Complementarity Relations Research Articles

Influence of polarization and the environment on wave–particle duality

Wave-particle duality is certainly one of the most curious concepts of contemporary physics, which ascribes mutually exclusive behaviors to quantum systems that cannot be observed simultaneously. In the context of two-path interferometers, these two behaviors are usually described in terms of the visibility of interference fringes and the path distinguishability. Here, we use quantum information-theoretic tools to derive quantifiers of these two properties, which account for the combined influence of path probability and polarization, and demonstrate that they satisfy a complementarity relation. We further show that the derived quantities can work as probes in the study of open quantum dynamics by revealing interesting facets of environment actions, such as: decoherence, depolarization and scattering.

Bending and Stretching in a Narrow Ribbon of Nematic Polymer Networks

AbstractWe study the spontaneous out-of-plane bending of a planar untwisted ribbon composed of nematic polymer networks activated by a change in temperature. Our theory accounts for both stretching and bending energies, which compete to establish equilibrium. We show that when equilibrium is attained these energy components obey acomplementarityrelation: one is maximum where the other is minimum. Moreover, we identify ableachingregime: for sufficiently large values of an activation parameter (which measures the mismatch between the degrees of order in polymer organization in the reference and current configurations), the ribbon’s deformation is essentially independent of its thickness.

Local predictability and coherence versus distributed entanglement in entanglement swapping from partially entangled pure states

Complete complementarity relations, as e.g. P(ρA)2+C(ρA)2+E(|Ψ〉AB)2=1, constrain the local predictability and local coherence and the entanglement of bipartite pure states. For pairs of qubits prepared initially in a particular class of partially entangled pure states with null local coherence, these relations were used in Ref. [6] (Basso and Maziero (2022)) to provide an operational connection between local predictability of the pre-measurement states with the probability of the maximally entangled components of the states after the Bell-basis measurement of the entanglement swapping protocol (ESP). In this article, we extend this result for general pure initial states. We obtain a general relation between pre-measurement local predictability and coherence and the distributed entanglement in the ESP. We use IBM's quantum computers to verify experimentally some instances of these general theoretical results.

Operational connection between predictability and entanglement in entanglement swapping from partially entangled pure states

Complementarity and entanglement are fundamental features of Quantum Mechanics that were recently related in triality equalities that involve quantum coherence, the wave aspect of a quanton, and quantum predictability and quantum entanglement, the particle aspect of a quanton. In this article, we give an operational connection between predictability and entanglement in entanglement swapping from initially partially entangled states. For this, we show that the predictability of the pre-measurement one-qubit density matrix is directly related to the probability of the partially entangled component of the post-measurement state. Going even further, we analyze the entanglement swapping for partially entangled states in the light of complementarity relations and show that, in the cases where the entanglement increases after a Bell-basis measurement, the predictability is consumed when compared to the initially prepared state.

Quantifying Complementarity via 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.

Hiding the which-path information of photons

Which-path information of a quantum particle in interferometers is the key to infer the past of quantum particle. It arises many extensive discussions including quantum complementarity and path-visibility relation. The basic of these discussions are the description, detection and control of which-path information. In this article, we focus on the investigation of multidimensional which-path information in nested Mach-Zehnder interferometer. A general expression of which-path information is given and can be partially extracted by different detection method. Further analysis shows that the which-path information can be controlled by the phase differences and beam splitting ratios between the arms of nested Mach-Zehnder interferometer. Moreover, a new which-path information elimination phenomenon has been predicted and demonstrated experimentally. Our work can help to understand the physics of quantum particles, potentially apply to quantum information process and quantum metrology.

Entanglement monotones from complementarity relations

Bohr’s complementarity and Schrödinger’s entanglement are two prominent physical characters of quantum systems. In this article, we formally connect them. It is known that complementarity relations for wave-particle duality are saturated only for pure, single-quanton, quantum states. For mixed states, the wave-particle quantifiers never saturate a complementarity relation and can even reach zero for a maximally mixed state. To fully characterize a quanton, it is not enough to consider its wave-particle aspect; we have also to regard its quantum correlations with other systems. Here we prove that for any complete complementarity relation involving predictability and visibility measures that satisfy the criteria established in the literature, the corresponding quantum correlations are entanglement monotones. Therefore, we formally connect entanglement monotones with complementarity relations without appealing to a particular measure.

