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arXiv:quant-ph/0402190

Entanglement of a Multiparticle Schroedinger Cat State

A. N. Salgueiro

Published Feb 25, 2004• 0 citations
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Abstract

We characterize the degree of entanglement of a subsystem of $k$ particles in a $N$-two level system ($k\leq N/2$) initially prepared in a mesoscopic superposition $|ψ>=\int dθf(θ) (|ϕ_{1}(θ)>^{\otimes N}+|ϕ_{2}(θ)>^{\otimes N})$, where $f(θ)$ is a gaussian or a delta function, subject to the time evolution described by a dephasing channel. Negativity is used as a measure of entanglement for such system. For an arbitrary number of particles $N$, numerical results are given for the full time evolution up to ten particles. Analytical results are obtained for short times and asymptotic time regimes. We show that negativity is initially proportional to the square root of the product of the number of particles in each partition, the overlap ${||}^2$ and the coupling to the environment. Asymptotically, negativity tends to zero, a necessary condition for separability.

DOI
10.48550/arxiv.quant-ph/0402190

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We characterize the degree of entanglement of a subsystem of kkk particles in a NNN-two level system (k≤N/2k\leq N/2k≤N/2) initially prepared in a mesoscopic superposition ∣ψ>=∫dθf(θ)(∣ϕ1(θ)>⊗N+∣ϕ2(θ)>⊗N)|ψ>=\int dθf(θ) (|ϕ_{1}(θ)>^{\otimes N}+|ϕ_{2}(θ)>^{\otimes N})∣ψ>=∫dθf(θ)(∣ϕ1​(θ)>⊗N+∣ϕ2​(θ)>⊗N), where f(θ)f(θ)f(θ) is a gaussian or a delta function, subject to the time evolution described by a dephasing channel. Negativity is used as a measure of entanglement for such system. For an arbitrary number of particles NNN, numerical results are given for the full time evolution up to ten particles. Analytical results are obtained for short times and asymptotic time regimes. We show that negativity is initially proportional to the square root of the product of the number of particles in each partition, the overlap ∣∣2{||}^2∣∣2 and the coupling to the environment. Asymptotically, negativity tends to zero, a necessary condition for separability.
Entanglement of an open bipartite system is investigated. In particular, we calculate the entanglement of a subsystem AAA with kkk particles in an NNN atoms of two-level system $(k\ensuremath{\leqslant}N∕2)$ initially prepared in a mesoscopic superposition $\ensuremath{\mid}\ensuremath{\psi}⟩=[{\ensuremath{\mid}{\ensuremath{\phi}}_{1}({\ensuremath{\theta}}_{0})⟩}^{\ensuremath{\bigotimes}N}+{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}N}]$, with $\ensuremath{\mid}{\ensuremath{\phi}}_{1}({\ensuremath{\theta}}_{0})⟩=\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\theta}}_{0}\ensuremath{\mid}0⟩+\mathrm{sin}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\theta}}_{0}\ensuremath{\mid}1⟩$, subject to the time evolution described by a dephasing channel. For an arbitrary number of particles NNN, numerical results are given for the full time evolution up to ten particles and $1\ensuremath{\leqslant}k\ensuremath{\leqslant}5$. Analytical results are obtained for short times and asymptotic time regimes. Entanglement is maximum when ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}∕2$ [Greenberger-Horne-Zeilinger (GHZ) state], independently of the number of particles in each partition. As NNN gets larger, entanglement tends to 1∕21∕21∕2, and becomes independent of ${\ensuremath{\theta}}_{0}$. The ``thermodynamic'' regime ($N\ensuremath{\rightarrow}\ensuremath{\infty}$ with fNfNfN finite, where f=k∕Nf=k∕Nf=k∕N) is reached already for N=1000N=1000N=1000 particles. In the regime of k⪡Nk⪡Nk⪡N and $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the entanglement has its maximum value at ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}∕2$ and for ${\ensuremath{\theta}}_{0}=0,\ensuremath{\pi}$ its minimum value. The time evolution reduces the maximum value of entanglement, showing that the subsystem AAA loses coherence to the environment. Finally, we point out the differences between the entanglement rate obtained here from that one in Phys. Rev. Lett. 89, 210402 (2002), where only one NNN-particle subsystem is used.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we solve in the asymptotic long-time regime the master equation for two independent harmonic oscillators interacting with an environment. We give a description of the continuous-variable asymptotic entanglement in terms of the covariance matrix of the considered subsystem for an arbitrary Gaussian input state. Using Peres–Simon necessary and sufficient condition for separability of two-mode Gaussian states, we show that for certain classes of environments the initial state evolves asymptotically to an entangled equilibrium bipartite state, while for other values of the coefficients describing the environment, the asymptotic state is separable. We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuousvariable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment.Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state.For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled.In other cases, entanglement generation or entanglement collapse (entanglement sudden death) take place or even a periodic collapse and revival of entanglement.We show that for certain classes of environments the initial state evolves asymptotically to an entangled equilibrium bipartite state, while for other values of the coefficients describing the environment, the asymptotic state is separable.We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we solve in the asymptotic long-time regime the master equation for two independent harmonic oscillators interacting with an environment. We give a description of the continuous-variable asymptotic entanglement in terms of the covariance matrix of the considered subsystem for an arbitrary Gaussian input state. Using Peres--Simon necessary and sufficient condition for separability of two-mode Gaussian states, we show that for certain classes of environments the initial state evolves asymptotically to an entangled equilibrium bipartite state, while for other values of the coefficients describing the environment, the asymptotic state is separable. We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous-variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state. For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation or entanglement collapse (entanglement sudden death) take place or even a periodic collapse and revival of entanglement. We show that for certain classes of environments the initial state evolves asymptotically to an entangled equilibrium bipartite state, while for other values of the coefficients describing the environment, the asymptotic state is separable. We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.