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arXiv:quant-ph/0302055 · Physical Review A

Entanglement between a qubit and the environment in the spin-boson model

Theo A. Costi, Ross H. McKenzie

Published Sep 5, 2003• 104 citations
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Abstract

The quantitative description of the quantum entanglement between a qubit and its environment is considered. Specifically, for the ground state of the spin-boson model, the entropy of entanglement of the spin is calculated as a function of $\ensuremath{\alpha},$ the strength of the ohmic coupling to the environment, and $\ensuremath{\varepsilon},$ the level asymmetry. This is done by a numerical renormalization group treatment of the related anisotropic Kondo model. For $\ensuremath{\varepsilon}=0,$ the entanglement increases monotonically with $\ensuremath{\alpha},$ until it becomes maximal for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}{1}^{\ensuremath{-}}.$ For fixed $\ensuremath{\varepsilon}>0,$ the entanglement is a maximum as a function of $\ensuremath{\alpha}$ for a value, $\ensuremath{\alpha}={\ensuremath{\alpha}}_{M}<1.$

DOI
10.1103/physreva.68.034301

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The quantitative description of the quantum entanglement between a qubit and its environment is considered. Specifically, for the ground state of the spin-boson model, the entropy of entanglement of the spin is calculated as a function of \ensuremathα,\ensuremath{\alpha},\ensuremathα, the strength of the ohmic coupling to the environment, and \ensuremathε,\ensuremath{\varepsilon},\ensuremathε, the level asymmetry. This is done by a numerical renormalization group treatment of the related anisotropic Kondo model. For \ensuremathε=0,\ensuremath{\varepsilon}=0,\ensuremathε=0, the entanglement increases monotonically with \ensuremathα,\ensuremath{\alpha},\ensuremathα, until it becomes maximal for \ensuremathα\ensuremath→1\ensuremath−.\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}{1}^{\ensuremath{-}}.\ensuremathα\ensuremath→​1\ensuremath−. For fixed \ensuremathε>0,\ensuremath{\varepsilon}>0,\ensuremathε>0, the entanglement is a maximum as a function of \ensuremathα\ensuremath{\alpha}\ensuremathα for a value, \ensuremathα=\ensuremathαM<1.\ensuremath{\alpha}={\ensuremath{\alpha}}_{M}<1.\ensuremathα=\ensuremathαM​<1.
We study the entanglement between a qubit and its environment from the spin-boson model with Ohmic dissipation. Through a mapping to the anisotropic Kondo model, we derive the entropy of entanglement of the spin $E(\ensuremath{\alpha},\ensuremath{\Delta},h)$, where $\ensuremath{\alpha}$ is the dissipation strength, $\ensuremath{\Delta}$ is the tunneling amplitude between qubit states, and hhh is the level asymmetry. For $1\ensuremath{-}\ensuremath{\alpha}\ensuremath{\gg}\ensuremath{\Delta}/{\ensuremath{\omega}}_{c}$ and $(\ensuremath{\Delta},h)\ensuremath{\ll}{\ensuremath{\omega}}_{c}$, we show that the Kondo energy scale TK{T}_{K}TK​ controls the entanglement between the qubit and the bosonic environment (${\ensuremath{\omega}}_{c}$ is a high-energy cutoff). For $h\ensuremath{\ll}{T}_{K}$, the disentanglement proceeds as (h/TK)2(h/{T}_{K}{)}^{2}(h/TK​)2; for $h\ensuremath{\gg}{T}_{K}$, EEE vanishes as $({T}_{K}/h{)}^{2\ensuremath{-}2\ensuremath{\alpha}}$, up to a logarithmic correction. For a given hhh, the maximum entanglement occurs at a value of $\ensuremath{\alpha}$ which lies in the crossover regime $h\ensuremath{\sim}{T}_{K}$. We emphasize the possibility of measuring this entanglement using charge qubits subject to electromagnetic noise.
The concept of entanglement entropy appears in multiple contexts, from black hole physics to quantum information theory, where it measures the entanglement of quantum states. We investigate the entanglement entropy in a simple model, the spin-boson model, which describes a qubit (two-level system) interacting with a collection of harmonic oscillators that models the environment responsible for decoherence and dissipation. The entanglement entropy allows to make a precise unification between entanglement of the spin with its environment, decoherence, and quantum phase transitions. We derive exact analytical results which are confirmed by Numerical Renormalization Group arguments both for an ohmic and a subohmic bosonic bath. Those demonstrate that the entanglement entropy obeys universal scalings. We make comparisons with entanglement properties in the quantum Ising model and in the Dicke model. We also emphasize the possibility of measuring this entanglement entropy using charge qubits subject to electromagnetic noise; such measurements would provide an empirical proof of the existence of entanglement entropy.
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We develop a systematic variational coherent-state expansion for the many-body ground state of the spin-boson model, in which a quantum two-level system is coupled to a continuum of harmonic oscillators. Energetic constraints at the heart of this technique are rationalized in terms of polarons (displacements of the bath states in agreement with classical expectations) and antipolarons (counterdisplacements due to quantum tunneling effects). We present a comprehensive study of the ground-state two-level system population and coherence as a function of tunneling amplitude, dissipation strength, and bias (akin to asymmetry of the double-well potential defining the two-state system). The entanglement among the different environmental modes is investigated by looking at spectroscopic signatures of the bipartite entanglement entropy between a given environmental mode and all the other modes. We observe a drastic change in behavior of this entropy for increasing dissipation, indicative of the entangled nature of the environmental states. In addition, the entropy spreads over a large energy range at strong dissipation, a testimony to the wide entanglement window characterizing the underlying Kondo state. Finally, comparisons to accurate numerical renormalization-group calculations and to the exact Bethe ansatz solution of the model demonstrate the rapid convergence of our variationally optimized multipolaron expansion, suggesting that it should also be a useful tool for dissipative models of greater complexity, as relevant for numerous systems of interest in quantum physics and chemistry.