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  • 1. Outline Entanglement Dynamics of a Two-qubit System in a Spin Star Configuration Y. Hamdouni1 M. Fannes2 F. Petruccione1 1 School of Physics University of KwaZulu-Natal2 Institute of Theoretical Physics, University of Leuven, Celestijnenlaan 200D B-3001 Heverlee, Belgium SAIP, 2006 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 2. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionOutline 1 Introduction 2 The Model The two-qubit system The environment The Hamiltonian 3 Exact reduced dynamics Initial conditions Time evolution 3 The limit N → ∞ Correlation functions Time evolution 4 Entanglement evolution concurrence Main results 5 Conclusion Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 3. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionIntroduction Classical bit The canonical building block of classical Information Processing (IP) is the bit. A bit can be put into two possible states, ”0” or ”1”. Examples 1 Two different voltages across a transistor on a chip. 2 Two different orientations of the magnetic domain on a disc 3 Two different classical light pulses traveling down an optical fibre Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 4. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionIntroduction Classical bit The canonical building block of classical Information Processing (IP) is the bit. A bit can be put into two possible states, ”0” or ”1”. Examples 1 Two different voltages across a transistor on a chip. 2 Two different orientations of the magnetic domain on a disc 3 Two different classical light pulses traveling down an optical fibre Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 5. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionIntroduction Classical bit The canonical building block of classical Information Processing (IP) is the bit. A bit can be put into two possible states, ”0” or ”1”. Examples 1 Two different voltages across a transistor on a chip. 2 Two different orientations of the magnetic domain on a disc 3 Two different classical light pulses traveling down an optical fibre Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 6. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionIntroduction Classical bit The canonical building block of classical Information Processing (IP) is the bit. A bit can be put into two possible states, ”0” or ”1”. Examples 1 Two different voltages across a transistor on a chip. 2 Two different orientations of the magnetic domain on a disc 3 Two different classical light pulses traveling down an optical fibre Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 7. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionIntroduction Classical bit The canonical building block of classical Information Processing (IP) is the bit. A bit can be put into two possible states, ”0” or ”1”. Examples 1 Two different voltages across a transistor on a chip. 2 Two different orientations of the magnetic domain on a disc 3 Two different classical light pulses traveling down an optical fibre Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 8. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionNowdays, components of electronic devices are shrinking(Moore’s first law!). Inevitably, we are led to deal with newmaterials, nano-devices and molecular electronics ⇒ scaleswhere quantum phenomena predominate: particle-wave duality the superposition principle probabilistic nature of quantum measurement outcomes entanglement...etc.The unification of classical information theory and quantummechanics gave rise to a new field of physics called quantuminformation theory. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 9. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionNowdays, components of electronic devices are shrinking(Moore’s first law!). Inevitably, we are led to deal with newmaterials, nano-devices and molecular electronics ⇒ scaleswhere quantum phenomena predominate: particle-wave duality the superposition principle probabilistic nature of quantum measurement outcomes entanglement...etc.The unification of classical information theory and quantummechanics gave rise to a new field of physics called quantuminformation theory. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 10. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionNowdays, components of electronic devices are shrinking(Moore’s first law!). Inevitably, we are led to deal with newmaterials, nano-devices and molecular electronics ⇒ scaleswhere quantum phenomena predominate: particle-wave duality the superposition principle probabilistic nature of quantum measurement outcomes entanglement...etc.The unification of classical information theory and quantummechanics gave rise to a new field of physics called quantuminformation theory. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 11. