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-Natal
2 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
Conclusion
Outline
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
Conclusion
Introduction
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
Conclusion
Introduction
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
Conclusion
Introduction
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
Conclusion
Introduction
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
Conclusion
Introduction
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
Conclusion
Nowdays, components of electronic devices are shrinking
(Moore’s first law!). Inevitably, we are led to deal with new
materials, nano-devices and molecular electronics ⇒ scales
where quantum phenomena predominate:
particle-wave duality
the superposition principle
probabilistic nature of quantum measurement outcomes
entanglement...etc.
The unification of classical information theory and quantum
mechanics gave rise to a new field of physics called quantum
information theory.
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

9. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
Nowdays, components of electronic devices are shrinking
(Moore’s first law!). Inevitably, we are led to deal with new
materials, nano-devices and molecular electronics ⇒ scales
where quantum phenomena predominate:
particle-wave duality
the superposition principle
probabilistic nature of quantum measurement outcomes
entanglement...etc.
The unification of classical information theory and quantum
mechanics gave rise to a new field of physics called quantum
information theory.
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

10. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
Nowdays, components of electronic devices are shrinking
(Moore’s first law!). Inevitably, we are led to deal with new
materials, nano-devices and molecular electronics ⇒ scales
where quantum phenomena predominate:
particle-wave duality
the superposition principle
probabilistic nature of quantum measurement outcomes
entanglement...etc.
The unification of classical information theory and quantum
mechanics gave rise to a new field of physics called quantum
information theory.
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

11. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
Nowdays, components of electronic devices are shrinking
(Moore’s first law!). Inevitably, we are led to deal with new
materials, nano-devices and molecular electronics ⇒ scales
where quantum phenomena predominate:
particle-wave duality
the superposition principle
probabilistic nature of quantum measurement outcomes
entanglement...etc.
The unification of classical information theory and quantum
mechanics gave rise to a new field of physics called quantum
information theory.
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

12. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
Nowdays, components of electronic devices are shrinking
(Moore’s first law!). Inevitably, we are led to deal with new
materials, nano-devices and molecular electronics ⇒ scales
where quantum phenomena predominate:
particle-wave duality
the superposition principle
probabilistic nature of quantum measurement outcomes
entanglement...etc.
The unification of classical information theory and quantum
mechanics gave rise to a new field of physics called quantum
information theory.
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

13. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
Nowdays, components of electronic devices are shrinking
(Moore’s first law!). Inevitably, we are led to deal with new
materials, nano-devices and molecular electronics ⇒ scales
where quantum phenomena predominate:
particle-wave duality
the superposition principle
probabilistic nature of quantum measurement outcomes
entanglement...etc.
The unification of classical information theory and quantum
mechanics gave rise to a new field of physics called quantum
information theory.
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

14. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
Nowdays, components of electronic devices are shrinking
(Moore’s first law!). Inevitably, we are led to deal with new
materials, nano-devices and molecular electronics ⇒ scales
where quantum phenomena predominate:
particle-wave duality
the superposition principle
probabilistic nature of quantum measurement outcomes
entanglement...etc.
The unification of classical information theory and quantum
mechanics gave rise to a new field of physics called quantum
information theory.
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

15. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
Nowdays, components of electronic devices are shrinking
(Moore’s first law!). Inevitably, we are led to deal with new
materials, nano-devices and molecular electronics ⇒ scales
where quantum phenomena predominate:
particle-wave duality
the superposition principle
probabilistic nature of quantum measurement outcomes
entanglement...etc.
The unification of classical information theory and quantum
mechanics gave rise to a new field of physics called quantum
information theory.
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

16. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
Nowdays, components of electronic devices are shrinking
(Moore’s first law!). Inevitably, we are led to deal with new
materials, nano-devices and molecular electronics ⇒ scales
where quantum phenomena predominate:
particle-wave duality
the superposition principle
probabilistic nature of quantum measurement outcomes
entanglement...etc.
The unification of classical information theory and quantum
mechanics gave rise to a new field of physics called quantum
information theory.
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

17. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
Nowdays, components of electronic devices are shrinking
(Moore’s first law!). Inevitably, we are led to deal with new
materials, nano-devices and molecular electronics ⇒ scales
where quantum phenomena predominate:
particle-wave duality
the superposition principle
probabilistic nature of quantum measurement outcomes
entanglement...etc.
The unification of classical information theory and quantum
mechanics gave rise to a new field of physics called quantum
information theory.
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

18. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
prepare
operations on qubits evolve
measure
!!
Leakage away of quantum information by decoherence (due to
uncontroled 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
prepare
operations on qubits evolve
measure
!!
Leakage away of quantum information by decoherence (due to
uncontroled 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 to
uncontroled 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
prepare
operations on qubits evolve
measure
!!
Leakage away of quantum information by decoherence (due to
uncontroled 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
prepare
operations on qubits evolve
measure
!!
Leakage away of quantum information by decoherence (due to
uncontroled 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
The 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
Conclusion
Heisenberg 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
Conclusion
we 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 subspace
C ⊗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
Conclusion
we 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 subspace
C ⊗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
Conclusion
Hamiltonian
   
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
Conclusion
Hamiltonian
   
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
Conclusion
Hamiltonian
   
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 0
where
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 0
where
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 0
where
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 0
where
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
Conclusion
The 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
Conclusion
Initial 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
Conclusion
Time 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
Conclusion
Time 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
Conclusion
Time 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
Conclusion
Time 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
Conclusion
Time 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
Conclusion
where
 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 23
S2n +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 33
with the environment correlation functions
Correlation 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
Conclusion
Exact 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
Conclusion
Off-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
Conclusion
with
√
−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
Conclusion
The 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
Conclusion
The 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
Conclusion
Time 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
Conclusion
Time 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
Conclusion
Time 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
Conclusion
Time 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
Conclusion
The function ζ is given by
αt αt
ζ(t ) = − D+ (− )
2 2
1
lim ζ(t ) = −
t →∞ 2
where we have introduced the Dawson function
Dawson 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
Conclusion
The function ζ is given by
αt αt
ζ(t ) = − D+ (− )
2 2
1
lim ζ(t ) = −
t →∞ 2
where we have introduced the Dawson function
Dawson 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
Conclusion
The function ζ is given by
αt αt
ζ(t ) = − D+ (− )
2 2
1
lim ζ(t ) = −
t →∞ 2
where we have introduced the Dawson function
Dawson 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 . Initial
state 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 a
function 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 . The
initial 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
Conclusion
Entanglement
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
Conclusion
Entanglement
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
Conclusion
Concurrence
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
Conclusion
Concurrence 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
Conclusion
Results
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
Conclusion
Results
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
Conclusion
Results
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 states
1
√ (| + − + | − + ) (solid curve) and
√ (| − − + | + + ) (dashed
1
2 2
curve).
Hamdouni, Fannes, Petruccione Entanglement Dynamics...

89. Introduction
The Model
Exact reduced dynamics
The limit N → ∞
Entanglement evolution
Conclusion
Conclusion
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
Conclusion
Conclusion
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
Conclusion
Conclusion
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
Conclusion
Conclusion
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
Conclusion
Conclusion
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 YOU
Hamdouni, Fannes, Petruccione Entanglement Dynamics...