Complementarity Relations Research Articles

An updated quantum complementarity principle

Bohr’s complementarity principle has long been a fundamental concept in quantum mechanics, positing that, within a given experimental set-up, a quantum system (or quanton) can exhibit either its wave-like character, denoted as W , or its particle-like character, denoted as P , but not both simultaneously. Modern interpretations of Bohr’s complementarity principle acknowledge the coexistence of these aspects in the same experiment while introducing the constraint W + P ≤ α . Notably, estimations of W or P frequently rely on indirect retrodiction methods, a practice that has led to the claim of the violation of Bohr’s complementarity principle. By taking a different route, recent advancements demonstrate that quantum complementarity relations can be rigorously derived from the axioms of quantum mechanics. To reconcile these observations and eliminate potential paradoxes or violations, we propose an updated formulation for the quantum complementarity principle, which is stated as follows: For a given quantum state preparation ρ t at a specific instant of time t , the wave and particle behaviours of a quanton are constrained by a complementarity relation 𝔚 ( ρ t ) + 𝔓 ( ρ t ) ≤ α ( d ) , which is derived directly from the axioms of quantum mechanics.

Coherence and entropy complementarity relations of generalized wave-particle duality

Coherence and entropy complementarity relations of generalized wave-particle duality

Complementarity of subjective and objective realities: An experimental investigation of variations of subjective self-reported intelligence data when objective data are erased or not-erased

The Complementarity Principle (CP), introduced by Nils Bohr, described the wave-particle duality of quantum phenomena and its dependence on the measurement setup exploring the quantum states. In general, this principle summarized the fact that two mutually exclusive and contradictory states of quantum events can be reconciled as a measurement-dependent occurrence of these states. Originally, the CP was postulated only for the realm of quantum mechanics, but recently the Generalized Quantum Theory (GQT) extended its area of applicability to the macroscopic domain. According to GQT complementarity relations are also hypothesized to exist between macroscopic objective and subjective measurements of the same concept. This implies objective measurement-dependent variations of subjective experience. To test this hypothesis, seven studies were conducted in which individual variations in subjective intelligence ratings were tested in relation to the availability or non-availability of corresponding objective individual intelligence data. Only one pre-registered study (Study 2) showed strong Bayesian evidence for H1 (BF10 > 10), indicating, as predicted, higher subjective intelligence ratings when objective data were erased compared to a condition in which objective data were available within a male sample. This effect could not be replicated in direct replication attempts, nor did a moderator search in subsequent studies find any robust systematic variation in the data. The results seem to question the validity of the macroscopic complementarity conjecture derived from the GQT. On the other hand, they could also be interpreted as the “effect and decline” data pattern that the Model of Pragmatic Information would predict when conducting empirical confirmations of macroscopic complementarity relations. Possible future research strategies to clarify these different interpretations are discussed.

Nonsmooth dynamic modeling and simulation of spatial mechanisms with frictional translational clearance joints

This paper presents a nonsmooth dynamic approach of modeling and computation for spatial mechanism including spatial frictional translational joints (SFTJs) with small clearances. In this research, the dynamic formulation is derived in the Lie group setting, which leads to a coordinate-free and compact formulation. In the following, the normal contact interaction between the slider and guide is described by the complementarity relations between the constraint force in the normal direction. The tangent contact interaction in the SFTJ is characterized by the Coulomb’s friction law in the type of a set-valued map. Based on the horizontal linear complementarity problem (HLCP), the non-smooth dynamics can be established and calculated by combining the Lemke’s algorithm and the RK-MK time integration algorithm. Finally, A spatial crank-slider mechanism with SFTJs is shown as a numerical case. The simulation results demonstrate the correctness and effectiveness of the proposed method which can capture the nonsmooth dynamical behavior of the system.