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionNowdays, components of electronic devices are shrinking(Moore’s first law!). Inevitably, we are led to deal with newmaterials, nano-devices and molecular electronics ⇒ scaleswhere quantum phenomena predominate: particle-wave duality the superposition principle probabilistic nature of quantum measurement outcomes entanglement...etc.The unification of classical information theory and quantummechanics gave rise to a new field of physics called quantuminformation theory. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 12. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionNowdays, components of electronic devices are shrinking(Moore’s first law!). Inevitably, we are led to deal with newmaterials, nano-devices and molecular electronics ⇒ scaleswhere quantum phenomena predominate: particle-wave duality the superposition principle probabilistic nature of quantum measurement outcomes entanglement...etc.The unification of classical information theory and quantummechanics gave rise to a new field of physics called quantuminformation theory. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 13. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionNowdays, components of electronic devices are shrinking(Moore’s first law!). Inevitably, we are led to deal with newmaterials, nano-devices and molecular electronics ⇒ scaleswhere quantum phenomena predominate: particle-wave duality the superposition principle probabilistic nature of quantum measurement outcomes entanglement...etc.The unification of classical information theory and quantummechanics gave rise to a new field of physics called quantuminformation theory. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 14. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionNowdays, components of electronic devices are shrinking(Moore’s first law!). Inevitably, we are led to deal with newmaterials, nano-devices and molecular electronics ⇒ scaleswhere quantum phenomena predominate: particle-wave duality the superposition principle probabilistic nature of quantum measurement outcomes entanglement...etc.The unification of classical information theory and quantummechanics gave rise to a new field of physics called quantuminformation theory. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 15. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionNowdays, components of electronic devices are shrinking(Moore’s first law!). Inevitably, we are led to deal with newmaterials, nano-devices and molecular electronics ⇒ scaleswhere quantum phenomena predominate: particle-wave duality the superposition principle probabilistic nature of quantum measurement outcomes entanglement...etc.The unification of classical information theory and quantummechanics gave rise to a new field of physics called quantuminformation theory. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 16. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionNowdays, components of electronic devices are shrinking(Moore’s first law!). Inevitably, we are led to deal with newmaterials, nano-devices and molecular electronics ⇒ scaleswhere quantum phenomena predominate: particle-wave duality the superposition principle probabilistic nature of quantum measurement outcomes entanglement...etc.The unification of classical information theory and quantummechanics gave rise to a new field of physics called quantuminformation theory. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 17. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionNowdays, components of electronic devices are shrinking(Moore’s first law!). Inevitably, we are led to deal with newmaterials, nano-devices and molecular electronics ⇒ scaleswhere quantum phenomena predominate: particle-wave duality the superposition principle probabilistic nature of quantum measurement outcomes entanglement...etc.The unification of classical information theory and quantummechanics gave rise to a new field of physics called quantuminformation theory. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 18. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionThe quantum bit The Building block The fundamental unit of information in the quntum theory of information processing is the quantum bit or the qubit, the quantum counterpart of the classical bit. It is a two-level system living in a two-dimensional Hilbert space spaned by the basis state vectors {|+ , |− }. The qubit enjoys more freedom: its state is a linear combination of the basis state vectors. Examples 1 spin- 1 particle 2 2 two-level atom 3 photon polarization states Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 19. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionThe quantum bit The Building block The fundamental unit of information in the quntum theory of information processing is the quantum bit or the qubit, the quantum counterpart of the classical bit. It is a two-level system living in a two-dimensional Hilbert space spaned by the basis state vectors {|+ , |− }. The qubit enjoys more freedom: its state is a linear combination of the basis state vectors. Examples 1 spin- 1 particle 2 2 two-level atom 3 photon polarization states Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 20. