Entanglement evolution for quantum trajectories Dominique Spehner

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Entanglement evolution for quantum trajectories Dominique Spehner Laboratoire de Physique et Modelisation des Milieux Condenses & Institut Fourier, Grenoble, France Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 1

  • modelisation des milieux condenses

  • convex decompositions

  • e?

  • quantum optics

  • over all

  • optimal decomposition

  • joint work


Publié le : lundi 1 novembre 2010
Lecture(s) : 21
Source : www-fourier.ujf-grenoble.fr
Nombre de pages : 29
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Entanglement evolution for quantum trajectories
Dominique Spehner
´ ´
Laboratoire de Physique et Modelisation des Milieux Condenses
& Institut Fourier, Grenoble, France
Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 1Outlines
• Evolution of entanglement in the presence of couplings
with an environment
• Average concurrence for quantum trajectories
• Conclusions & Perspectives
Jointwork with: Sylvain Vogelsberger (I.F. Grenoble)
Ref.: arXiv:1006.1317 [quant-ph]
Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 2Evolution of entanglement
EntanglementofformationE between 2 subsystemsA &B
ρ
in amixedstateρ: by definition,E is aninfimum over all
ρ
P
convex decompositionsρ = p |ψ ihψ | (withp ≥ 0),
k k k k
k
X

E = inf p E , E =S tr|ψ ihψ |
ρ k ψ ψ vonNeuman k k
k k
A
k
[Bennett et al. PRA 54 (’96)].
Ifρ evolves with time, so does the optimal decomposition
{p ,|ψ i} realizing the minimum.
k k
Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 3Evolution of entanglement
EntanglementofformationE between 2 subsystemsA &B
ρ
in amixedstateρ: by definition,E is aninfimum over all
ρ
P
convex decompositionsρ = p |ψ ihψ | (withp ≥ 0),
k k k k
k
X

E = inf p E , E =S tr|ψ ihψ |
ρ k ψ ψ vonNeuman k k
k k
A
k
[Bennett et al. PRA 54 (’96)].
Ifρ evolves with time, so does the optimal decomposition
{p ,|ψ i} realizing the minimum.
k k
When the 2 subsystemsinteractwiththeirenvironment, the
entanglement getsshared betweenA,B, and the environment
֒→E typically decreases (entanglement loss betweenA &B).
ρ
Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 3Evolution of entanglement
EntanglementofformationE between 2 subsystemsA &B
ρ
in amixedstateρ: by definition,E is aninfimum over all
ρ
P
convex decompositionsρ = p |ψ ihψ | (withp ≥ 0),
k k k k
k
X

E = inf p E , E =S tr|ψ ihψ |
ρ k ψ ψ vonNeuman k k
k k
A
k
[Bennett et al. PRA 54 (’96)].
Ifρ evolves with time, so does the optimal decomposition
{p ,|ψ i} realizing the minimum.
k k
When the 2 subsystemsinteractwiththeirenvironment, the
entanglement getsshared betweenA,B, and the environment
֒→E typically decreases (entanglement loss betweenA &B).
ρ
Q1: Can theA-B entanglement disappear completely?
Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 3Evolution of entanglement
EntanglementofformationE between 2 subsystemsA &B
ρ
in amixedstateρ: by definition,E is aninfimum over all
ρ
P
convex decompositionsρ = p |ψ ihψ | (withp ≥ 0),
k k k k
k
X

E = inf p E , E =S tr|ψ ihψ |
ρ k ψ ψ vonNeuman k k
k k
A
k
[Bennett et al. PRA 54 (’96)].
Ifρ evolves with time, so does the optimal decomposition
{p ,|ψ i} realizing the minimum.
k k
When the 2 subsystemsinteractwiththeirenvironment, the
entanglement getsshared betweenA,B, and the environment
֒→E typically decreases (entanglement loss betweenA &B).
ρ
Q1: Can theA-B entanglement disappear completely?
Q2: Can one extract information from the environment (by mea-
suring it) in order to “know” the optimal decomposition?
Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 3Entanglement sudden death
ENTANGLEMENT TYPICALLY DISAPPEARS BEFORE COHERENCES ARE LOST!
ρ
It can disappear after a finite time
T=0
ρ
S
• always the case if the qubits relax to a
common
bath
Gibbsstateρ atpositivetemperature

T>0
• otherwise depends on the initial state.
ρ
0
[Diosi ’03], [Dodd & Halliwell PRA 69 (’04)], [Yu et Eberly PRL 93 (’04)]
Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 4Entanglement sudden death
ENTANGLEMENT TYPICALLY DISAPPEARS BEFORE COHERENCES ARE LOST!
ρ
It can disappear after a finite time
T=0
ρ
S
• always the case if the qubits relax to a
common
bath
Gibbsstateρ atpositivetemperature

T>0
• otherwise depends on the initial state.
ρ
0
[Diosi ’03], [Dodd & Halliwell PRA 69 (’04)], [Yu et Eberly PRL 93 (’04)]
C(t)
If the two qubits are coupled to a
1
common bath, entanglement can
also suddently reappear
due to effective (bath-mediated) qubit
interaction creating entanglement
t
t
t
ESD ESD
sudden birth
[Ficek & Tana´s PRA 74 (’06)], [Hernandez &
Orszag PRA 78 (’08)], [Mazzola et al. PRA (’09)]
Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 4Quantum trajectories
As a result of continuous measurements on the environment, the
bipartite system remains in a pure state|ψ(t)i at all timest> 0
t∈R 7!|ψ(t)i quantum trajectory
+
Reason: each measurement disentangle the system and the
environment (by wavepacket reduction).
Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 5Quantum trajectories
As a result of continuous measurements on the environment, the
bipartite system remains in a pure state|ψ(t)i at all timest> 0
t∈R 7!|ψ(t)i quantum trajectory
+
Reason: each measurement disentangle the system and the
environment (by wavepacket reduction).
Averaging over the measurements, one gets the density matrix:
Z
ρ(t) =|ψ(t)ihψ(t)| = dp[ψ]|ψ(t)ihψ(t)| .
Quantum Optics V, Cozumel, Mexico 16/11/2010 – p. 5

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