Uncertainties Influencing Transportation System Performances

The design and operation of transportation systems, as with any large complex technical system, are marked by indetermination—risks and uncertainties (scientific/methodologic and/or socio-economic). This paper analyzes the occurrence and consequences of uncertainties, defined as completely unknown random events (“unknown unknowns”), on transportation system performances. Interest in the topic is justified by the considerable value and long life of transportation system components. In order to reduce the effects of uncertainties, a holistic approach to all technical infrastructures in society, regardless of the flow category (material, energy, information), is necessary. Technological progress and changes in territorial activity systems historically confirm the dynamism of the competition and complementarity relations between civil and industrial infrastructures and transport infrastructures, as well as among different modal transport/traffic infrastructures. Declining discount rates are applied to compensate for the effects of uncertainties on investment project opportunities on long time horizons. There is no unanimous agreement on the discount rate values. Unforeseen exogenous events are considered differentiated/non-systemic or undifferentiated/systemic uncertainties. They can have significant consequences on the performance of a transport system, including a change in the transport market share. Therefore, an adaptive policy is required to reduce the methodological/scientific and socio-economic uncertainties that affect the design and operation of any transportation system.

The effects of proximities on the evolving structure of intercity innovation networks in the Guangdong–Hong Kong–Macao Greater Bay Area: comparison between scientific and technology knowledge

ABSTRACT The various knowledge flows shape and change the regional innovation patterns, which are also influenced by regional conditions. What are the similarities, differences and connections between science and technology linkage, as two different types of knowledge network, deserve to be explored in depth. Drawing on scientific paper co-publications and patent transfer data, we constructed two different types of intercity innovation networks during 2006–2016 in the Guangdong–Hong Kong–Macao Greater Bay Area (GBA), a city region special for its ‘one country, two systems’ structure. After that, we explored the evolutionary characteristics of the networks and further examined the different impacts of multidimensional proximity on scientific collaboration and technology transfer. Our results show that technology transfer is more sensitive to spatial factors, institutional barriers caused by ‘one country, two systems’ is a bigger obstacle to technology transfer between cities, cultural proximity and cognitive proximity have a more significant impact on paper cooperation network. Moreover, geographical proximity can indirectly affect knowledge spillover by acting on the proximity of other dimensions. As for scientific collaboration, social, cognitive and institutional proximities can compensate for the lack of geographical proximity, and cultural proximity frequently goes along with geographical proximity; as for technology transfer, geographical proximity has neither substitutional or complementarity relations with cultural and cognitive proximities, the interrelatedness between geographical and institutional and social proximities are complementarity which is opposed to paper co-publications. This study explores the differences in spillover mechanisms of different knowledge types and contributes to enriching the empirical framework of multidimensional proximities and innovation network researches.

Estimating the Shannon Entropy and (Un)certainty Relations for Design-Structured POVMs

Complementarity relations between various characterizations of a probability distribution are at the core of information theory. In particular, lower and upper bounds for the entropic function are of great importance. In applied topics, we often deal with situations, where the sums of certain powers of probabilities are known. The main question is how to convert the imposed restrictions into two-sided estimates on the Shannon entropy. It is addressed in two different ways. The more intuitive of them is based on truncated expansions of the Taylor type. Another method is based on the use of coefficients of the shifted Chebyshev polynomials. We propose here a family of polynomials for estimating the Shannon entropy from below. As a result, estimates are more uniform in the sense that errors do not become too large in particular points. The presented method is used for deriving uncertainty and certainty relations for positive operator-valued measures assigned to a quantum design. Quantum designs are currently the subject of active researches due to potential use in quantum information science. It is shown that the derived estimates are applicable in quantum tomography and detecting steerability of quantum states.

Predictability as a quantum resource

Just recently, complementarity relations (CRs) have been derived from the basic rules of Quantum Mechanics. The complete CRs are equalities involving quantum coherence, $C$, quantum entanglement, and predictability, $P$. While the first two are already quantified in the resource theory framework, such a characterization lacks for the last. In this article, we start showing that, for a system prepared in a state $\rho$, $P$ of $\rho$, with reference to an observable $X$, is equal to $C$, with reference to observables mutually unbiased (MU) to $X$, of the state $\Phi_{X}(\rho)$, which is obtained from a non-revealing von Neumann measurement (NRvNM) of $X$. We also show that $P^X(\rho)>C^{Y}(\Phi_{X}(\rho))$ for observables not MU. Afterwards, we provide quantum circuits for implementing NRvNMs and use these circuits to experimentally test these (in)equalities using the IBM's quantum computers. Furthermore, we give a resource theory for predictability, identifying its free quantum states and free quantum operations and discussing some predictability monotones. Besides, after applying one of these predictability monotones to study bipartite systems, we discuss the relation among the resource theories of quantum coherence, predictability, and purity.

