transmission of entanglement in three cldcoupled qbitqubit...
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Transmission of Entanglement in Three C l d Q bit S tCoupled Qubit Systems
With Heat Bath
Saeed PegahanPhysics DepartmentU i it f I f h
6th Asia PacificConference & Workshop In
University of IsfahanSupervisors: Professor F. Kheirandish,
Assistance Professor M. SoltaniConference & Workshop In
Quantum Information Science2012
Isfahan Quantum Optics Group
Outline
Motivations Basic ConceptsMotivations, Basic Concepts.
The Evolution of a three partite system coupled to thermal reservoir.
Transmission of Entanglement in the presence of thermal reservoirTransmission of Entanglement in the presence of thermal reservoir.Monogamy Inequality in Terms of Negativity.Effects of External Fields on Transmission of Entanglement andControl of Entanglement.
Conclusions and Future Directions.
Motivations
Any observed disappearance of quantum coherence and interference canAny observed disappearance of quantum coherence and interference canbe attributed to interaction with the environment, that is, Decoherence.But is it always true to say that Reservoir is a destroyer ofEntanglement?g
To understand how to control and transfer entanglement as aresource of quantum communication and information, we need toresource of quantum communication and information, we need tostudy the mechanism by which this process occurs.
There has been a lot of interest and work in this regard but most ofThere has been a lot of interest and work in this regard, but most ofthis work involves the coupling between two qubits ,We will insteadconcentrate on the coupling between three Qubits.
S iti P i i lSuperposition Principle, Basic Concepts
n nCψ ψ=∑ 1 21 ( )2
ψ ψ ψ= +2
( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 21 2 1 2 1 2
1 1 1 = Re{ }2 2 2
x x x x x x xρ ψ ψ ψ ψ ψ ψ ∗+ = + +
( ) ( )2 21 1
Interference Term
( ) ( )1 21 12 2
x xρ ψ ψ∝ +
D hDecoherence, Basic Concepts
What is Decoherence ?
Measurement by the environment
Destruction of quantummechanical behavior
hallmark of the crossover from the quantum to
Caused by the environment
ec a ca be av ofrom the quantum to the classical world
Caused by the environment,Environment Noise
Collapse of Wave function
Destruction of Entangled state
Q t E t l tQuantum Entanglement, Basic Concepts
A pure state is separable if it can be expressed as a tensor product of its subsystems: ψ ψ ψ= ⊗of its subsystems:
A mixed state is separable if it can be represented as a mixture of product states:
1 2ψ ψ ψ= ⊗
i i i i ip a a b bρ = ⊗∑of product states: i i i i ii
p a a b bρ ⊗∑State that is not separable is entangled
Bell states (pure entangled states)
1 ( 0 1 ) 12
ψ ′ = + ⊗Separable state
Werner states (mixed entangled states )1 01 112
ψ ′ = +Separable state
1 2ψ ψ ψ≠ ⊗ Entangled State
D hDecoherence, Basic Concepts
Closed quantum systems
Unitary evolution keep the system in pure state, and prevent from mixed-state.
Open quantum systemsp q y
R l t t b f tl i l t d f it di ThReal system cannot be perfectly isolated from its surroundings. The open system evolution is characterized by non‐Unitary evolution.
