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Chaos and Entanglement

Chaos and Entanglement. friends or enemies?. Rafał Demkowicz-Dobrzański (CFT PAN). p.  (t). x.  (0). l – positive Lyapunov exponent. Classical Chaos. Deterministic laws. Exponential sensitivity to initial perturbation . Quantum world:. Classical world:. Regular systems.

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Chaos and Entanglement

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  1. Chaos and Entanglement friends or enemies? Rafał Demkowicz-Dobrzański (CFT PAN)

  2. p (t) x (0) l – positive Lyapunov exponent Classical Chaos Deterministic laws Exponential sensitivity to initial perturbation

  3. Quantum world: Classical world: Regular systems Chaotic systems Quantum Chaos • Do quantum systems which in the classical limit are chaotic differ form the ones which are classically regular? • What are the quantum signatures of chaos? • Distribution of energy levels • Stability of quantum motion under perturbed Hamiltonian • Entanglement ???

  4. Bipartite quantum system: H1 H2 • Arbitrary pure state: • Separable state (product state): • Entangled state: • (more than one term in the Schmidt decomposition) Entanglement

  5. Reduced density matrix of the first subsystem: : • Mixedness of the reduced density matrix (linear entropy): : product state maximally entangled state Measure of entanglement

  6. U – unitary operator acting in U - entanglement produced Su(d) – Haar measure • Does chaos influence entangling properties? Entangling properties of quantum evolutions • How an initially product state is being entangled? • Entangling power of U

  7. k k weak coupling chaoticity parameter in the Hamiltonian of the system Two weakly coupled chaotic systems • How entanglement production changes when varying k, • under fixed coupling? - initial entanglement growth rate - long time behaviour of entanglement • Does it depend on the type of initial product state chosen?

  8. j – total spin - basis Kicked Top • Hamiltonian: • One period evolution:

  9. For large j they commute Kicked Top • Heisenberg picture: • Direction operators:

  10. discreet dynamics on a sphere Classical limit for the kicked top

  11. Classical limit for the kicked top discreet dynamics on a sphere

  12. Classical limit for the kicked top discreet dynamics on a sphere

  13. Classical limit for the kicked top discreet dynamics on a sphere

  14. minimal uncertainty with respect • to angular momentum components Hussimi function: Most classical quantum states • Spin-coherent states: - overcomplete set • Phase space picture of quantum states Hussimi function of a spin-coherent state (j=20):

  15. j=20 k=1 k=6 Evolution of a spin-coherent state

  16. Hamiltonian: • Entanglement evolution j=20 e=0.01 Coupled kicked tops • One period evolution operator: Chaos enhances initial production rate!?

  17. U0 • Entanglement of to the second order in e: time correlation function Short time behaviour of entanglement • Perturbative formula (Tanaka et al. 2002) Interaction picture:

  18. very low for coherent state: for random state: Chaos increases C(t,t) ! but kills corelations for Chaotic motion: Regular motion: Chaos and time correlation function

  19. linear increase quadratic increase Chaos induces initial linear entanglement increase • Chaotic regime: • Regular regime:

  20. Chaos always diminishes initial entangling power! Initial entangling power for the coupled kicked tops • Initial entanglement growth rate, averaged over either random or coherent states:

  21. j=20 e=0.01 Chaos helps achieving high asymptotic entanglement! Long time entangling properties

  22. Averaged asymptotic behaviour and eigenvectors entanglement • Asymptotic entanglement, averaged over either random or coherent states: • Averaged asymptotic entanglement and eigenvectors entanglement

  23. Friends: • Enemies: Conclusions Chaos and entanglement are.... a) Chaos drives low-uncertainty states into highly smeared statesand thus increases initial entanglement growth rate b) Chaos assures high asymptotic entanglement c) In different approaches, where chaos is not ,,localized’’ in the subsystems and the coupling is strong, chaos helps entanglement growth. a) For certain choices of parameters (j, e), regular dynamics, thanks to non-vanishing time correlations, outperforms chaotic dynamics in terms of initial entanglement production. b) For weakly coupled systems initial entangling power is always worst in chaotic case c) In the case of coupled kicked tops, very regular dynamics has equally high (even a little bit higher) asymptotic entanglement than chaotic cases.

  24. Chaos and Entanglement friends or enemies? Rafał Demkowicz-Dobrzański (CFT PAN)

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