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.
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.
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.
The so-called preparation uncertainty that occurs in the quantum world can be understood well in purely operational terms, and its existence in any given theory, perhaps differently than in quantum mechanics, can be verified by examining only measurement statistics. Namely, one says that uncertainty occurs in some theory when for some pair of observables, there is no preparation that would exhibit deterministic statistics for both of them. However, the right-hand side of the uncertainty relation is not operational anymore if we do not insist that it is just the minimum of the left-hand side for a given theory. For example, in quantum mechanics, it is some function of two observables that must be computed within the quantum formalism. Also, while joint nonmeasurability of observables is an operational notion, the complementarity in Bohr's sense (i.e., in terms of information needed to describe the system) has not yet been expressed in purely operational terms. Here we propose a general operational framework that provides answers to the above issues. We introduce an operational definition of complementarity and further postulate that complementary observables have to exhibit uncertainty, which means that we propose to put the (operational) complementarity as the right-hand side of the uncertainty relation. In particular, we identify two different notions of uncertainty and complementarity for which the above principle holds in the quantum-mechanical realm. We also introduce postulates for the general measures of uncertainty and complementarity. In order to define quantifiers of complementarity we first turn to the simpler notion of independence that is defined solely in terms of the statistics of two observables. Importantly, for clean and extremal observables---i.e., ones that cannot be simulated irreducibly by other observables---any measure of independence reduces to the proper complementary measure. Finally, as an application of our general framework we define a number of complementarity indicators and show that they can be used to state uncertainty relations. One of them, assuming some natural symmetries, leads to the Tsirelson bound for the Clauser-Horne-Shimony-Holt (CHSH) inequality. Lastly, we show that for a single system a variant of information causality called the information content principle, under the above symmetries, can be interpreted as an uncertainty relation in the above sense.
The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.
Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.
One of the most remarkable features of quantum physics is that attributes of quantum objects, such as the wavelike and particlelike behaviors of single photons, can be complementary in the sense that they are equally real but cannot be observed simultaneously. Quantum measurements, serving as windows providing views into the abstract edifice of quantum theory, are basic tools for manifesting the intrinsic behaviors of quantum objects. However, a quantitative formulation of complementarity that highlights its manifestations in general measurement scenarios remains elusive. Here we develop a general framework for demonstrating quantum complementarity in the form of information exclusion relations (IERs), which incorporates the wave-particle duality relations as particular examples. Moreover, we explore the applications of our theory in entanglement detection and elucidate that our IERs lead to an extended form of entropic uncertainty relations, providing insights into the connection between quantum complementarity and the preparation uncertainty.
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.