Complex physical systems contain information which, under some well-defined processes can differentiate between local and nonlocal information. Both these fundamental aspects of information are defined operationally. Local information is locally accessible and allows one to perform processes, such as physical work, while nonlocal information allows one to perform processes such as teleportation. It is shown that these two kinds of information are complementary in the sense that two parties can either gain access to the nonlocal information or to the local information but not both. This complementarity has a form similar to that expressed by entropic uncertainty relations. For pure states, the entanglement plays the role of Planck's constant. We also find another class of complementarity relations which applies to operators and is induced when two parties can only perform local operations and communicate classical (LOCC). In particular, observables such as the parity and phase of two qubits commute but under LOCC, they are complementary observables. It is also found this complementarity is pure in the sense that it can be ``decoupled'' from the uncertainty principle. It is suggested that these complementarities represent an essential extension of Bohr's complementarity to complex (distributed) systems which are entangled.
Complex physical systems contain information which, under some well-defined processes can differentiate between local and nonlocal information. Both these fundamental aspects of information are defined operationally. Local information is locally accessible and allows one to perform processes, such as physical work, while nonlocal information allows one to perform processes such as teleportation. It is shown that these two kinds of information are complementary in the sense that two parties can either gain access to the nonlocal information or to the local information but not both. This complementarity has a form similar to that expressed by entropic uncertainty relations. For pure states, the entanglement plays the role of Planck's constant. We also find another class of complementarity relations which applies to operators and is induced when two parties can only perform local operations and communicate classical (LOCC). In particular, observables such as the parity and phase of two qubits commute but under LOCC, they are complementary observables. It is also found this complementarity is pure in the sense that it can be ``decoupled'' from the uncertainty principle. It is suggested that these complementarities represent an essential extension of Bohr's complementarity to complex (distributed) systems which are entangled.
In spite of many results in quantum information theory, the complex nature of compound systems is far from clear. In general the information is a mixture of local and nonlocal (``quantum'') information. It is important from both pragmatic and theoretical points of view to know the relationships between the two components. To make this point more clear, we develop and investigate the quantum-information processing paradigm in which parties sharing a multipartite state distill local information. The amount of information which is lost because the parties must use a classical communication channel is the deficit. This scheme can be viewed as complementary to the notion of distilling entanglement. After reviewing the paradigm in detail, we show that the upper bound for the deficit is given by the relative entropy distance to so-called pseudoclassically correlated states; the lower bound is the relative entropy of entanglement. This implies, in particular, that any entangled state is informationally nonlocal---i.e., has nonzero deficit. We also apply the paradigm to defining the thermodynamical cost of erasing entanglement. We show the cost is bounded from below by relative entropy of entanglement. We demonstrate the existence of several other nonlocal phenomena which can be found using the paradigm of local information. For example, we prove the existence of a form of nonlocality without entanglement and with distinguishability. We analyze the deficit for several classes of multipartite pure states and obtain that in contrast to the GHZ state, the Aharonov state is extremely nonlocal. We also show that there do not exist states for which the deficit is strictly equal to the whole informational content (bound local information). We discuss the relation of the paradigm with measures of classical correlations introduced earlier. It is also proved that in the one-way scenario, the deficit is additive for Bell diagonal states. We then discuss complementary features of information in distributed quantum systems. Finally we discuss the physical and theoretical meaning of the results and pose many open questions.
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.
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.
The uncertainty principle can be expressed in entropic terms, also taking into account the role of entanglement in reducing uncertainty. The information exclusion principle bounds instead the correlations that can exist between the outcomes of incompatible measurements on one physical system, and a second reference system. We provide a more stringent formulation of both the uncertainty principle and the information exclusion principle, with direct applications for, e.g., the security analysis of quantum key distribution, entanglement estimation, and quantum communication. We also highlight a fundamental distinction between the complementarity of observables in terms of uncertainty and in terms of information.
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.
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.