I t ti f t l l t ithInteraction of a two level atom with a single mode field
† †1ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )( )2z f xH a a g a aωσ ω σ= + + + +h h h
J C i M d l2
†ˆ ˆ ˆ ˆ ˆ( ) ( exp( ) exp( ))IH t g a i t a i tσ σ− += Δ + − Δh
Janynes-Cumming Model
Rotation Wave Approximation-Interaction Picturefω ωΔ = −
pp†ˆ ˆ ˆ ˆ ˆ( ) ( exp( ) exp( ))I k k k
kH t g a i t a i tσ σ− += Δ + − Δ∑h
Thermal Reservoir-Rotation Wave Approximation
I t ti f t l l t ithInteraction of a two level atom with a single mode field
ˆ ˆ ˆˆ ˆ ˆ ˆ/ [ ( )] [ ( )] [ ( )]d dt L t i H t D tρ ρ ρ ρ′ +/ [ ( )] [ , ( )] [ ( )]S S S S Sd dt L t i H t D tρ ρ ρ ρ= = − +
Master Equation
Unitary Evolution
Non-Unitary evolutionNon-Unitary evolution
Decoherence and possibly also Dissipation
I t ti f t l l t ithInteraction of a two level atom with a thermal Reservoir
1ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ/ ([ ( ) ] ( ) [ ( ) ]]tid dt Tr H t Tr H t H t dtρ ρ ρ ρ ρ− ′ ′= ⊗ ⊗∫
Born-Markov Master Equation
( ) ( ) ( ) ( )2/ ([ ( ), ] ( ),[ ( ), ]]i i i
i
R I S t R t R I I S t R tt
d dt Tr H t Tr H t H t dtρ ρ ρ ρ ρ= ⊗ − ⊗∫h h
ΓBorn Approximation Markov Approximation
ˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ/ [ , ( )] [ ( ) ( ) 2 ( ) ]2S s S th S S Sd dt H t n t t tρ ρ σ σ ρ ρ σ σ σ ρ σ− + − + + −
+ + +
Γ′= − + −
Γ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ( 1) [ ( ) ( ) 2 ( ) ]2th S S Sn t t tσ σ ρ ρ σ σ σ ρ σ+ − + − − +Γ
− + + −
Absorbing energy from the environment1
Emission of energy to the environment1
exp( ) 1th
k
B
n wk T
=−
h
0thn =0T =
I t ti f t l l t ithInteraction of a two level atom with a thermal Reservoir
Results:Results:
at T=0, the energy of excited atom emitted to the reservoir.
Reservoir acts as a disturbing factor which absorbs energy of the excited atom.
After the short run, exited atom decay to its lower level.
Due to dissipation, We expect that at T=0 thermal reservoir is a disadvantage for atoms.
I t ti f N id ti l Q bitInteraction of N identical Qubits with a thermal Reservoir at T=0
† †1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ( ))N N
zi l l l l i l i lH a a g a aωσ ω σ σ+ −= + + + +∑ ∑ ∑∑h h h
1( ) ( ( ))
2i l l l l i l i li l l i
H a a g a aωσ ω σ σ=
= + + + +∑ ∑ ∑∑h h h
ˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ/ [ , ( )] [ ( ) ( ) 2 ( ) ]S s S S S Sd dt H t t t tρ ρ σ σ ρ ρ σ σ σ ρ σ+ − + − − +Γ′= − + −/ [ , ( )] [ ( ) ( ) 2 ( ) ]2
ˆ ˆ
S s S S S S
N
i
d dt H t t t tρ ρ σ σ ρ ρ σ σ σ ρ σ
σ σ± ±
+
=∑ 3 2 30/ (3 )cω μ πεΓ = h
i∑ 0( )μ
Weisskopf-Wigner Approximationˆˆ ˆ ˆ/ [ , ] / [ ]d dt H i Lρ ρ ρ= +h
Environment-Induced Entanglement with a single photon, Physical Review A, 2009, L. Davidovich et. al.
Transmission of Entanglement
1 2 1 2 3(( ) / 2)↑↓ − ↓↑ ⊗ ↓ˆˆ ˆ ˆ/ [ , ] / [ ]d dt H i Lρ ρ ρ= +h 1,2 1,2 3(( ) )/ [ , ] / [ ]d dt H i Lρ ρ ρ+h
Initial StateMaster Equation
{ , }↑ ↓ zσEigenstates
T i i f E t l tTransmission of Entanglement, Negativity Measure
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ/ [ ( ) ( ) 2 ( ) ]d dt t t t+ − + − − +Γ
3
/ [ ( ) ( ) 2 ( ) ]2
ˆ ˆ
S S S S
i
d dt t t tρ σ σ ρ ρ σ σ σ ρ σ
σ σ
+ + +
± ±
= − + −
=∑2
ii=∑
1 / 2BTρ⎛ ⎞= −⎜ ⎟⎝ ⎠
†T T TB B BTrρ ρ ρ=1
1 / 2ρ⎜ ⎟⎝ ⎠ 1
ρ ρ ρ
8-dimensional Hilbert space8 dimensional Hilbert space64-Coupled differential Equations
T i i f E t l tTransmission of Entanglement between Second and Third Qubits
910
Γ=
5Γ
10Γ=
310
Γ=
Dissipation rate increases, Transmission of Entanglement
reaches to Stationary state
T i i f E t l tTransmission of Entanglement between First and Third Qubits
