The concept of wave-particle duality holds significant importance in the field of quantum mechanics, as it elucidates the dual nature encompassing both wavelike and particlelike properties exhibited by microscopic particles. In this paper, we construct generalized measures for the predictability and visibility of $n$-path interference fringes to quantify the wave and particle properties in quantum high-dimensional systems. By employing the Morozova-Chentsov function, we ascertain that the wave-particle relationship can be delineated by the average coherence. This function exhibits a close correlation with the metric-adjusted skew information, thereby we establish complementary relations between visibility, predictability, and quantum $f$ entropy, which reveal deep connections between wave-particle duality and other physical quantities. Through our methodology, diverse functions can be selected to yield corresponding complementary relationships.
The concept of wave-particle duality holds significant importance in the field of quantum mechanics, as it elucidates the dual nature encompassing both wavelike and particlelike properties exhibited by microscopic particles. In this paper, we construct generalized measures for the predictability and visibility of n-path interference fringes to quantify the wave and particle properties in quantum high-dimensional systems. By employing the Morozova-Chentsov function, we ascertain that the wave-particle relationship can be delineated by the average coherence. This function exhibits a close correlation with the metric-adjusted skew information, thereby we establish complementary relations between visibility, predictability, and quantum f entropy, which reveal deep connections between wave-particle duality and other physical quantities. Through our methodology, diverse functions can be selected to yield corresponding complementary relationships.
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.
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 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 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, an important and fundamental concept established upon pure quantum systems, is central to the complementarity principle in quantum mechanics. However, due to the environment effects or even the entanglement between the quanton and the which-way detector (WWD), the quanton should be described by a mixed quantum state but not a pure quantum state. Although there are some attempts to clarify the complementarity principle for mixed quantum systems, it is still unclear how the mixedness affects the complementary relation. Here, we give a ternary complementary relation (TCR) among wave, particle and mixedness aspects for arbitrary multi-state systems, which are respectively quantified by the l1 measure for quantum coherence, the which-path predictability, and the linear entropy. In particular, we show how a WWD can transform entropy into predictability and coherence. Through modifying the POVM (positive-operator valued measure) measurement on WWD, our TCR can be simplified as the wave-mixedness and particle-mixedness duality relations. Beyond enclosing the wave-particle duality relation [PRL 116, 160406 (2016)], our TCR relates to the wave-particle-entanglement complementarity relation [PRL 98, 250501 (2008); Opt. Commun. 283, 827 (2010)].
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.