On complementarity and distribution of imaginarity in finite dimensions

We investigate the complementarity relation among the l1 norm of imaginarity, linear entropy and the Brukner–Zeilinger invariant information in the presence of conjugate symmetry. By using an equality derived from the complementarity relation, we define a quantity named index of imaginarity distribution with respect to the imaginarity, linear entropy and invariant information in finite dimensional quantum systems. We show that the index of imaginarity distribution is nonnegative and upper bounded by the square of the l1 norm of imaginarity, and serves as an invaluable tool in characterizing effectively the imaginarity distribution. Detailed examples are presented to illustrate our theoretical analysis.

Quantifying quantumness in three-flavor neutrino oscillations

We characterize quantum correlations encoded in a three-flavor oscillating neutrino system by using both plane-wave and wave-packet approach. By means of the Complete Complementarity Relations (CCR) we study the trade-off of predictability, local coherence and non-local correlations in terms of the relevant parameters, chosen from recent neutrino experiments. Although the CCR describe very well the contributions associated to bipartite correlations, an attempt of promoting these relations to include the genuine tri-partite contributions in the pure-state case leads to a not completely meaningful result. However, we provide an analysis of the genuine tripartite contributions both for the pure instance and for the mixed case, independently of CCR.

Complete complementarity relations for quantum correlations in neutrino oscillations with CP violation

We characterize quantum correlations encoded in a three-flavor oscillating neutrino system by taking into account CP violation. By means of the complete complementarity relations (CCR), we study the trade-off of predictability, local coherence and nonlocal correlations by using relevant parameters given by neutrino oscillation experiments. Although the CCR describe very well the contributions associated to bipartite correlations, we explore the possible presence of genuine tripartite correlations, by exploiting polygamy relations. We also provide some observations about the nonlocal advantage of quantum coherence, a mazy quantum correlation characterizing the most valuable, nonlocal, portion of quantum coherence in a multi-partite system.

Quantum Dynamics of Cavity–Bose–Einstein Condensates in a Gravitational Field

We theoretically studied the quantum dynamics of a cavity–Bose–Einstein condensate (BEC) system in a gravitational field, which is composed of a Fabry–Pérot cavity and a BEC. We also show how to deterministically generate the transient macroscopic quantum superposition states (MQSSs) of the cavity by the use of optomechanical coupling between the cavity field and the BEC. The quantum dynamics of the cavity–BEC system specifically include phase space trajectory dynamics, system excitation number dynamics, quantum entanglement dynamics, and quantum coherence dynamics. We found that the system performs increasingly complex trajectories for larger values of the Newtonian gravity parameter. Moreover, the number of phonon excitations of the system can be increased by coupling the cavity–BEC system to Newtonian gravity, which is analogous to an external direct current drive. The scattering of atoms inside the BEC affects the periodicity of the quantum dynamics of the system. We demonstrate a curious complementarity relation between the quantum entanglement and quantum coherence of cavity–BEC systems and found that the complementarity property can be sustained to some extent, despite being in the presence of the cavity decay. This phenomenon also goes some way to show that quantum entanglement and quantum coherence can be referred to together as quantum resources.

Complementarity relations of a delayed-choice quantum eraser in a quantum circuit

Complementarity relations of a delayed-choice quantum eraser in a quantum circuit

Robustness of Wave-Particle Duality under Unruh Effect.

By considering a uniformly accelerated two-level system in an initial superposition state of a qubit, we investigate the loss of coherence induced by the acceleration. In addition, we investigate the impact of acceleration on the complementarity relation in a quantum interferometric circuit or quantum scattering circuit. We present an alternative approach to exploring acceleration effects through examination of quantum coherence decay and degradation in the interference pattern. Our investigations help to provide understanding of the consequences of decoherence induced by the Unruh effect on the wave-particle duality of a uniformly accelerated qubit.

