Abstract In order to measure the inherent information content in a quantum state, Brukner and Zeilinger developed the idea of operationally invariant information in terms of the outcome probabilities of measuring a whole collection of mutually complementary observables. In this work, by partitioning the orthonormal operator basis of the associated operator Hilbert space into two complementary subsets, we establish a new decomposition of the Brukner-Zeilinger information. We further demonstrate that the two complementary components of the Brukner-Zeilinger information capture nicely the wave and particle features encoded in the quantum state after an n -path interferometer. Thus, we establish a universal complementarity relation involving the interference (wave feature) and the predictability (particle feature) via the unified concept of the Brukner-Zeilinger invariant information. These findings provide new characterizations of the wave-particle duality via the fundamental Brukner-Zeilinger invariant information.
Abstract In order to measure the inherent information content in a quantum state, Brukner and Zeilinger developed the idea of operationally invariant information in terms of the outcome probabilities of measuring a whole collection of mutually complementary observables. In this work, by partitioning the orthonormal operator basis of the associated operator Hilbert space into two complementary subsets, we establish a new decomposition of the Brukner-Zeilinger information. We further demonstrate that the two complementary components of the Brukner-Zeilinger information capture nicely the wave and particle features encoded in the quantum state after an n -path interferometer. Thus, we establish a universal complementarity relation involving the interference (wave feature) and the predictability (particle feature) via the unified concept of the Brukner-Zeilinger invariant information. These findings provide new characterizations of the wave-particle duality via the fundamental Brukner-Zeilinger invariant information.
To quantify the intrinsic information content in a quantum state, Brukner and Zeilinger introduced the concept of operationally invariant information in terms of the outcome probabilities of measuring a complete set of mutually complementary observables [\ifmmode \check{C}\else \v{C}\fi{}. Brukner and A. Zeilinger, Phys. Rev. Lett. 83, 3354 (1999)]. This information quantity has basic significance and implications, and the present work is devoted to some further studies of it. We first introduce the Brukner-Zeilinger invariant information in the presence of conjugate symmetry or antisymmetry, which are motivated by considerations of fundamental issues concerning conjugate symmetry in quantum mechanics. Then we prove that both the Brukner-Zeilinger invariant information with conjugate symmetry and that with conjugate antisymmetry are convex in the quantum state, and we show that they constitute a natural decomposition of the Brukner-Zeilinger invariant information. We further relate them to the imaginarity (i.e., the usage of a complex number field) of quantum mechanics and evaluate their extreme values.
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 address the problem of properly quantifying information in quantum theory. Brukner and Zeilinger proposed the concept of an operationally invariant measure based on measurement statistics. Their measure of information is calculated with probabilities generated in a complete set of mutually complementary observations. This approach was later criticized for several reasons. We show that some critical points can be overcome by means of natural extension or reformulation of the Brukner–Zeilinger approach. In particular, this approach is connected with symmetric informationally complete measurements. The ‘total information’ of Brukner and Zeilinger can further be treated in the context of mutually unbiased measurements as well as general symmetric informationally complete measurements. The Brukner–Zeilinger measure of information is also examined in the case of detection inefficiencies. It is shown to be decreasing under the action of bistochastic maps. The Brukner–Zeilinger total information can be used for estimating the map norm of quantum operations.
Abstract We investigate the total variance of a quantum state with respect to a complete set of mutually complementary measurements and its relation to the Brukner–Zeilinger invariant information. By summing the variances over any complete set of mutually unbiased measurements and general symmetric informationally complete measurements respectively, we show that the Brukner–Zeilinger invariant information associated with such types of quantum measurements is equal to the difference between the maximal variance and the total variance obtained. These results provide an operational link between the previous interpretations of the Brukner–Zeilinger invariant information.
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.
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.