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionThe quantum bit The Building block The fundamental unit of information in the quntum theory of information processing is the quantum bit or the qubit, the quantum counterpart of the classical bit. It is a two-level system living in a two-dimensional Hilbert space spaned by the basis state vectors {|+ , |− }. The qubit enjoys more freedom: its state is a linear combination of the basis state vectors. Examples 1 spin- 1 particle 2 2 two-level atom 3 photon polarization states Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 21. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionThe quantum bit The Building block The fundamental unit of information in the quntum theory of information processing is the quantum bit or the qubit, the quantum counterpart of the classical bit. It is a two-level system living in a two-dimensional Hilbert space spaned by the basis state vectors {|+ , |− }. The qubit enjoys more freedom: its state is a linear combination of the basis state vectors. Examples 1 spin- 1 particle 2 2 two-level atom 3 photon polarization states Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 22. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionThe quantum bit The Building block The fundamental unit of information in the quntum theory of information processing is the quantum bit or the qubit, the quantum counterpart of the classical bit. It is a two-level system living in a two-dimensional Hilbert space spaned by the basis state vectors {|+ , |− }. The qubit enjoys more freedom: its state is a linear combination of the basis state vectors. Examples 1 spin- 1 particle 2 2 two-level atom 3 photon polarization states Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 23. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionThe quantum bit The Building block The fundamental unit of information in the quntum theory of information processing is the quantum bit or the qubit, the quantum counterpart of the classical bit. It is a two-level system living in a two-dimensional Hilbert space spaned by the basis state vectors {|+ , |− }. The qubit enjoys more freedom: its state is a linear combination of the basis state vectors. Examples 1 spin- 1 particle 2 2 two-level atom 3 photon polarization states Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 24. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution Conclusion prepareoperations on qubits evolve measure!!Leakage away of quantum information by decoherence (due touncontroled interactions with the surounding environment). ⇓ experimetal and theoretical challenge Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 25. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution Conclusion prepareoperations on qubits evolve measure!!Leakage away of quantum information by decoherence (due touncontroled interactions with the surounding environment). ⇓ experimetal and theoretical challenge Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 26. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution Conclusion prepare −→operations on qubits evolve measure!!Leakage away of quantum information by decoherence (due touncontroled interactions with the surounding environment). ⇓ experimetal and theoretical challenge Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 27. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution Conclusion prepareoperations on qubits evolve measure!!Leakage away of quantum information by decoherence (due touncontroled interactions with the surounding environment). ⇓ experimetal and theoretical challenge Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 28. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution Conclusion prepareoperations on qubits evolve measure!!Leakage away of quantum information by decoherence (due touncontroled interactions with the surounding environment). ⇓ experimetal and theoretical challenge Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 29. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe model We consider a pair of two spin- 1 particles (our qubits) in a spin 2 1 star configuration: a structure composed of N spin- 2 particles located on a the surface of a sphere whose center is occupied by the qubits. We assume that the central spins do not interact with each other and we neglect mutual interactions between the outer spins The main contribution to the total Hamiltonian comes from the coupling of the qubits to the spin star configuration. The qubits are treated as an open system coupled to an environment of many degrees of freedom. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 30. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe model We consider a pair of two spin- 1 particles (our qubits) in a spin 2 1 star configuration: a structure composed of N spin- 2 particles located on a the surface of a sphere whose center is occupied by the qubits. We assume that the central spins do not interact with each other and we neglect mutual interactions between the outer spins The main contribution to the total Hamiltonian comes from the coupling of the qubits to the spin star configuration. The qubits are treated as an open system coupled to an environment of many degrees of freedom. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 31. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe model We consider a pair of two spin- 1 particles (our qubits) in a spin 2 1 star configuration: a structure composed of N spin- 2 particles located on a the surface of a sphere whose center is occupied by the qubits. We assume that the central spins do not interact with each other and we neglect mutual interactions between the outer spins The main contribution to the total Hamiltonian comes from the coupling of the qubits to the spin star configuration. The qubits are treated as an open system coupled to an environment of many degrees of freedom. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 32. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe model We consider a pair of two spin- 1 particles (our qubits) in a spin 2 1 star configuration: a structure composed of N spin- 2 particles located on a the surface of a sphere whose center is occupied by the qubits. We assume that the central spins do not interact with each other and we neglect mutual interactions between the outer spins The main contribution to the total Hamiltonian comes from the coupling of the qubits to the spin star configuration. The qubits are treated as an open system coupled to an environment of many degrees of freedom. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 33. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution Conclusion Figure: 1 Spin star configuration.Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 34. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe qubits The total spin operator of the two-qubit system is ˆ ˆ I I ˆ S = S1 ⊗ ˆ + ˆ ⊗ S2 (1) The corresponding spin state is given C2 ⊗ C2 = C ⊕ C3 . (2) eigenbasis vectors eigenbasis vectors {| + + , | + − , | − + , | − − } {|0, 0 , |1, 1 , |1, 0 , |1, −1 } where | 1 2 = | 1 ⊗ | 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 35. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe qubits The total spin operator of the two-qubit system is ˆ ˆ I I ˆ S = S1 ⊗ ˆ + ˆ ⊗ S2 (1) The corresponding spin state is given C2 ⊗ C2 = C ⊕ C3 . (2) eigenbasis vectors eigenbasis vectors {| + + , | + − , | − + , | − − } {|0, 0 , |1, 1 , |1, 0 , |1, −1 } where | 1 2 = | 1 ⊗ | 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 36. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe qubits The total spin operator of the two-qubit system is ˆ ˆ I I ˆ S = S1 ⊗ ˆ + ˆ ⊗ S2 (1) The corresponding spin state is given C2 ⊗ C2 = C ⊕ C3 . (2) eigenbasis vectors eigenbasis vectors {| + + , | + − , | − + , | − − } {|0, 0 , |1, 1 , |1, 0 , |1, −1 } where | 1 2 = | 1 ⊗ | 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 37. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe qubits The total spin operator of the two-qubit system is ˆ ˆ I I ˆ S = S1 ⊗ ˆ + ˆ ⊗ S2 (1) The corresponding spin state is given C2 ⊗ C2 = C ⊕ C3 . (2) eigenbasis vectors eigenbasis vectors {| + + , | + − , | − + , | − − } {|0, 0 , |1, 1 , |1, 0 , |1, −1 } where | 1 2 = | 1 ⊗ | 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 38. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe qubits The total spin operator of the two-qubit system is ˆ ˆ I I ˆ S = S1 ⊗ ˆ + ˆ ⊗ S2 (1) The corresponding spin state is given C2 ⊗ C2 = C ⊕ C3 . (2) eigenbasis vectors eigenbasis vectors {| + + , | + − , | − + , | − − } {|0, 0 , |1, 1 , |1, 0 , |1, −1 } where | 1 2 = | 1 ⊗ | 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 39. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe environment Spin space of the environment: N 2 2 ⊗N (C ) = ν(j, N ) Cdj . j =0 N dj = 2j + 1, 0 j 2 and the degeneracy N /2 N N ν(j, N ) = − ν(j, N )(2j + 1) = 2N N /2 − j N /2 − j − 1 j =0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 40. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe environment Spin space of the environment: N 2 2 ⊗N (C ) = ν(j, N ) Cdj . j =0 N dj = 2j + 1, 0 j 2 and the degeneracy N /2 N N ν(j, N ) = − ν(j, N )(2j + 1) = 2N N /2 − j N /2 − j − 1 j =0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 41. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe environment Spin space of the environment: N 2 2 ⊗N (C ) = ν(j, N ) Cdj . j =0 N dj = 2j + 1, 0 j 2 and the degeneracy N /2 N N ν(j, N ) = − ν(j, N )(2j + 1) = 2N N /2 − j N /2 − j − 1 j =0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 42. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe environment Spin space of the environment: N 2 2 ⊗N (C ) = ν(j, N ) Cdj . j =0 N dj = 2j + 1, 0 j 2 and the degeneracy N /2 N N ν(j, N ) = − ν(j, N )(2j + 1) = 2N N /2 − j N /2 − j − 1 j =0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 43. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionThe environment Spin space of the environment: N 2 2 ⊗N (C ) = ν(j, N ) Cdj . j =0 N dj = 2j + 1, 0 j 2 and the degeneracy N /2 N N ν(j, N ) = − ν(j, N )(2j + 1) = 2N N /2 − j N /2 − j − 1 j =0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 44. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionHeisenberg XY interaction The interaction between the central qubits and the environment is described by the Heisenberg XY interaction: N N H = α[(σ1 + σ2 ) ⊗ x x σix + (σ1 + σ2 ) ⊗ y y σiy ]. i i α is the coupling constant. Defining the lowering and raising operators N σ1,2 = σ1,2 ± i σ1,2 , ± x y J± = σi± . i =1 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 45. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution Conclusionwe can write H = α[(σ1 + σ2 ) ⊗ J− + (σ1 + σ2 ) ⊗ J+ ] + + − − = α[σ+ ⊗ J− + σ− ⊗ J+ ]. (3)where σ± = σ1 + σ2 . ± ±Due to symmetry, we can restrict ourselves to the subspaceC ⊗N ⊗ C 3 H (|00 ⊗ |ΦB ) = 0. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 46. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution Conclusionwe can write H = α[(σ1 + σ2 ) ⊗ J− + (σ1 + σ2 ) ⊗ J+ ] + + − − = α[σ+ ⊗ J− + σ− ⊗ J+ ]. (3)where σ± = σ1 + σ2 . ± ±Due to symmetry, we can restrict ourselves to the subspaceC ⊗N ⊗ C 3 H (|00 ⊗ |ΦB ) = 0. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 47. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionHamiltonian     0 0 0 0 1 0 σ+ = 1 0 0 σ− = 0 0 1 0 1 0 0 0 0   0 J+ 0 H = α J− 0 J+  . (4) 0 J− 0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 48. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionHamiltonian     0 0 0 0 1 0 σ+ = 1 0 0 σ− = 0 0 1 0 1 0 0 0 0   0 J+ 0 H = α J− 0 J+  . (4) 0 J− 0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 49. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution ConclusionHamiltonian     0 0 0 0 1 0 σ+ = 1 0 0 σ− = 0 0 1 0 1 0 0 0 0   0 J+ 0 H = α J− 0 J+  . (4) 0 J− 0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 50. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution Conclusion J+ K n−1 J− 0 J+ K n−1 J+   H2n = α2n  0 Kn 0 , J− K n−1 J 0 J− K n−1 J − + J+ K n   0 0 H 2n+1 = α2n+1 K n J− 0 K n J+  , 0 J− K n 0where K = J+ J− + J− J+ = 2(J 2 − Jz ). 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 51. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution Conclusion J+ K n−1 J− 0 J+ K n−1 J+   H2n = α2n  0 Kn 0 , J− K n−1 J 0 J− K n−1 J − + J+ K n   0 0 H 2n+1 = α2n+1 K n J− 0 K n J+  , 0 J− K n 0where K = J+ J− + J− J+ = 2(J 2 − Jz ). 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 52. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution Conclusion J+ K n−1 J− 0 J+ K n−1 J+   H2n = α2n  0 Kn 0 , J− K n−1 J 0 J− K n−1 J − + J+ K n   0 0 H 2n+1 = α2n+1 K n J− 0 K n J+  , 0 J− K n 0where K = J+ J− + J− J+ = 2(J 2 − Jz ). 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 53. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution Conclusion J+ K n−1 J− 0 J+ K n−1 J+   H2n = α2n  0 Kn 0 , J− K n−1 J 0 J− K n−1 J − + J+ K n   0 0 H 2n+1 = α2n+1 K n J− 0 K n J+  , 0 J− K n 0where K = J+ J− + J− J+ = 2(J 2 − Jz ). 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 54. Introduction The Model The two-qubit system Exact reduced dynamics The environment The limit N → ∞ The Hamiltonian Entanglement evolution Conclusion  √ √ √  1 + J+ cos(αtK K )−1 J− −iJ+ sin(αtK K ) √ J+ cos(αtK K )−1 J+  √ √ √ U (t ) =  −i sin(αt √ J−K) cos(αt K ) −i sin(αt K ) J+ √     K √ √ K√  cos(αt K )−1 J− K J− −iJ− sin(αt √ K) 1 + J− cos(αt K )−1 K J+ K (5) Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 55. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution ConclusionThe Liouvillian The state of the composite system is fully described by its total density matrix ρ. The evolution in time of ρ is given by ρ(t ) = exp(Lt )ρ(0). (6) ρ(0) is the initial density matrix. Lρ(t ) = −i [H, ρ(t )]. (7) The dynamics of the reduced system is obtained by tracing over the environment degrees of freedom: ρS (t ) = trB {ρ(t )}. (8) Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 56. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution ConclusionInitial condition Initial state factorizes: ρ(0) = ρS (0) ⊗ ρB (0), (9)  0 ρ0 ρ0  ρ11 12 13 ρS (0) = ρ0∗ 12 ρ0 22 ρ0  . 23 (10) ρ0∗ 13 ρ0∗ 23 ρ0 33 ρB (0) = 2−N IB = lim (e−HB /kB T )/Z . (11) T →∞ Z = trB e−HB /kB T is the partition function. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 57. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution ConclusionTime evolution of ρS ∞ tk ρS (t ) = tr {Lk ρS (0) ⊗ 2−N IB }. (12) k! B k =1 n n n L ρ=i (−1) H ρH n − . (13) n =0 trB {L2n+1 ρS (0) ⊗ 2−N IB } = 0. n−1 n−1 2n 2n trB {L2n ρS (0) ⊗ 2−N IB } = 2k S2n − 2k 2k +1 S2n +1 + F2n 2k k =1 k =0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 58. