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Decoherence-free sub-space Decoherence-free sub-space and quantum error-rejection and quantum error-rejection

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Lecture Note 7

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DecoherenceDecoherence

open system dynamics

System

Environment

)(00 0)( tEE tU )(11 1

)( tEE tU

( )0 1 0 0 1 10 1 0 ( ) 1 ( )U tE E t E t

2 *0 0 1 1 0

2*1 0 0 1 1

( )q E q E

E Et Tr

E E

0 10 1

The off-diagonal element of the qubit density matrix will drop down with the rate depends on the coupling between qubit and environment.

More generally ...

How to guide the dynamics of system-environment coupling?

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Quantum Error Correction for QC

Active (Error correction): deal well with independent errors on qubits

Quantum Entanglement Purification for QC

Entanglement Purification (any unknown mixed state) Local Filtering (known state) Entanglement Concentration (unknown state)

QC based on Decoherence-free Subspace

Passive (error avoidance): find a subspace of the system space over which evolution stays unitary, unperturbed, correlated noise

Error-free Transfer in QC Active (error rejection): reject the contaminated information

Possible solutions to overcome decoherencePossible solutions to overcome decoherencein long-distance quantum communication (QC)in long-distance quantum communication (QC)

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QC based on Decoherence-

free Subspace

Error-free Transfer in QC

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Decoherence-free subspace Decoherence-free subspace (DFS)(DFS)

( ) gei ti tU te g e e e g

( )

=

g ge e

e g

i t i ti t i tU t

i t

e g g e

e e e g e g e e

e e g g e

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Decoherence Free Subspace General Definitions, Collective Decoherence

Robustness to perturbing error processes

Use of DF subspace for concatenation into a Quantum Error Correcting Code (QECC)

Relationship between DF subspace and QECC

Existential universality results on DF subspaces/symmetrization methods

Subsystem Generalization

How do we perform quantum communication in a DFS?

[Phys. Rev. Lett. 79, 1953 (1997); Phys. Rev. Lett. 79, 3306 (1997); Phys. Rev. Lett. 81, 2594 (1998)]

[Phys. Rev. Lett. 81, 2594 (1998); Phys. Rev. A 60, 1944 (1999)]

[Phys. Rev. Lett. 82, 4556 (1999)]

[Phys. Rev. A 60 729(R) (1999)]

[Phys. Rev. Lett. 84, 2525(2000)]

1997

Symmetrization/Bang-bang methods [Phys. Rev. A 58, 2733 (1998); Phys.Lett. A 258, 77 (1999) ] 1998

1999

2000

DFS HistoryDFS History

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DFS under Collective DFS under Collective NoiseNoise

Collective Rotation Noise ： Noise can be seen as some unitary transformation as U(θ,Φ), if for all the channel, the unitary is the same, then it is called collective noise. If Φ is 0, i.e., U ＝ U(θ), it is called collective rotation noise

, : i

i

U H Cos H e Sin V

V e Sin H Cos V

2 2 2 2

1

2

1

2

1

21

2

i i

i i

H V V H

Cos H e Sin V e Sin H Cos V

e Sin H Cos V Cos H e Sin V

Cos Sin H V Cos Sin V H

H V V H

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2 2 2 2

1

2

1

2

1

21

2

H H V V

Cos H Sin V Cos H Sin V

Cos V Sin H Cos V Sin H

Cos Sin H H Cos Sin V V

H H V V

[P. G. Kwiat et al., Science 290, 498(2000); J. B. Altepeter, et al., Phys. Rev. Lett. 92, 147901(2004)]

DFS under DFS under Collective Rotation Collective Rotation

NoiseNoise :U H Cos H Sin V

V Sin H Cos V

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DFS for Collective Rotation DFS for Collective Rotation NoiseNoise

The two state are invariant under the collective rotation noise. All the linear superposition of the two states constitute a subspace that is decoherence free to the noise.

)(2

1HVVH

)(2

1VVHH

[P. G. Kwiat et al., Science 290, 498(2000);

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Similar to BB84, +,- respect the diagonal state and anti-diagonal state respectively.

The four state can be used to encode key and the security bound is the same as BB84 protocol.

)(2

1HVVHV

)(2

1VVHHH

)(2

1)(

2

1 VHVH

)(2

1)(

2

1 VHVH

Application in Application in quantum key quantum key distribution using a distribution using a DFSDFS

[X.B.Wang, Phys. Rev. A 72, 050304(R) (2005)][X.B.Wang, Phys. Rev. A 72, 050304(R) (2005)]

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Experimental Experimental SetupSetup

[Q. Zhang, PRA 73, 020301 (R) 2006]

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Experimental Experimental ResultResult

QBER of DFS and traditional BB84

under the collective rotation noise.

|θ| > π/8, QBERBB84>11%

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DrawbackDrawback

• DFS only for Collective Rotation Noise

• Other noiseFree space phase drifting caused by

temperature differenceLong distance in optical fibers will cause

a redoubtable obstacle

Noise not only in H/V basis!Noise not only in H/V basis!

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Collective NoiseCollective Noise

, : i

i

U H Cos H e Sin V

V e Sin H Cos V

2 2

1 2 3

1

21

2

1

2

i i

i i

i i

i i

H V V H

Cos H e Sin V e Sin H Cos V

e Sin H Cos V Cos H e Sin V

Cos Sin H V V H Cos Sin e e H H V V

Cos Sin e e H H V V

HV VH HH VV HH VV

2 2 2

1 2 3 1

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A new protocolA new protocolFirst apply a time delay between H and V, the state will be

T THV VH HV V H

After a collective noise

1

2 3

1

2 3

2

2

T T

T T T T

T T T T

T T T T

T T T T

HV V H

H V V H H V V H

H H V V H H V V

V H H V V H H V

H H V V H H V V

2 2

HV VH

Bob can measure in any direction (H’/V’) which also can be considered as part of the collective noise.

