investigated by many authors, and various duality relations between them are established. Such relations not only provide insights into fundamental problems of quantum mechanics, such as the understanding of quantum coherence and wave-particle duality; but also are important in applications of quantum technologies. In this paper, based on the Bures distance and unambiguous quantum state discrimination, systematic analysis of the complementarity between the quantum coherence and path information in two path interferometers is carried out. Similarly as other related works, the wave aspect, or the visibility of the interferometer, is quantified by the l1-norm measure of quantum coherence, and the path information is considered via unambiguous quantum state discrimination. In this way, a novel duality relation in two path interferometers is obtained. Compared with known results, our work considers mixed states as well as pure states; considers the path predictability resulting from the intrinsic path asymmetry of the quantum state, as well as the path distinguishability resulting from the use of a which-path detector. Therefore, our work systematically generalizes known results in two path interferometers by removing all the unnecessary restrictions. Specifically, the most general form of quantum states in two path interferometers is considered and the duality relation between quantum coherence and path information is proved based on the positivity property of density matrices. The cases of path predictability and path distinguishability are considered separately. For path predictability, the proof is straightforward; whereas some advanced mathematical techniques, such as the Schur-Weyl inequality, properties of the fidelity and properties of positive matrices, are required in order to give a rigorous proof of the duality relation between coherence and path distinguishability. Concrete examples are provided to illustrate the abstract method and results. Our work concerns about two path interferometers exclusively and depends heavily that the dimensionality is two, therefore it would be an interesting task to generalize the results in this paper to n-path interferometers.