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution ConclusionTime evolution of ρS ∞ tk ρS (t ) = tr {Lk ρS (0) ⊗ 2−N IB }. (12) k! B k =1 n n n L ρ=i (−1) H ρH n − . (13) n =0 trB {L2n+1 ρS (0) ⊗ 2−N IB } = 0. n−1 n−1 2n 2n trB {L2n ρS (0) ⊗ 2−N IB } = 2k S2n − 2k 2k +1 S2n +1 + F2n 2k k =1 k =0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 59. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution ConclusionTime evolution of ρS ∞ tk ρS (t ) = tr {Lk ρS (0) ⊗ 2−N IB }. (12) k! B k =1 n n n L ρ=i (−1) H ρH n − . (13) n =0 trB {L2n+1 ρS (0) ⊗ 2−N IB } = 0. n−1 n−1 2n 2n trB {L2n ρS (0) ⊗ 2−N IB } = 2k S2n − 2k 2k +1 S2n +1 + F2n 2k k =1 k =0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 60. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution ConclusionTime evolution of ρS ∞ tk ρS (t ) = tr {Lk ρS (0) ⊗ 2−N IB }. (12) k! B k =1 n n n L ρ=i (−1) H ρH n − . (13) n =0 trB {L2n+1 ρS (0) ⊗ 2−N IB } = 0. n−1 n−1 2n 2n trB {L2n ρS (0) ⊗ 2−N IB } = 2k S2n − 2k 2k +1 S2n +1 + F2n 2k k =1 k =0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 61. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution ConclusionTime evolution of ρS ∞ tk ρS (t ) = tr {Lk ρS (0) ⊗ 2−N IB }. (12) k! B k =1 n n n L ρ=i (−1) H ρH n − . (13) n =0 trB {L2n+1 ρS (0) ⊗ 2−N IB } = 0. n−1 n−1 2n 2n trB {L2n ρS (0) ⊗ 2−N IB } = 2k S2n − 2k 2k +1 S2n +1 + F2n 2k k =1 k =0 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 62. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution Conclusionwhere  0 (ρ11 − ρ0 )Rn−2 + ρ33 Pn 33 ρ0 Qk 12 n ρ0 O k 13 n S2n 2k = (−α)n  ρ0∗ Qn−k 12 n ρ0 Fn 22 ρ0 Qk +1  23 n 0∗ O n ρ13 n−k ρ0∗ Qn−k +1 n ∆n 23∆n = −(ρ0 − ρ0 )Rn−2 + ρ0 Fn + (ρ0 − 2ρ0 )Pn 11 33 33 11 33 ρ0 Pn ρ0 Qk +1 n 0   22 23S2n +1 = (−α)n ρ0∗ Qn−k 2k 23 n 0 − ρ0 )P + ρ0 F (ρ11 33 n 33 n ρ0 Qk +1  12 n 0 ρ0∗ Qn−k 12 n ρ0 (Fn − Pn ) 22 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 63. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution Conclusion 2ρ0 Pn ρ0 (Fn + Pn ) ρ0 Fn   11 12 13 F2n = (−α)n ρ0∗ (Fn + Pn ) 12 2ρ0 Fn 22 ρ0 (2Fn − Pn ) 23 ρ0 Fn 13 ρ0∗ (2Fn − Pn ) 2ρ0 (Fn − Pn ) 23 33with the environment correlation functionsCorrelation functions Rn = 2−N trB {(J− J+ )2 K n−2 }, Qk = 2−N trB {J+ K k −1 J− K n−k }, n Ok = 2−N trB {J+ J+ K k −1 J− J− K n−k −1 }, n (14) −N n−1 Pn = 2 trB {J− J+ K }, Fn = 2−N trB {K n }. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 64. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution ConclusionExact solution Diagonal elements ρ11 (t ) = ρ0 (1 + 2g (t )) + (ρ0 − ρ0 )f (t ) + ρ0 e(t ) + ρ0 h(t ), 11 11 33 33 22 ρ22 (t ) = ρ0 + (ρ0 − ρ0 )h(t ) + (ρ0 − ρ0 ) (t ), 22 11 33 33 22 (15) ρ33 (t ) = ρ0 (1 − 2g (t )) − (ρ0 − ρ0 )f (t ) − ρ0 h(t ) 33 11 33 22 0 0 0 0 + (ρ11 − 2ρ33 )e(t ) + (ρ22 − ρ33 ) (t ). Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 65. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution ConclusionOff-diagonal elements ρ12 (t ) = ρ0 [ (t ) + e1 (t )] + ρ0 h(t ), 12 23 ρ13 (t ) = ρ0 [ (t ) + f (t )], 13 ρ23 (t ) = ρ0 [ (t ) + e2 (t )] + ρ0 h(t ), 23 12 (16) ρ14 (t ) = ρ0 (1 + g (t )), 14 ρ24 (t ) = ρ0 (t ), 24 ρ34 (t ) = ρ0 (1 + g (t )). 34 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 66. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution Conclusionwith √ −N cos(αt K ) − 1 2 f (t ) = 2 trB J− J+ , (17) K √ cos(αt K ) − 1 g (t ) = 2−N trB J− J+ , (18) K √ sin2 (αt K ) h(t ) = 2−N trB J− J+ , (19) K √ (cos(αt K ) − 1)2 e(t ) = 2−N trB J− J+ , (20) K √ (t ) = 2−N trB sin2 (αt K ) , (21) √ (t ) = 2−N trB cos(αt K ) , (22) Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 67. Introduction The Model Exact reduced dynamics Initial conditions The limit N → ∞ Time evolution Entanglement evolution Conclusion √ √ −N 2 cos(αt K) − 1 2 cos(αt K) − 1 f (t ) = 2 trB J− J+ , (23) K √ K cos(αt K ) − 1 √e1 (t ) = 2−N trB J+ J− cos(αt K ) , (24) K √ cos(αt K ) − 1 √e2 (t ) = 2−N trB J− J+ cos(αt K ) , (25) K √ √ sin(αt K ) sin(αt K ) h(t ) = 2−N trB J+ √ J− √ . (26) K K Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 68. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution ConclusionThe limit of infinite number of spins As N → ∞ trB (J+ J− )n ∼ trB (J+ J− )n− (J− J+ ) 2N N n n! ≈ . 2n Then n N n n! Ok ∼ Rn ≈ , (27) 4 n N n n! Qk ∼ Pn ≈ , (28) 2 Fn ≈ N n n!. (29) Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 69. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution ConclusionThe limit of infinite number of spins As N → ∞ trB (J+ J− )n ∼ trB (J+ J− )n− (J− J+ ) 2N N n n! ≈ . 2n Then n N n n! Ok ∼ Rn ≈ , (27) 4 n N n n! Qk ∼ Pn ≈ , (28) 2 Fn ≈ N n n!. (29) Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 70. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution ConclusionTime evolution Rescaling! α α→ √ . (30) N 1 f (t ) = f (t ) = ζ(2t ) − ζ(t ), (t ) = 1 + 2ζ(t ), 4 g (t ) = ζ(t ), 1 e (t ) = ζ(2t ) − 2ζ(t ), 1 2 h(t ) = h(t ) = − ζ(2t ), 1 2 e1,2 (t ) = ζ(2t ) − ζ(t ). (t ) = −ζ(2t ), 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 71. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution ConclusionTime evolution Rescaling! α α→ √ . (30) N 1 f (t ) = f (t ) = ζ(2t ) − ζ(t ), (t ) = 1 + 2ζ(t ), 4 g (t ) = ζ(t ), 1 e (t ) = ζ(2t ) − 2ζ(t ), 1 2 h(t ) = h(t ) = − ζ(2t ), 1 2 e1,2 (t ) = ζ(2t ) − ζ(t ). (t ) = −ζ(2t ), 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 72. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution ConclusionTime evolution Rescaling! α α→ √ . (30) N 1 f (t ) = f (t ) = ζ(2t ) − ζ(t ), (t ) = 1 + 2ζ(t ), 4 g (t ) = ζ(t ), 1 e (t ) = ζ(2t ) − 2ζ(t ), 1 2 h(t ) = h(t ) = − ζ(2t ), 1 2 e1,2 (t ) = ζ(2t ) − ζ(t ). (t ) = −ζ(2t ), 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 73. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution ConclusionTime evolution Rescaling! α α→ √ . (30) N 1 f (t ) = f (t ) = ζ(2t ) − ζ(t ), (t ) = 1 + 2ζ(t ), 4 g (t ) = ζ(t ), 1 e (t ) = ζ(2t ) − 2ζ(t ), 1 2 h(t ) = h(t ) = − ζ(2t ), 1 2 e1,2 (t ) = ζ(2t ) − ζ(t ). (t ) = −ζ(2t ), 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 74. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution ConclusionThe function ζ is given by αt αt ζ(t ) = − D+ (− ) 2 2 1 lim ζ(t ) = − t →∞ 2where we have introduced the Dawson functionDawson function x 2 2 D+ (x ) = e−x et dt 0 √ π −x 2 = e erfi(x ). 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 75. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution ConclusionThe function ζ is given by αt αt ζ(t ) = − D+ (− ) 2 2 1 lim ζ(t ) = − t →∞ 2where we have introduced the Dawson functionDawson function x 2 2 D+ (x ) = e−x et dt 0 √ π −x 2 = e erfi(x ). 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 76. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution ConclusionThe function ζ is given by αt αt ζ(t ) = − D+ (− ) 2 2 1 lim ζ(t ) = − t →∞ 2where we have introduced the Dawson functionDawson function x 2 2 D+ (x ) = e−x et dt 0 √ π −x 2 = e erfi(x ). 2 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 77. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution Conclusion 1 ρ∞ = S × 43 0 + ρ0 + 2 ρ0 ) ρ0 + ρ0 3 0 2ρ0  2 (ρ11 33 3 22 12 23 2 ρ13 14 ρ0∗ + ρ0∗ 12 23 (ρ0 + ρ0 + 2ρ0 ) 11 33 22 ρ0 + ρ0 23 12 0   3 0∗ 3 0 0 + 2 ρ0 ) 2ρ0  2 ρ13 ρ0∗ + ρ0∗ 23 12 2 (ρ11 + ρ33 3 22 34 2ρ0∗ 14 0 2ρ0∗ 34 4ρ0 44 Partial decoherence Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 78. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution Conclusion Figure: 2 The time evolution of the density matrix element ρ11 . Initialstate of the two-qubit system is the pure state | − − . The figure shows the plots obtained for N = 100, N = 400 and the limit N → ∞. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 79. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution Conclusion Figure: 3 The evolution in time of the density matrix element ρ22 as afunction of time. The initial condition of the two-qubit system is the pure state | − − . The figure shows the plots obtained for N = 100, N = 400 and the limit N → ∞. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 80. Introduction The Model Exact reduced dynamics Correlation functions The limit N → ∞ Time evolution Entanglement evolution Conclusion Figure: 4 The evolution in time of the density matrix element ρ13 . Theinitial condition of the two-qubit system is the entangled state √ (| + + + | − − ). The figure shows the plots obtained for 1 2 N = 100, N = 400 and the limit N → ∞. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 81. Introduction The Model Exact reduced dynamics concurrence The limit N → ∞ Main results Entanglement evolution ConclusionEntanglement Definition By definition, entanglementrefers to nonlocal correlations that exist between quantum systems. A quantum state ρ describing a bipartite system A + B and living in the finite Hilbert space HA ⊗ HB is said to be entangled (or not separable) if it cannot be written as a convex combination of product states: ρ= ck ρk ⊗ ρk A B k A state which is not entangled is called separable. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 82. Introduction The Model Exact reduced dynamics concurrence The limit N → ∞ Main results Entanglement evolution ConclusionEntanglement Definition By definition, entanglementrefers to nonlocal correlations that exist between quantum systems. A quantum state ρ describing a bipartite system A + B and living in the finite Hilbert space HA ⊗ HB is said to be entangled (or not separable) if it cannot be written as a convex combination of product states: ρ= ck ρk ⊗ ρk A B k A state which is not entangled is called separable. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 83. Introduction The Model Exact reduced dynamics concurrence The limit N → ∞ Main results Entanglement evolution ConclusionConcurrence There exist many entanglement measures which quantify the amount of entanglement in a given state. For mixed states, the concurrence is used: C (ρ) = max{0, λ1 − λ2 − λ3 − λ4 }. (31) λ1 λ2 λ3 λ4 are the eigenvalues of the operator ρ(σy ⊗ σy )ρ∗ (σy ⊗ σy ) C (ρ) ranges from 0 for separable states to 1 for maximally entangled states. