Besides Heisenberg uncertainty, entanglement is another mechanism that can enforce complementarity in multipartite quantum systems. Analysing its effects in a bipartite system of two qubits, M. Jakob and J. Bergou had shown that the usual wave-particle duality relation, in interferometric systems, regarding which-way information and interference visibility, should be extended to a ¿triality¿ relation containing, in addition, a quantitative entanglement measure. Employing a model for the neutral K-meson propagation in free space in which its weak decay products are included as a second party, we study its effect in the neutral kaon interferometry. We show that a new quantitative triality relation can also be established in this case. The state of the total system remains pure in the course of dynamics, so, we can use the von Neumann entropy of a reduced party as a quantitative measure of entanglement. The other two quantities in the triality relation are the distinguishability between the decay products states corresponding to the two distinct kaon propagation modes KS and KL, and the wave-like path interference between these states. These two distinct modes are the analogues to the two separated paths in usual interferometric devices. The inequality obtained here can account for the complementarity between strangeness oscillations and lifetime information previously considered in the literature. Moreover, it allows us to visualise through the K0 ¿ K0 oscillations the fundamental role of entanglement in quantum complementarity.
Besides Heisenberg uncertainty, entanglement is another mechanism that can enforce complementarity in multipartite quantum systems. Analysing its effects in a bipartite system of two qubits, M. Jakob and J. Bergou had shown that the usual wave-particle duality relation, in interferometric systems, regarding which-way information and interference visibility, should be extended to a ¿triality¿ relation containing, in addition, a quantitative entanglement measure. Employing a model for the neutral K-meson propagation in free space in which its weak decay products are included as a second party, we study its effect in the neutral kaon interferometry. We show that a new quantitative triality relation can also be established in this case. The state of the total system remains pure in the course of dynamics, so, we can use the von Neumann entropy of a reduced party as a quantitative measure of entanglement. The other two quantities in the triality relation are the distinguishability between the decay products states corresponding to the two distinct kaon propagation modes KS and KL, and the wave-like path interference between these states. These two distinct modes are the analogues to the two separated paths in usual interferometric devices. The inequality obtained here can account for the complementarity between strangeness oscillations and lifetime information previously considered in the literature. Moreover, it allows us to visualise through the K0 ¿ K0 oscillations the fundamental role of entanglement in quantum complementarity.
In bipartite quantum systems, entanglement correlations between the parties exerts direct influence in the phenomenon of wave-particle duality. This effect has been quantitatively analyzed in the context of two qubits by Jakob and Bergou [Opt. Commun. 283, 827 (2010)]. Employing a description of the K-meson propagation in free space where its weak decay states are included as a second party, we study here this effect in the kaon-antikaon oscillations. We show that a new quantitative ``triality'' relation holds, similar to the one considered by Jakob and Bergou. In our case, it relates the distinguishability between the decay-product states corresponding to the distinct kaon propagation modes KS,KL, the amount of wave-like path interference between these states, and the amount of entanglement given by the reduced von Neumann entropy. The inequality can account for the complementarity between strangeness oscillations and lifetime information previously considered in the literature, therefore allowing one to see how it is affected by entanglement correlations. As we will discuss, it allows one to visualize clearly through the ${K}^{0}\ensuremath{-}\overline{K}{\phantom{\rule{0.16em}{0ex}}}^{0}$ oscillations the fundamental role of entanglement in quantum complementarity.
The aim of this contribution is to illustrate two basic aspects of quantum mechanics applied to the neutral kaon system. We first describe a recent quantitative formulation of Bohr's complementarity principle for free--space evolution of single kaons and entangled kaon pairs. We then show that the neutral kaon system is also suitable for an optimal demonstration of the ``quantum eraser'', including its operation in the ``delayed choice'' mode. In our discussions, strangeness oscillations play the role of the traditional interference pattern linked to wave--like behaviour. The role of the two interferometric paths taken by particle--like objects is played by the differently propagating KS and KL components. Their distinct decay widths provide a quantum ``mark'' which can be erased by appropriate strangeness measurements.
