Universal and Measurable Entanglement Entropy in the Spin-Boson Model

Abstract

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(α, ∆, h), where α is the dissipation strength, ∆ is the tunneling amplitude between qubit states, and h is the level asymmetry. For 1 − α ≫ ∆/ωc and (∆, h) ≪ ωc, we show that the Kondo energy scale TK controls the entanglement between the qubit and the bosonic environment (ωc is a high-energy cutoff). For h ≪ TK, the disentanglement proceeds as (h/TK ) 2 ; for h ≫ TK, E vanishes as (TK/h) 2−2α , up to a logarithmic correction. For a given h, the maximum entanglement occurs at a value of α which lies in the crossover regime h ∼ TK . We emphasize the possibility of measuring this entanglement using charge qubits subject to electromagnetic noise. The concept of quantum entropy appears in multiple contexts, from black hole physics 1 to quantum information theory, where it measures the entanglement of quantum states.2 Prompted by the link between entanglement and quantum criticality, 3 a number of researchers have begun to study the entanglement entropy of condensed matter systems. In this Letter, we employ the spin-boson model 4,5 to describe the entanglement between a qubit (two-level system) and an infinite collection of bosons. With an Ohmic bosonic bath, the spin-boson model undergoes a quantum phase transition of Kosterlitz-Thouless type when α − 1 = ∆/ω c , where α is the strength of the coupling to the environment, ∆ is the tunneling amplitude between the qubit states, and ω c ≫ ∆ is an ultraviolet cutoff.6,7 When the two levels of the qubit are degenerate, the entanglement between the qubit and the bosons is discontinuous at this transition.8,9 Here we report the first rigorous analytical results for the entanglement (quantum entropy) in the strongly entangled regime 1 − α ≫ ∆/ω c .We exploit a mapping between the spin-boson model and the anisotropic Kondo model; our results follow from the Bethe ansatz solution of the equivalent interacting resonant level model. 10,11 We show that the entropy of entanglement (E) of the qubit with the environment is controlled by the Kondo energy scale T K

Entanglement in many-body systems

“…III, in low dimensional spin system there exists a particular choice of the coupling constants for which the ground state is factorized (Kurmann et al, 1982). This is a special point also from the perspective of investigating the entanglement in the ground state.…”

“…The previous expression allows At the transition point the entropy is discontinuous with a jump given by (Kopp et al, 2006) ∆S = ln 2 + δ/4δ c ln(δ/δ c ) (δ c is a high energy cutoff). A systematic analisys of the entropy in the spin-boson model for different coupling regimes was pursued recently in (K.Le Hur et al, 2007;Kopp and Le Hur, 2007). We finally mention the interesting connection between entanglement and energy fluctuations introduced by (Jordan and Buttiker, 2004) and exploited in details both for a spin and for an harmonic oscillators coupled to a bath.…”

“…(23) is in general entangled. It exists however, for any value of the coupling constants γ and ∆, J > 0 a point in d = 1, 2 (for bipartite lattices) where the ground state is factorized (Kurmann et al, 1982;Roscilde et al, 2005b). It occurs at the so called factorizing field h f given by…”

Bipartite fluctuations as a probe of many-body entanglement

“…For a pure state S α (ρ A ) = S α (ρ B ) and it makes no difference which subsystem is analyzed; indeed, this symmetry is an important feature of entanglement entropy. The scaling of entanglement entropy with subsystem size has yielded a wealth of interesting results for both gapped and critical systems [3,[6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Alternatively, detailed information about the structure of entanglement can be obtained by studying the full spectrum of eigenvalues of the reduced density matrix, specifically through the eigenvalues of the "entanglement Hamiltonian"Ĥ A defined byρ A = e −ĤA /Tr(e −ĤA ) [20][21][22][23][24][25][26][27][28][29][30][31][32].…”

Entanglement and level crossings in frustrated ferromagnetic rings

“…A variety of different partitionings, and correspondingly different entanglements, have been used for entanglement studies in condensed matter models. Many studies have focused on the entanglement between two sites as measured by the concurrence or negativity (e.g., [2,3,4,5,6,7,8]), or the entanglement between one spin/site with the rest (e.g., [9,10,11]). Such measures generically are not expected to provide information different from traditional condensed matter quantities like 2-point correlation functions, since the entanglement measure can be related easily to correlation functions (e.g., [5,6,7,8,9]).…”