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 84. Introduction The Model Exact reduced dynamics concurrence The limit N → ∞ Main results Entanglement evolution ConclusionConcurrence evolution The operator V = σy ⊗ σy is a linear skew-adjoint operator in C2 ⊗ C2 , i.e., VV = −I. In C ⊕ C3 it reads 0 0 1 0   0 −1 0 0 V ⊗V = 1 0 . (32) 0 0 0 0 0 1 Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 85. Introduction The Model Exact reduced dynamics concurrence The limit N → ∞ Main results Entanglement evolution ConclusionResults The concurrence corresponding to the initial separable state | − − is equal to C (ρ) = max{0, −ρ22 (t )} = 0. the initial state is the maximally entangled state |Ψ = √ (| + − + | − + ), then the concurrence takes the form 1 2 C (ρ) = max{0, ρ22 (t ) − 2 ρ11 (t )ρ33 (t )}. the maximally entangled state |Φ = 1 √ 2 (| − − + | + + ) has the concurrence C (ρ) = max{0, 2ρ13 (t ) − ρ22 (t )}. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 86. Introduction The Model Exact reduced dynamics concurrence The limit N → ∞ Main results Entanglement evolution ConclusionResults The concurrence corresponding to the initial separable state | − − is equal to C (ρ) = max{0, −ρ22 (t )} = 0. the initial state is the maximally entangled state |Ψ = √ (| + − + | − + ), then the concurrence takes the form 1 2 C (ρ) = max{0, ρ22 (t ) − 2 ρ11 (t )ρ33 (t )}. the maximally entangled state |Φ = 1 √ 2 (| − − + | + + ) has the concurrence C (ρ) = max{0, 2ρ13 (t ) − ρ22 (t )}. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 87. Introduction The Model Exact reduced dynamics concurrence The limit N → ∞ Main results Entanglement evolution ConclusionResults The concurrence corresponding to the initial separable state | − − is equal to C (ρ) = max{0, −ρ22 (t )} = 0. the initial state is the maximally entangled state |Ψ = √ (| + − + | − + ), then the concurrence takes the form 1 2 C (ρ) = max{0, ρ22 (t ) − 2 ρ11 (t )ρ33 (t )}. the maximally entangled state |Φ = 1 √ 2 (| − − + | + + ) has the concurrence C (ρ) = max{0, 2ρ13 (t ) − ρ22 (t )}. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 88. Introduction The Model Exact reduced dynamics concurrence The limit N → ∞ Main results Entanglement evolution Conclusion Figure: 5 Concurrence as a function of time for initial states1√ (| + − + | − + ) (solid curve) and √ (| − − + | + + ) (dashed 12 2 curve). Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 89. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionConclusion Any pure state of the two-qubit system evolves into a mixed state. The pure entangled state √2 (| + − − | − + ) of the two 1 qubits is a decoherence-free state. The environment has no effect on the separability of pure separable states. The environment has the tendency to decrease the degree of entanglement of initially entangled states of the qubits. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 90. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionConclusion Any pure state of the two-qubit system evolves into a mixed state. The pure entangled state √2 (| + − − | − + ) of the two 1 qubits is a decoherence-free state. The environment has no effect on the separability of pure separable states. The environment has the tendency to decrease the degree of entanglement of initially entangled states of the qubits. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 91. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionConclusion Any pure state of the two-qubit system evolves into a mixed state. The pure entangled state √2 (| + − − | − + ) of the two 1 qubits is a decoherence-free state. The environment has no effect on the separability of pure separable states. The environment has the tendency to decrease the degree of entanglement of initially entangled states of the qubits. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 92. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionConclusion Any pure state of the two-qubit system evolves into a mixed state. The pure entangled state √2 (| + − − | − + ) of the two 1 qubits is a decoherence-free state. The environment has no effect on the separability of pure separable states. The environment has the tendency to decrease the degree of entanglement of initially entangled states of the qubits. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 93. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution ConclusionConclusion Any pure state of the two-qubit system evolves into a mixed state. The pure entangled state √2 (| + − − | − + ) of the two 1 qubits is a decoherence-free state. The environment has no effect on the separability of pure separable states. The environment has the tendency to decrease the degree of entanglement of initially entangled states of the qubits. Hamdouni, Fannes, Petruccione Entanglement Dynamics...
  • 94. Introduction The Model Exact reduced dynamics The limit N → ∞ Entanglement evolution Conclusion THANK YOUHamdouni, Fannes, Petruccione Entanglement Dynamics...