Chaos and Entanglement

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Chaos and Entanglement

• friends or enemies?

Rafal Demkowicz-Dobrzanski (CFT PAN)
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Classical Chaos
Deterministic laws
Exponential sensitivity to initial perturbation
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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 ???

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Entanglement
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Measure of entanglement
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Entangling properties of quantum evolutions

• How an initially product state is being entangled?

• Entangling power of U

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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?

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Kicked Top

• Hamiltonian

• One period evolution

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Kicked Top

• Heisenberg picture

• Direction operators

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Classical limit for the kicked top
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Classical limit for the kicked top

discreet dynamics on a sphere
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Classical limit for the kicked top

discreet dynamics on a sphere
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Classical limit for the kicked top

discreet dynamics on a sphere
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Most classical quantum states

• Spin-coherent states

- overcomplete set

• Phase space picture of quantum states

Hussimi function of a spin-coherent state (j20)
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Evolution of a spin-coherent state
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Coupled kicked tops

• One period evolution operator

Chaos enhances initial production rate!?
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Short time behaviour of entanglement

• Perturbative formula (Tanaka et al. 2002)

Interaction picture
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Chaos and time correlation function
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Chaos induces initial linear entanglement increase

• Chaotic regime

• Regular regime

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Initial entangling power for the coupled kicked
tops

• Initial entanglement growth rate, averaged over
  either random or coherent states

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Long time entangling properties
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Averaged asymptotic behaviour and eigenvectors
entanglement

• Asymptotic entanglement, averaged over either
  random or coherent states

• Averaged asymptotic entanglement and
  eigenvectors entanglement

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Conclusions
Chaos and entanglement are....
a) Chaos drives low-uncertainty states into
highly smeared states and 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.
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Chaos and Entanglement

• friends or enemies?

Rafal Demkowicz-Dobrzanski (CFT PAN)