Dissipation rate increases,
9Γ=
510
Γ=
310
Γ=
pTransmission of Entanglement time
Decreases3
10Γ=
10Γ=
910
Γ=
510
Γ=
T i i f E t l tTransmission of Entanglement between First and Second Qubits
Dissipation rate increases, pTransmission of Entanglement time
Decreases
510
Γ=
310
Γ=
910
Γ=10
M I lit i t fMonogamy Inequality in terms of Negativity, Concurrence Measure
( ) ( )y y y yρ ρ σ σ ρ σ σ∗′ = ⊗ ⊗
1 2 3 4( ) max{0, }C ρ λ λ λ λ′ ≡ − − −
( ) ( )y y y yρ ρ ρFor Two-Qubit System
2 2 2( )AB AC A BCC C C+ ≤ Coffman-Kundu-Wootters
CKWCKW
( ) 2 detA BC AC ρ= For Three-Qubit System
M I lit i t fMonogamy Inequality in terms of Negativity
2 2 2+ ≤2 2 2C C C+ ≤ ( )AB AC A BC+ ≤
AB ABC=
( )AB AC A BCC C C+ ≤
Monogamy InequalityAB ABC
AC ACC=( ) ( )A BC A BCC=
Monogamy Inequality In terms of Negativity for three Qubit Physical Review AMonogamy Inequality In terms of Negativity for three-Qubit, Physical Review A,2007, Yong-Cheng Ou and Heng Fan.
M I lit i t fMonogamy Inequality in terms of Negativity
5
310
Γ=
510
Γ=
910
Γ=
Eff t f E t l Fi ldEffects of External Fields on Transmission of Entanglement
ˆˆ ˆ ˆ/ [ , ] / [ ]S S Sd dt H i Lρ ρ ρ= +h ˆ ˆ xH Bσ=
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ[ ] [2 ( ) ( ) ( ) ]2S S S SL t t tρ σ ρ σ σ σ ρ ρ σ σ− + + − + −Γ
= − −
T i i f E t l tTransmission of Entanglement between Second and Third Qubits
3108B
Γ =
=
3100B
Γ =
=
T i i f E t l tTransmission of Entanglement between First and Third Qubits
3
Γ =
3100B
Γ =
=
108B =
T i i f E t l tTransmission of Entanglement between First and Second Qubits
3108B
Γ =3
100B
Γ =
=8B = 0B =
M I lit i t fMonogamy Inequality in terms of Negativity
3Γ
310
Γ =
3100B
Γ =
=
108B =
T i i f E t l tTransmission of Entanglement between Second and Third Qubits
3108B
Γ =
=
3100B
Γ =
=
ˆ ˆ yH Bσ=
T i i f E t l tTransmission of Entanglement between First and Third Qubits
3108B
Γ =
=
3100B
Γ =
=
ˆ ˆ yH Bσ=
T i i f E t l tTransmission of Entanglement between First and Second Qubits
3108B
Γ =
=
3100B
Γ =
=
ˆ ˆ yH Bσ=
M I lit i t fMonogamy Inequality in terms of Negativity
3Γ
108B
Γ =
=3
100B
Γ =
=
ˆ ˆ yH Bσ=
Conclusion and Future Direction
Creation of Entanglement between Second and ThirdCreation of Entanglement between Second and ThirdQubits in the common thermal reservoir.
T i i f t l t f b tTransmission of entanglement from on subsystemconsists of First and Second Qubits to another one thatcontains First and Third Qubits as a result of the aboveeffect.
Realizing the effect of External magnetic Field as ang gobject for control of entanglement.
Future Directions
To explore a theoretical set‐up for optimum controland Transmission of Entanglement in three Partite OpenQuantum system.
The mechanism which give rise to the desire EntangledThe mechanism which give rise to the desire Entangledstate even in presence of Noise source.To change the Nature of Reservoir with memory effectand investigate the effect of it on the control andTransmission of Entanglement.New techniques for generating long lived EntanglementNew techniques for generating long lived Entanglementby means of Reservoir Engineering.
g{tÇ~ çÉâ yÉÜ çÉâÜ tààxÇà|ÉÇ
Experimental Set-Up
Consider two flux qubit that are coupled to a transitionline resonator An external voltage field is applied toline resonator. An external voltage field is applied tosystem. It can be shown that we can control theentanglement between qubit and resonator mode byapplying detuning between external voltage field andqubit‐resonator resonance frequency.