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arXiv:quant-ph/0209017 · Physical Review A

Decoherence of entangled kaons and its connection to entanglement measures

Reinhold A. Bertlmann, Katharina Durstberger-Rennhofer, Beatrix C. Hiesmayr

Published Jul 30, 2003• 79 citations
arXivHTMLPDF

Abstract

We study the time evolution of the entangled kaon system by considering the Liouville--von Neumann equation with an additional term which allows for decoherence. We choose, as generators of decoherence, the projectors to the two-particle eigenstates of the Hamiltonian. Then we compare this model with the data of the CPLEAR experiment and find in this way an upper bound on strength $\ensuremath{\lambda}$ of the decoherence. We also relate $\ensuremath{\lambda}$ to an effective decoherence parameter $\ensuremath{\zeta}$ considered previously in literature. Finally we discuss our model in the light of different measures of entanglement, i.e., von Neumann entropy S, entanglement of formation E, and concurrence C, and we relate decoherence parameter $\ensuremath{\zeta}$ to the loss of entanglement: $1\ensuremath{-}E.$

DOI
10.1103/physreva.68.012111

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We study the time evolution of the entangled kaon system by considering the Liouville--von Neumann equation with an additional term which allows for decoherence. We choose, as generators of decoherence, the projectors to the two-particle eigenstates of the Hamiltonian. Then we compare this model with the data of the CPLEAR experiment and find in this way an upper bound on strength \ensuremathλ\ensuremath{\lambda}\ensuremathλ of the decoherence. We also relate \ensuremathλ\ensuremath{\lambda}\ensuremathλ to an effective decoherence parameter \ensuremathζ\ensuremath{\zeta}\ensuremathζ considered previously in literature. Finally we discuss our model in the light of different measures of entanglement, i.e., von Neumann entropy S, entanglement of formation E, and concurrence C, and we relate decoherence parameter \ensuremathζ\ensuremath{\zeta}\ensuremathζ to the loss of entanglement: 1\ensuremath−E.1\ensuremath{-}E.1\ensuremath−E.
In the context of the entangled B0B0{B}^{0}{B}^{0}B0B0 state produced at the $\ensuremath{\Upsilon}(4S)$ resonance, we consider a modification of the usual quantum-mechanical time evolution with a dissipative term, which contains only one parameter denoted by $\ensuremath{\lambda}$ and respects complete positivity. In this way a decoherence effect is introduced in the time evolution of the two-particle B0B0{B}^{0}{B}^{0}B0B0 state, which becomes stronger with increasing distance between the two particles. While our model of time evolution has decoherence for the two-particle system, we assume that, after the decay of one of the two B mesons, the resulting one-particle state obeys the purely quantum-mechanical time evolution. From the data on dilepton events we derive an upper bound on $\ensuremath{\lambda}.$ We also show how $\ensuremath{\lambda}$ is related to the so-called ``decoherence parameter'' $\ensuremath{\zeta},$ which parametrizes decoherence in neutral flavored meson-antimeson systems.
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Abstract In this chapter we revisit the creation of an entangled kaon state where the particles comprising the relevant pair propagate in to opposite directions. Experimental tests of Bell inequalities in particle physics are difficult and feature loopholes for local realistic models. Although such Bell tests are of fundamental interest, the examination of entanglement rather than non-locality for meson-antimeson pairs might be more feasible. To examine the possibilities and challenges for such tests, it is useful to study possible mechanisms for decoherence that might influence the quality of the entanglement. We consider possible decoherence effects arising from interactions of the quantum system with its environment: strong-interaction scatterings of kaons with nucleons, weak-interaction decays, as well as experimental noise. We concentrate on a specific model of decoherence developed by Bertlmann and Grimus that allows us to quantify the strength of such possible decoherence effects by the data of existing experiments
Control of open quantum dynamics is of great interest for realizing quantum technologies. Therefore, it is an important task to quantify and characterize the entropy for open quantum systems under decoherence. In this paper, we study the time evolution of the von Neumann entropy for open quantum systems described by the Lindblad master equation. Note that, in particular, when the decoherence corresponds to the measurement for the observable in the system, the von Neumann entropy tends to monotonically increases as the variance becomes larger. Furthermore, we present a lower bound of the von Neumann entropy in the long-time limit. This lower bound has advantages of being straightforwardly calculated and applicable to a general Markovian open quantum system.
We study two different decoherence modes for entangled qubits by considering a Liouville--von Neumann master equation. Mode A is determined by projection operators onto the eigenstates of the Hamiltonian and mode B by projectors onto rotated states. We present solutions for general and for Bell diagonal states and calculate for the latter the mixedness and the amount of entanglement given by the concurrence. We propose a realization of the decoherence modes within neutron interferometry by applying fluctuating magnetic fields. An experimental test of the Kraus operator decomposition describing the evolution of the system for each mode is presented.
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.