Complementarity plays a central role in the conceptual development of quantum mechanics, and also provides practical applications in quantum information technologies. How to properly quantify it is an important problem in quantum foundations, and there exists different types of complementarity relations. In this paper, a complementarity relation is established with the robustness of asymmetry. Specifically, the two complementary aspects are quantified by applying the robustness of asymmetry corresponding to two cyclic groups whose generators are linked by the Fourier matrix. This complementarity relation is compared with known results and considered in other perspectives, especially its operational meaning regarding quantum state discrimination. We conclude that the internal asymmetry of quantum states is closely related to other fundamental concepts, such as complementarity and coherence, and it is possible to quantitatively investigate complementarity and quantum state discrimination using the robustness of asymmetry.
Complementarity plays a central role in the conceptual development of quantum mechanics, and also provides practical applications in quantum information technologies. How to properly quantify it is an important problem in quantum foundations, and there exists different types of complementarity relations. In this paper, a complementarity relation is established with the robustness of asymmetry. Specifically, the two complementary aspects are quantified by applying the robustness of asymmetry corresponding to two cyclic groups whose generators are linked by the Fourier matrix. This complementarity relation is compared with known results and considered in other perspectives, especially its operational meaning regarding quantum state discrimination. We conclude that the internal asymmetry of quantum states is closely related to other fundamental concepts, such as complementarity and coherence, and it is possible to quantitatively investigate complementarity and quantum state discrimination using the robustness of asymmetry.
Quantum states may exhibit asymmetry with respect to the action of a given group. Such an asymmetry of states can be considered as a resource in applications such as quantum metrology, and it is a concept that encompasses quantum coherence as a special case. We introduce explicitly and study the robustness of asymmetry, a quantifier of asymmetry of states that we prove to have many attractive properties, including efficient numerical computability via semidefinite programming, and an operational interpretation in a channel discrimination context. We also introduce the notion of asymmetry witnesses, whose measurement in a laboratory detects the presence of asymmetry. We prove that properly constrained asymmetry witnesses provide lower bounds to the robustness of asymmetry, which is shown to be a directly measurable quantity itself. We then focus our attention on coherence witnesses and the robustness of coherence, for which we prove a number of additional results; these include an analysis of its specific relevance in phase discrimination and quantum metrology, an analytical calculation of its value for a relevant class of quantum states, and tight bounds that relate it to another previously defined coherence monotone.
While Bohr's complementarity principle constitutes a bedrock of quantum mechanics with profound implications, coherence, as a defining feature of the quantum realm originating from the superposition principle, pervades almost every quantum consideration. By exploiting the algebraic and geometric structure of state-channel interaction, we show that an information-theoretic measure of coherence and a quantitative symmetry-asymmetry complementarity emerge naturally from the formalism of quantum mechanics. This is achieved by decomposing the state-channel interaction into a symmetric part and an asymmetric part, which satisfy a conservation relation. The symmetric part is represented by the symmetric Jordan product, and the asymmetric part is synthesized by the skew-symmetric Lie product. The latter further leads to a significant extension of the celebrated Wigner--Yanase skew information, and has an operational interpretation as quantum coherence of a state with respect to a channel. This not only presents a basic and alternative framework for addressing complementarity, but also puts the study of coherence in a broad context involving channels. Fundamental properties of the symmetry-asymmetry complementarity are revealed, and applications and implications are illustrated via several prototypical channels as well as the Mach--Zehnder interferometry, in which the fringe visibility is linked to symmetry and the which-path is linked to asymmetry.
Observables in quantum mechanics are generally complementary, that is, they reveal mutually incompatible pieces of information about a given system. This property is not only a fundamental tenet of the quantum formalism, but also a key component in quantum cryptographic protocols. As such it has fuelled much research into finding sets of highly complementary observables. In its strongest form -- the one we consider in this work -- the information between complementary observables is not merely incompatible but mutually exclusive: maximal information about one observable implies no information about the other, and vice versa. Maximal sets of non-degenerate complementary observables are known to exist in systems of prime power dimension such as n-qubit systems. Here, we study complementarity of degenerate observables, specifically we prove that observables associated with the n-qubit Pauli group also exhibit complementarity under this restriction: first, we obtain a criterion for two such observables to be complementary and, second, we relate maximal sets of complementary observables with informational pure state complementarity equalities, further studied in two companion papers. Equivalently, these results can be formulated in terms of maximal complementary sets of (not necessarily maximal) Abelian subgroups of the Pauli group linking with (possibly coarse-grained) mutually unbiased bases. Finally, we prove that these complementarity sets entail strong entropic uncertainty relations.
Bohr’s complementarity principle was introduced as a qualitative statement about quantum systems, which have properties that are equally real, but mutually exclusive. This principle, together with the uncertainty principle, is at the origin of Quantum Mechanics (QM), following the development of the theory since then. Another intriguing aspect of QM is entanglement, which is a type of correlation that is only possible within the mathematical formalism of QM. Its central importance in the Quantum foundations, as well as its important role in the fields of quantum information and quantum computation, has made the theory of entanglement achieve great progress in recent decades. In this dissertation we discuss the aspects and relations among complementarity relations, uncertainty, and entanglement. Such aspects are also investigated in relativistic scenarios, since the interest on how entanglement behaves in relativistic scenarios has grown more and more. First, by exploring the properties of the density operator, we obtain complementarity relations, using coherence measures well known in the literature. Next, we show that it is possible to obtain complete complementarity relations, i.e., relations that also take into account entanglement. For this, we explore the purity of a multipartite quantum system. In addition, we discuss the relation between complementarity and uncertainty of an observable, since it is possible to decompose uncertainty into its classical and quantum parts. We also obtain complementarity relations for uncertainty. Finally, we study the complete complementary relations and its Lorentz invariance. Although it is known that entanglement entropy does not remain invariant under Lorentz transformations, and neither does the measures that quantifies complementarity, we show that theses measures when taken together, in a complete complementarity relation, are Lorentz invariant.
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.
Quantum complementarity is a fundamental feature of quantum systems and has captivated the physics research community for nearly a century, with significant advancements emerging in recent decades. This review traces the historical evolution of the concept of complementarity, beginning with Bohr's original formulation. It then explores its modern quantification through complementarity relations and its profound connection to the foundational postulates of quantum theory. Furthermore, it delves into key related developments, such as the operational definition of complementarity in the context of incompatible observables, its potential links with quantum uncertainty relations and contextuality, its various applications, and other pertinent topics. This review aims to serve physicists interested in quantum resources, quantum correlations, and the foundational principles of quantum mechanics.