We introduce a complete set of complementary quantities in bipartite, two-dimensional systems. Complementarity then relates the quantitative entanglement measure concurrence which is a bipartite property to the single-particle quantum properties predictability and visibility, for the most general quantum state of two qubits. Consequently, from an interferometric point of view, the usual wave-particle duality relation must be extended to a ``triality'' relation containing, in addition, the quantitative entanglement measure concurrence, which has no classical counterpart and manifests a genuine quantum aspect of bipartite systems. A generalized duality relation, that also governs possible violations of the Bell's inequality, arises between single- and bipartite properties.
The study and characterisation of quantum entanglement represents an extended effort within the field of quantum information. This thesis addresses two problems. The first regards entanglement in open systems, where it’s more difficult to define a dichotomic (two outcome) observable due to population loss. The second problem concerns genuine multipartite entanglement, that is entanglement between many particles in states that cannot be separable with respect to any possible bipartition of a system.\nThe first chapter consists of an introduction to the basic elements of quantum mechanics and quantum information theory. We briefly describe a few historical facts about quantum mechanics and the study of entanglement, to help put things into perspective.\nThe second chapter is a detailed discussion of the neutral kaon system. These particles are produced in pairs and are described as a particle-antiparticle system. This system displays some unique properties like the violation of CP-symmetry, and shares some characteristics with other mesons, such as neutral particle oscillation and decay. This makes formulating Bell-type inequalities for them more difficult, but also provides the opportunity for new tests.\nThe third chapter continues with the presentation of an already existent effective formalism for performing Bell tests on neutral kaons. The essence of the formalism is a switch to the Heisenberg picture, transferring the time evolution and dependence on measurement directions to a so-called effective operator. This has many advantages, some of the most important being a proper normalization during population loss (due to decay), an easy application to\nthe case of multiple particles and the fact that it allows one to observe entanglement within the system for a relatively long amount of time. The final part presents a way to simulate neutral kaons with atomic systems, namely with Ytterbium isotopes and discuss what are the differences between these atomic systems and neutral kaons.\nThe second part of the thesis treats the problem of proving genuine multipartite entanglement from separable two-body marginals. For this, an introduction to semidefinite programming is necessary due to the fact that the problem under study can be formulated in terms of this method. We will briefly discuss linear and semidefinite programming as well as the concept of duality and provide some basic examples.\nThe formal description of this problem and previous results are provided in the fifth chapter. Our result shows that, through a cyclic iteration between two SDPs, one obtains states with the desired properties. Using this method we managed to go up to six qubit states for various configurations. From this we can construct higher-dimensional states having the same properties.
This chapter provides an in-depth analysis of the properties and phenomena associated with neutral K-mesons. Kaons are quantum systems illustrating strange behaviors. We begin by examining the significance of strangeness and charge parity violation in understanding these particles. The concept of strangeness oscillations is then introduced, explaining oscillations between K0 and K¯0 states. The regeneration of KS is investigated, uncovering the underlying mechanisms involved. The discussion moves on to quasi-spin space, exploring its bases and their implications. The entangled states of kaon pairs K0K¯0 are considered, with a focus on maximally entangled neutral kaons and nonmaximally entangled states. Decoherence effects on entangled kaons are examined, utilizing the density matrix description to capture the dynamics. A dedicated decoherence parameter is introduced to quantify the impact of decoherence. Furthermore, the chapter investigates the loss of entanglement through measures such as von Neumann entanglement entropy, entanglement of formation, and concurrence. These measures provide insights into quantifying and characterizing entanglement in the context of neutral kaons. Through this comprehensive exploration of properties, phenomena, and entanglement dynamics, this chapter aims to pointing out recent works on neutral kaons, contributing to advancements in particle physics.
Abstract 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.