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.
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.
We show that the amount of complementarity that a quantum system can exhibit depends on which complementary properties one is considering. Consider a set of mutually unbiased bases (MUBs) which corresponds to the spectral decomposition of some maximally complementary properties: The perfect knowledge of one, i.e. the state of the system is one of the basis states, implies that all the others are completely unknown, namely the measurement of another property will find any of its possible outcomes with uniform probability. We show that, depending on which three of the MUBs we choose, a 5-dimensional system can have different degrees of complementarity, measured using the minimum over all the system states of the sum of the entropies and of the variances of the measurement outcomes related to the three chosen observables. This property was first found experimentally, and demonstrates that an experimental discovery can disclose quantum information effects.
Quantum uncertainty relations impose fundamental limits on the joint knowledge that can be acquired from complementary observables: Perfect knowledge of a quantum state in one basis implies maximal indetermination in all other mutually unbiased bases (MUBs). Uncertainty relations derived from joint properties of the MUBs are generally assumed to be uniform, irrespective of the specific observables chosen within a set. In this work, we demonstrate instead that the uncertainty relations can depend on the choice of observables. Through both experimental observation and numerical methods, we show that selecting different sets of three MUBs in a five-dimensional quantum system results in distinct uncertainty bounds, i.e., in varying degrees of complementarity, in terms of both entropy and variance.
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.
Complementarity is a phenomenon explaining several core features of quantum theory, such as the well-known uncertainty principle. Roughly speaking, two objects are said to be complementary if being certain about one of them necessarily forbids useful knowledge about the other. Two quantum measurements that do not commute form an example of complementary measurements, and this phenomenon can also be defined for ensembles of states. Although a key quantum feature, it is unclear whether complementarity can be understood more operationally, as a necessary resource in some quantum information task. Here we show this is the case, and relates to a novel task which we term η-unambiguous exclusion. As well as giving complementarity a clear operational definition, this also uncovers the foundational underpinning of unambiguous exclusion tasks for the first time. We further show that a special type of measurement complementarity is equivalent to advantages in certain encryption tasks. Finally, our analysis suggest that complementarity of measurement and state ensemble can be interpreted as strong forms of measurement incompatibility and quantum steering, respectively.
We present a family of quadratics in Pauli expectation values, and prove that they constitute state-independent invariants for all n-qubit pure states. This family generalises the two-qubit `pentagon identities', discovered in the reconstruction programme of [P. A. Höhn, Quantum 1, 38 (2017), P. A. Höhn and C. S. P. Wever, Phys. Rev. A 95, 012102 (2017)], where they characterise the space of pure states, as well as the unitary group, and are interpreted as complementarity equalities in the Brukner-Zeilinger information measure. The generalisation to arbitrarily many qubits is nontrivial as it requires new tools which in turn reveal novel structural properties that are absent in the two-qubit case. A thorough analysis of these properties, and their relation with mutual unbiasedness and complementarity in the n-qubit Pauli group, can be found in two companion papers.
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.