Calculation of Pairwise Thermal Entanglement for Odd Qubit XX Chain

Using Jordan-Winger transformation and finite Fourier transformation, the Hamiltonian of XX chain can be transformed to the diagonal form. If the total qubits number N=odd, theinternal energy U, and correlation function G_zz in thermalequilibrium can be calculated. Then the pairwise concurrence and thecritical value β_cr can be calculated. For N≥5 XXchains, there are pairwise concurrences in thermal equilibrium forantiferromagnetic and ferromagnetic cases when β>β_cr.