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.