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Giuseppe Florio Dipartimento di Fisica, Università di Bari, Italy

Multipartite Entanglement in a Quantum Phase Transition via Probability Density Function. Giuseppe Florio Dipartimento di Fisica, Università di Bari, Italy. P. Facchi Dipartimento di Matematica, Università di Bari, Italy S. Pascazio, G. Costantini

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Giuseppe Florio Dipartimento di Fisica, Università di Bari, Italy

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  1. Multipartite Entanglement in a Quantum Phase Transition via Probability Density Function Giuseppe Florio Dipartimento di Fisica, Università di Bari, Italy P. Facchi Dipartimento di Matematica, Università di Bari, Italy S. Pascazio, G. Costantini Dipartimento di Fisica, Università di Bari, Italy Palermo CEWQO 2007

  2. Objective Explore the link between ENTANGLEMENT and Quantum Phase Transitions (QPT) Osterloh, Amici, Falci, Fazio. Nature 2002 Osborne, Nielsen, PRA 2002 Calabrese, Cardy, J. Stat. Mech.: Theory Expt. 2004 Gu, Deng, Li, Ling, PRL 2004 Vidal, Latorre, Rico, Kitaev, PRL 2005 AND MANY MORE... Palermo CEWQO 2007

  3. An initial remark • Parisi: complex systems • Man’ko, Marmo, Sudarshan, Zaccaria: multipartite entanglement (J. Phys. A 02-03) • One (or a few) real number(s) is/are not enough • number of measures (i.e. real numbers) needed to quantify multipartite entanglement grows exponentially with n=number of qubits statistical methods! Palermo CEWQO 2007

  4. : Eigenvalues of : Dimension of the Hilbert space used to describe the system A measure of Entanglement Purity Palermo CEWQO 2007

  5. For Separable States… A measure of Entanglement Palermo CEWQO 2007

  6. B A Number of qubits in A Participation Number Palermo CEWQO 2007

  7. Objective: evaluate entanglement • Clearly, the quantity will depend on the bipartition, according to the distribution of entanglement among all possible bipartitions Palermo CEWQO 2007

  8. B A A B Objective: evaluate entanglement Palermo CEWQO 2007

  9. Objective: evaluate Entanglement • The distribution of characterizes multipartite entanglement. • The average will be a measure of the amount of entanglement in the system, while the variance will measure how well such entanglement is distributed: a smaller variance will correspond to a larger insensitivity to the choice of the partition. • P. Facchi, G.F., S. Pascazio, Phys. Rev. A 74, 042331 (2006) Palermo CEWQO 2007

  10. A 1) A 2) A 3) An example: GHZ [Greenberger, Horne, Zeilinger (1990)] For all bipartitions!! Well distributed entanglement Palermo CEWQO 2007

  11. (Classical) Phase Transitions • Discontinuity in one or more physical properties due to a change in a thermodynamic variable such as the temperature • Typical example: Ferromagnetic system • Below a critical temperature Tc, it exhibits spontaneous magnetization. • At T=0 the system is frozen in the ground state without fluctuations. Palermo CEWQO 2007

  12. Energy gap Correlation Length (Quantum) Phase Transitions • The transition describes a discontinuity in the ground state of a many-body system due to its quantum fluctuations (at 0 temperature). • Level crossing between ground state and excited states. • Examples of scaling laws: Palermo CEWQO 2007

  13. The system [Pfeuty (1976); Lieb et al. (1961); Katsura (1962)] • Quantum Ising model in a transverse field • It exhibits a QPT for Energy gap Correlation Length Palermo CEWQO 2007

  14. An example: 10 spins [Florio et al., J.Phys. A (in print), quant-ph/0612098] Palermo CEWQO 2007

  15. 11 7 Results (7-11 spins) n GHZ (approximately) Separable states Palermo CEWQO 2007

  16. 11 n 7 Results (7-11 spins) Palermo CEWQO 2007

  17. Results (7-11 spins) This shows that our entanglement characterization “sees” the Quantum Phase Transition! Analogous results for the width of the distribution Palermo CEWQO 2007

  18. Results (7-11 spins) What about the behavior of average and width? Palermo CEWQO 2007

  19. Results (7-11 spins) Palermo CEWQO 2007

  20. Results (7-11 spins) • A consequence: • At the QPT the entanglement of the ground state is insensitive to the bipartition. Therefore it could be a good tool for generating multipartite entanglement. • Is there a relation between QPT and chaotic systems (high value of entanglement, well distributed)? Palermo CEWQO 2007

  21. Conclusions • Entanglement can be characterized using its distribution over all possible bipartitions (average AND width). • This characterization “sees” the QPT of the Ising Model with transverse field. • From numerical evidences we obtain that the amount of entanglement AND the width diverge… Palermo CEWQO 2007

  22. Conclusions and perspectives • Quantum Phase Transition: analytical evaluation of entanglement (conformal field theory, renormalization group). • Is a QPT a good tool for distributing entanglement? More evidences needed… other models (XX, XY… in progress). • What is the effect of interactions beyond nearest neighbour? (in progress) Palermo CEWQO 2007

  23. Further details Palermo CEWQO 2007

  24. Typical states Independent uniformly distributed random variables Random point uniformly distributed on the hypersphere with distribution: Palermo CEWQO 2007

  25. For it is possible to obtain the distribution Palermo CEWQO 2007

  26. Quantum Ising model in 1D[Pfeuty (1976); Lieb et al. (1961); Katsura (1962)] • Quantum Ising model in a transverse field • Coupling limits Palermo CEWQO 2007

  27. Possible scenarios The last conclusion is particularly significant: the amount of entanglement goes to infinity but so does the width of the entanglement distribution. Two scenarios are possible: • σrel vanishes for larger n. This means that at the QPT the entanglement of the GS is macroscopically insensitive to the choice of the bipartitions. • σrel does notvanishes. In this case the strong divergence of σ(µmax) would imply that the distribution of entanglement is not optimal. This means that the amount of entanglement of non-contiguous spins partitions macroscopically differs from that of contiguous ones Palermo CEWQO 2007

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