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Scaling of entanglement close to a quantum phase transition

Abstract

Classical phase transitions occur when a physical system reaches a state below a critical temperature characterized by macroscopic order1. Quantum phase transitions occur at absolute zero; they are induced by the change of an external parameter or coupling constant2, and are driven by quantum fluctuations. Examples include transitions in quantum Hall systems3, localization in Si-MOSFETs (metal oxide silicon field-effect transistors; ref. 4) and the superconductor–insulator transition in two-dimensional systems5,6. Both classical and quantum critical points are governed by a diverging correlation length, although quantum systems possess additional correlations that do not have a classical counterpart. This phenomenon, known as entanglement, is the resource that enables quantum computation and communication8. The role of entanglement at a phase transition is not captured by statistical mechanics—a complete classification of the critical many-body state requires the introduction of concepts from quantum information theory9. Here we connect the theory of critical phenomena with quantum information by exploring the entangling resources of a system close to its quantum critical point. We demonstrate, for a class of one-dimensional magnetic systems, that entanglement shows scaling behaviour in the vicinity of the transition point.

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Figure 1: The change in the ground-state wavefunction in the critical region is analysed considering dC(1)/dλ as a function of the reduced coupling strength λ.
Figure 2: The finite size scaling is performed for the case of logarithmic divergences22.
Figure 3: As in the case of the nearest-neighbour concurrence, data collapse is also obtained for the next-nearest-neighbour concurrence C(2).
Figure 4: The universality hypothesis for the entanglement is checked by considering the model hamiltonian, defined in equation (1), for a different value of γ.

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Acknowledgements

We thank G.M. Palma, F. Plastina and J. Siewert for discussions. This work was supported by the European Community (IST-SQUBIT) and by INFM-PRA-SSQI.

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Correspondence to Rosario Fazio.

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Osterloh, A., Amico, L., Falci, G. et al. Scaling of entanglement close to a quantum phase transition. Nature 416, 608–610 (2002). https://doi.org/10.1038/416608a

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