From wave-particle duality to wave-particle-mixedness triality: an uncertainty approach

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.

Uncertainty relation of successive measurements based on Wigner–Yanase skew information

Wigner–Yanase skew information could quantify the quantum uncertainty of the observables that are not commuting with a conserved quantity. We present the uncertainty principle for two successive projective measurements in terms of Wigner–Yanase skew information based on a single quantum system. It could capture the incompatibility of the observables, i.e. the lower bound can be nontrivial for the observables that are incompatible with the state of the quantum system. Furthermore, the lower bound is also constrained by the quantum Fisher information. In addition, we find the complementarity relation between the uncertainties of the observable which operated on the quantum state and the other observable that performed on the post-measured quantum state and the uncertainties formed by the non-degenerate quantum observables performed on the quantum state, respectively.

Complete complementarity relations: Connections with Einstein-Podolsky-Rosen realism and decoherence, and extension to mixed quantum states

Wave-particle duality is an essential character of quantum systems. In the last few years, much progress has been made towards formally quantifying these quantum features. The properties of the quantum density matrix were shown to lead to duality inequalities for single-quanton mixed states and to triality identities, also known as complete complementarity relations (CCRs), for two-quanton pure states. In this article, we establish connections between CCRs and Einstein-Podolsky-Rosen (ir)realism, as formalized by Bilobran and Angelo (see Bilobran A. L. O. and Angelo R. M., EPL, 112 (2015) 40005). Besides, we show that, for tripartite pure states , CCRs can be applied to obtain the system-environment entanglement of formation, , through the local information of a system A and its classical correlation with an auxiliary system B. Moreover, we discuss CCRs in the context of the emergence of the pointer basis in the decoherence process identified via the constancy of system-apparatus classical correlations. We also obtain CCRs for purified mixed bipartite quantum-classical states. In addition, we derive CCRs for mixed-bipartite two-qudit quantum states and discuss its interpretations.

Complete complementarity relations and their Lorentz invariance

It is well known that entanglement under Lorentz boosts is highly dependent on the boost scenario in question. For single-particle states, a spin-momentum product state can be transformed into an entangled state. However, entanglement is just one of the aspects that completely characterizes a quantum system. The other two are known as the wave-particle duality. Although the entanglement entropy does not remain invariant under Lorentz boosts, and neither do the measures of predictability and coherence, we show here that these three measures taken together, in a complete complementarity relation (CCR), are Lorentz invariant. Peres et al. (Peres et al. 2002 Phys. Rev. Lett. 88 , 230402. ( doi:10.1103/PhysRevLett.88.230402 )) realized that even though it is possible to formally define spin in any Lorentz frame, there is no relationship between the observable expectation values in different Lorentz frames. Analogously, one can, in principle, define complementary relations in any Lorentz frame, but there is no obvious transformation law relating complementary relations in different frames. However, our result shows that the CCRs have the same value in any Lorentz frame, i.e. there is a transformation law connecting the CCRs. In addition, we explore relativistic scenarios for single and two-particle states, which helps in understanding the exchange of different aspects of a quantum system under Lorentz boosts.

Quantitative complementarity of wave-particle duality.

To test the principle of complementarity and wave-particle duality quantitatively, we need a quantum composite system that can be controlled by experimental parameters. Here, we demonstrate that a double-path interferometer consisting of two parametric downconversion crystals seeded by coherent idler fields, where the generated coherent signal photons are used for quantum interference and the conjugate idler fields are used for which-path detectors with controllable fidelity, is useful for elucidating the quantitative complementarity. We show that the quanton source purity μ s is tightly bounded by the entanglement E between the quantons and the remaining degrees of freedom by the relation [Formula: see text], which is experimentally confirmed. We further prove that the experimental scheme using two stimulated parametric downconversion processes is an ideal tool for investigating and understanding wave-particle duality and Bohr's complementarity quantitatively.

Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators

This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in optimization algorithms and the modeling of physical systems. The differential inclusion is described by a time-dependent set-valued mapping having the property that, for a given time instant, the set-valued mapping describes a maximal monotone operator. By successive application of a proximal operator, we construct a sequence of functions parameterized by the sampling time that corresponds to the discretization of the continuous-time system. Under certain mild assumptions on the regularity with respect to the time argument, and using appropriate tools from functional and variational analysis, this sequence is then shown to converge to the unique solution of the original differential inclusion. The result is applied to develop conditions for well-posedness of differential equations interconnected with nonsmooth time-dependent complementarity relations, using passivity of underlying dynamics (equivalently expressed in terms of linear matrix inequalities).

When does complementarity support pluralism about schools of economic thought?

ABSTRACT An intuitively appealing argument for pluralism in economics can be made on the grounds that schools of economic thought complement one another. Let us call this the complementarity-based argument for pluralism (CAP). The concepts of complementarity, pluralism, and school of thought are scrutinized in this paper to evaluate this argument. I argue that the complementarity of schools is relative to scientific goals, which implies that discussing complementarity of schools of economic thought requires discussing the goals of economic research. I also distinguish weak from strong complementarity and show that some alleged complementarity relations between schools are weak and thus provide little support for CAP. However, if strong complementarity relations, relative to a valuable goal, can be demonstrated to exist between specific schools, this is a strong reason for pluralism about those schools. Finally, I provide suggestions on how to distinguish strong from weak complementarity.

One-particle and two-particle visibilities in bipartite entangled Gaussian states

Complementarity between one- and two-particle visibility in discrete systems can be extended to bipartite quantum-entangled Gaussian states. The meaning of the two-particle visibility originally defined by Jaeger, Horne, Shimony, and Vaidman with the use of an indirect method that first corrects the two-particle probability distribution by adding and subtracting other distributions with varying degree of entanglement, however, deserves further analysis. Furthermore, the origin of complementarity between one-particle visibility and two-particle visibility is somewhat elusive and it is not entirely clear what is the best way to associate particular two-particle quantum observables with the two-particle visibility. Here, we develop a direct method for quantifying the two-particle visibility based on measurement of a pair of two-particle observables that are compatible with the measured pair of single-particle observables. For each of the two-particle observables the corresponding visibility is computed, after which the absolute difference of the latter pair of visibilities is considered as a redefinition of the two-particle visibility. Our approach reveals a mathematical symmetry as it treats the two pairs of one-particle or two-particle observables on equal footing by formally identifying all four observable distributions as rotated marginal distributions of the original two-particle probability distribution. The complementarity relation between one-particle visibility and two-particle visibility obtained with the direct method is exact in the limit of infinite Gaussian precision where the entangled Gaussian state approaches an ideal EPR state. The presented results demonstrate the theoretical utility of rotated marginal distributions for elucidating the nature of two-particle visibility and provide tools for the development of quantum applications employing continuous variables.

An uncertainty view on complementarity and a complementarity view on uncertainty

Since the uncertainty about an observable of a system prepared in a quantum state is usually described by its variance, when the state is mixed, the variance is a hybrid of quantum and classical uncertainties. Besides that, complementarity relations are saturated only for pure, single-quanton, quantum states. For mixed states, the wave-particle quantifiers never saturate the complementarity relation and can even reach zero for a maximally mixed state. So, to fully characterize a quanton it is not sufficient to consider its wave-particle aspect; one has also to regard its correlations with other systems. In this paper, we discuss the relation between quantum correlations and local classical uncertainty measures, as well as the relation between quantum coherence and quantum uncertainty quantifiers. We obtain a complete complementarity relation for quantum uncertainty, classical uncertainty, and predictability. The total quantum uncertainty of a d-paths interferometer is shown to be equivalent to the Wigner-Yanase coherence and the corresponding classical uncertainty is shown to be a quantum correlation quantifier. The duality between complementarity and uncertainty is used to derive quantum correlations measures that complete the complementarity relations for $l_1$-norm and $l_2$-norm coherences. Besides, we show that Brukner-Zeilinger's invariant information quantifies both the wave and particle characters of a quanton and we obtain a sum uncertainty relation for the generalized Gell Mann's matrices.