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### Abstract

The influence of the inhomogeneity on the macroscopic thermal pairwise entanglement for the system of coupled spins 1/2 (qubits) has been studied. The most important effect of the inhomogeneity on the thermal entanglement is in the new role of the external potential (magnetic field), which can produce nonzero entanglement for qubits, situated not far from the inhomogeneity.

I thank MPI PKS Dresden for kind hospitality. The support from the Institute of Chemistry of V. N. Karazin Kharkov National University is acknowledged.

### Key Topics

- Qubits
- 29.0
- Quantum entanglement
- 24.0
- Magnetic fields
- 10.0
- Wave functions
- 6.0
- Entropy
- 5.0

## Figures

Localization length ξ_{1} as the function of the relative strength of the impurity-host coupling *I* = *J′*/*J* and the relative strength of the magnetic field, which affects the spin of the impurity *x* = (*γ′ − γ*)*H*/*J*. One can see that for some values of the impurity-host strength and the field the correlation length can be of order of 500 sites of the chain.

Localization length ξ_{1} as the function of the relative strength of the impurity-host coupling *I* = *J′*/*J* and the relative strength of the magnetic field, which affects the spin of the impurity *x* = (*γ′ − γ*)*H*/*J*. One can see that for some values of the impurity-host strength and the field the correlation length can be of order of 500 sites of the chain.

The ground-state tangles τ_{av} (dotted line), τ_{0} (solid line), and τ_{1} (dashed line) as the function of the applied magnetic field *H* (we use units in which Planck’s and Boltzmann’s constants, gyromagnetic ratio γ and the exchange integral *J* in the chain are equal to 1 for *I* = 2.2 and α = 1.2). τ_{0} and τ_{1} manifest jumps at *H* _{0} = *I* ^{2} *J*/2γ(α(*I* ^{2} − α))^{1/2} which take place in the region of the parameters for the impurity (Ref. 20) *I* ^{2} > 2γ′/(1 + α), as the contribution of the local level. The average tangle shows no low-temperature jump at *H* _{0} comparing to the ones for the impurity site and near the impurity; instead τ_{av} manifests the kink (due to the second order quantum phase transition) at the critical value *H _{s} * =

*J*/γ, characteristic for the homogeneous system.

The ground-state tangles τ_{av} (dotted line), τ_{0} (solid line), and τ_{1} (dashed line) as the function of the applied magnetic field *H* (we use units in which Planck’s and Boltzmann’s constants, gyromagnetic ratio γ and the exchange integral *J* in the chain are equal to 1 for *I* = 2.2 and α = 1.2). τ_{0} and τ_{1} manifest jumps at *H* _{0} = *I* ^{2} *J*/2γ(α(*I* ^{2} − α))^{1/2} which take place in the region of the parameters for the impurity (Ref. 20) *I* ^{2} > 2γ′/(1 + α), as the contribution of the local level. The average tangle shows no low-temperature jump at *H* _{0} comparing to the ones for the impurity site and near the impurity; instead τ_{av} manifests the kink (due to the second order quantum phase transition) at the critical value *H _{s} * =

*J*/γ, characteristic for the homogeneous system.

The ground-state tangles τ_{4} (dashed line), and τ_{5} (solid line) as the function of the applied magnetic field *H* (parameters are the same as in Fig.2). We can see the jump at *H* _{0}, however the value of tangles are much smaller than in the vicinity of the inhomogeneity.

The ground-state tangles τ_{4} (dashed line), and τ_{5} (solid line) as the function of the applied magnetic field *H* (parameters are the same as in Fig.2). We can see the jump at *H* _{0}, however the value of tangles are much smaller than in the vicinity of the inhomogeneity.

The average concurrence *C* _{av} as the function of temperature *T* and the applied magnetic field *H* for *J* = 1.

The average concurrence *C* _{av} as the function of temperature *T* and the applied magnetic field *H* for *J* = 1.

The concurrence *C* _{01} at the impurity site as the function of temperature *T* and the applied magnetic field *H*. Parameters are the same as in Fig.2. The concurrence becomes smaller with the growth of the field.

The concurrence *C* _{01} at the impurity site as the function of temperature *T* and the applied magnetic field *H*. Parameters are the same as in Fig.2. The concurrence becomes smaller with the growth of the field.

The concurrence *C* _{45} in a short distance from the impurity as the function of temperature *T* and the applied magnetic field *H*. Parameters are the same as in Fig. 2. Notice much smaller scale for *C* _{45} comparing to *C* _{01}. The concurrence is zero at *H* = 0 and becomes nonzero for large enough *H*.

The concurrence *C* _{45} in a short distance from the impurity as the function of temperature *T* and the applied magnetic field *H*. Parameters are the same as in Fig. 2. Notice much smaller scale for *C* _{45} comparing to *C* _{01}. The concurrence is zero at *H* = 0 and becomes nonzero for large enough *H*.

The concurrence *C* _{01} at the impurity qubit as the function of temperature *T* and the parameter of the inhomogeneity *I* at zero field *H* = 0.

The concurrence *C* _{01} at the impurity qubit as the function of temperature *T* and the parameter of the inhomogeneity *I* at zero field *H* = 0.

The concurrence *C* _{12} for the nearest to the impurity qubits as the function of temperature *T* and the parameter of the inhomogeneity *I* at zero field *H* = 0.

The concurrence *C* _{12} for the nearest to the impurity qubits as the function of temperature *T* and the parameter of the inhomogeneity *I* at zero field *H* = 0.

The concurrence *C* _{01} at the impurity site as the function of temperature *T* at the applied magnetic field *H* = 0.5*J*/γ. Parameters are the same as in Fig. 2.

The concurrence *C* _{01} at the impurity site as the function of temperature *T* at the applied magnetic field *H* = 0.5*J*/γ. Parameters are the same as in Fig. 2.

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