Ignorance and Forgetfulness in Late Egyptian and Classical Egyptian from the New Kingdom until the 26th Dynasty: A Lexical Study

The conceptual domain of cognition in Ancient Egyptian is realized linguistically through numerous lexemes and expressions. Following Fortescue, these lexical units can be organized around five pivot-concepts that appear to consistently emerge cross-linguistically and define subdomains within cognition. These subdomains are: knowing, understanding, intending, remembering and thinking, to which a sixth notion attention has here been added. The present study focuses on three verbs with negative meanings in relation to the subdomains knowing and remembering: xm “be ignorant”, smx “forget” and mhj “be forgetful, forget”, as well as the negative constructions neg. + rx “not know”. The aim of this article is to show that the semantics of these lexical units are interconnected with contextual para-synonymy and complementarity relations.

Trade-off relations of geometric coherence

Quantum coherence is an important quantum resource and it is intimately related to various research fields. The geometric coherence is a coherence measure both operationally and geometrically. We study the trade-off relation of geometric coherence in qubit systems. We first derive an upper bound for the geometric coherence by the purity of quantum states. Based on this, a complementarity relation between the quantum coherence and the mixedness is established. We then derive the quantum uncertainty relations of the geometric coherence on two and three general measurement bases in terms of the incompatibility respectively, which turn out to be state-independent for pure states. These trade-off relations provide the limit to the amount of quantum coherence. As a byproduct, the complementarity relation between the minimum error probability for discriminating a pure-states ensemble and the mixedness of quantum states is established.

Tsallis relative α entropy of coherence dynamics in Grover′s search algorithm

Quantum coherence plays a central role in Grover’s search algorithm. We study the Tsallis relative entropy of coherence dynamics of the evolved state in Grover’s search algorithm. We prove that the Tsallis relative entropy of coherence decreases with the increase of the success probability, and derive the complementarity relations between the coherence and the success probability. We show that the operator coherence of the first relies on the size of the database the success probability and the target states. Moreover, we illustrate the relationships between coherence and entanglement of the superposition state of targets, as well as the production and deletion of coherence in Grover iterations.

Complete complementarity relations for three-flavor neutrino oscillations

We study quantum correlations encoded in a three-flavor neutrino system by using complete complementarity relations (CCR). Due to the presence of local coherence in two-flavor subsystems, the CCR has an additional contribution not present in the two flavor mixing case. We investigate such coherence for the three possible bipartite subsystems of the global state both for an electron and a muon neutrino system.

Complementarity between quantum coherence and mixedness: a majorization approach

Quantum coherence is a relevant resource for various quantum information processing tasks, but it is fragile since it is generally affected by environmental noise. This is reflected in the loss of purity of the system, which in turn limits the amount of quantum coherence of it. As a consequence, a complementarity relation between coherence and mixedness arises. Previous works characterize this complementarity through inequalities between the ℓ 1-norm of coherence and linear entropy, and between the relative entropy of coherence and von Neumann entropy. However, coherence–mixedness complementarity is expected to be a general feature of quantum systems, regardless of the measures used. Here, an alternative approach to coherence–mixedness complementarity, based on majorization theory, is proposed. Vectorial quantifiers of coherence and mixedness, namely the coherence vector and the spectrum, respectively, are used, instead of scalar measures. A majorization relation for the tensor product of both vectorial quantifiers is obtained, capturing general aspects of the trade-off between coherence and mixedness. The optimal bound for qubit systems and numerical bounds for qutrit systems are analyzed. Finally, coherence–mixedness complementarity relations are derived for a family of symmetric, concave and additive functions. These results provide a deeper insight into the relation between quantum coherence and mixedness.