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A new protocolA new protocolThen again, Bob apply a time delay between H and V, the state will be

1

2 3

1

2 3

2

2

T T TT T T TT

T TT T T TT T

T T TT T T TT

TT T T TT T T

H V V H H V V H

H H V V H H V V

V H H V V H H V

H H V V H H V V

1 1

2 3 2 3

1 1

2 2

2 2

TT TT

T TT TT T T T

T T T T VH V V H H H V

H H H H V V V V

The last operation is to project the state onto the subspacein which the photons arrive exactly at the same time

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• We will get with a probability

1/3 by a random unitary transformation

21|| (1 ) / 2 ||

T T T TH V V H

A new protocolA new protocol

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[T.-Y Chen et al., Phys. Rev. Lett. 96 150504 (2006)]

Experimental Setup

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Experimental Result4m fiber

without random rotations

with random rotations

average QBER

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Experimental Result1km fiber

[T.-Y Chen et al., Phys. Rev. Lett. 96 150504 (2006)]

without random rotations

with random rotations

average QBER

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QC based on Decoherence- free Subspace

Error-free Transfer in QC

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Bit-flip Error CorrectionBit-flip Error Correction

1 1123 123 123

2 3

CNot

CNot

with a probability occurs a

1 0

| 0 |1| 000 |111

| 0 | 0

0 1 000 111

1 0 11 0 1 1100 11

0 0 1 1 0 1 10

00 11

1 0

1 0

bit-flip error

10 01

U

p

CNot Operation Required!!![D. Bouwmeester, PRA 63, 040301(R) (2001).]

two bits flipping (p2) can’t be corrected

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Error-free transferError-free transfer

1234 1 1 234 234

Bell Measurement34 34Between 1 & 2

34 34

34 34

34 34

34 34

1| 0 |1 | 000 |111

2

| 00 |11

| 00 |11 | 0 |1 | 0 |1

| 0 | 1

| 0 |1

1 0

1 0

11 00| | | |0 1

No coincedence

0 1| |

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Problem in Experimental Realization

Possibility of two pair emission is in the same order and will cause four-fold coincidence!

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[X.-B. Wang, PRA 69, 022320 (2004)]

Error-free Error-free transfertransfer

2’

1’

2”

1”

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123 1 2 1 2 3 3

each photon in1' 2' 3' 1' 2' 3'the two arms of PBS

1' 2' 1' 2' 3' 1' 2' 1' 2' 3'

1' 2' 1' 2' 3'

Coincedence1" 2" 1" 2" 3'between 2" and 3"

1| |

21

21 1

2 2 2 2

H H VV H V

H H H V V V

H H V V H H V V

H H V V

H H V V

1" 1" 2" 3'H V Through a noisy channel with bit-flip error rate pnew the remaining QBER will be

2

2

2 2

2o

3 1

pp

p p ～

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Experimental Set-up

Trigged by D4 possibility of two pair emission will be much lower

[Y.-A. Chen et al., PRL 96, 220504 (2006)]

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4

4

cos(2 ) sin(2 )HWP :

sin(2 ) cos(2 )

sin( ) cos( ) 0 sin( ) cos( )QWP :

cos( ) sin( ) cos( ) sin( )0

1 cos(2 )2 cos( )sin( )

21 cos(2 )

2 cos( )sin( )2

i

i

i

e

e

ii

ii

By one HWP inside two QWP, any U-transmit can be implemented!

Bit-flip-error simulation

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i

cos(2 ) sin(2 )( ) ( ) ( )

sin(2 ) cos(2 )2 2

.

cos2

( ) ( ) ( ) ( )2 2

sin e2

1 1( ) ( ) ( ) ( )

2 2

1 co1

2

iQWP HWP QWP

i

Now set the angel of the HWP to and

if QWP HWP QWP

we can get

s(4 )cos( ) sin( ) cos( ) cos(4 ) sin( )

sin( ) cos( ) cos(4 ) sin( ) 1 cos(4 )cos( )

i

i

2 2

i

1 cos(4 )cos( ) sin( ) cos( ) cos(4 ) sin( )1

sin( ) cos( ) cos(4 ) sin

cos

(

(2 ) sin (2 )

cos2

sin e

) 1 cos(4 )cos

2

( )2

u u v v v u

if u we can g

i

i

et

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Quantum Noisy Channel

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Experimental Results

[Y.-A. Chen et al., PRL 96, 220504 (2006)]

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• The phase-shift error rejection can be realized. 1 1

0 0 1 , 1 0 12 2

phase shift error can be changed to bit flip error

| | | | | |

| | | | | |Phase shift

H V H V H V

H V H V H V

| |

| |

H V

H V

|

|

H

V

| |

| |

H V

H V

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• The higher order bit-flip error can be rejected.

encoding unknown quantum states into higher multi-photon entanglement (N), the higher order (up to N-1) error can be rejected

| |H V

| ... | ...HH H VV V

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• Applied to the quantum key distribution

the threshold of tolerable error rate overthe quantum noisy channel can be greatly improved. [X.-B. Wang, PRL 92, 077902 (2004)]