The exactness of the [formula omitted] penalty function for a class of mathematical programs with generalized complementarity constraints

In a Mathematical Program with Generalized Complementarity Constraints (MPGCC), complementarity relation is imposed between each pair of variable blocks. MPGCC includes the traditional Mathematical Program with Complementarity Constraints (MPCC) as a special case. On account of the disjunctive feasible region, MPCC and MPGCC are generally difficult to handle. The ℓ1 penalty method, often adopted in computation, opens a way of circumventing the difficulty. Yet it remains unclear about the exactness of the ℓ1 penalty function, namely, whether there exists a sufficiently large penalty parameter so that the penalty problem shares the optimal solution set with the original one. In this paper, we consider a class of MPGCCs that are of multi-affine objective functions. This problem class finds applications in various fields, e.g., the multi-marginal optimal transport problems in many-body quantum physics and the pricing problems in network transportation. We first provide an instance from this class, the exactness of whose ℓ1 penalty function cannot be derived by existing tools. We then establish the exactness results under rather mild conditions. Our results cover those existing ones for MPCC and apply to multi-block contexts.

An adaptive time stepping scheme for rate-independent systems with nonconvex energy

Abstract We investigate a local incremental stationary scheme for the numerical solution of rate-independent systems. Such systems are characterized by a (possibly) nonconvex energy and a dissipation potential, which is positively homogeneous of degree one. Due to the nonconvexity of the energy, the system does in general not admit a time-continuous solution. In order to resolve these potential discontinuities, the algorithm produces a sequence of state variables and physical time points as functions of a curve parameter. The main novelty of our approach in comparison to existing methods is an adaptive choice of the step size for the update of the curve parameter, depending on a prescribed tolerance for the residua in the energy-dissipation balance, and in a complementarity relation concerning the so-called local stability condition. It is proven that, for tolerance tending to zero, the piecewise affine approximations generated by the algorithm converge (weakly) to a so-called parametrized balanced viscosity solution. Numerical experiments illustrate the theoretical findings, and show that an adaptive choice of the step size indeed pays off as they lead to a significant increase of the step size during sticking and in viscous jumps.

Quantum simulation of the generalized-entangled quantum eraser and the related complete complementarity relations

We utilize IBM’s quantum computers to perform a full quantum simulation of the optical quantum eraser (QE) utilizing a Mach–Zehnder interferometer with a variable partially-polarizing beam splitter (VPPBS) at the input. The use of the VPPBS motivates us to introduce the entangled quantum eraser, for which the path information is erased using a Bell-basis measurement. We also investigate the behavior of the wave aspect, i.e., the quantum coherence, as well as the particle character, represented by the predictability and entanglement, as delineated in complete complementarity relations (CCRs). As we show in this article, the utilization of the VPPBS uncover interesting aspects of the QE and CCRs. For instance, we can recover the full wave-behavior by the erasure procedure even when we have only partial knowledge about the path through entanglement.

Degeneration of the Grover search algorithm with depolarization in the oracle-box wires

Grover’s search algorithm and similar techniques are widely used in quantum information science. Communication lines with the so-called oracle are one of inevitable vulnerabilities of quantum search. The impact of localized dephasing and amplitude damping on Grover’s algorithm had already been discussed in recent literature. In this paper, we study the influence of depolarization in the oracle-box wires on the search process. It is shown that even low level of noise is sufficient to degenerate Grover’s algorithm. Complementarity relations between the relative entropy of coherence and the success probability in the presence of depolarization are studied as well.

Quantum coherence versus interferometric visibility in a biased Mach–Zehnder interferometer

The double-slit interferometer and the Mach-Zehnder interferometer (MZI) with balanced beam splitters are prototypical setups for investigating the quantum wave-particle duality. These setups induced a quantitative association of interferometric visibility (IVI) with the wave aspect of a single quantum system (WAQ). Recently, it was realized that quantum coherence (QC) can be better suited than IVI for quantifying the WAQ in complementarity relations. In this article, we investigate a MZI with biased beam splitters both in the input and the output, and we show that in some cases the IVI is not adequate to quantify the WAQ since it does not reflect the behavior of the quantum coherence, even for a bi-dimensional closed quantum system. Using IBM quantum computers, we experimentally verify our theoretical findings by doing a full quantum simulation of the optical MZI with biased beam splitters.