Lecture Note 7 Decoherence-free sub-space space and quantum error-rejection rejection.06.006
open system dynamics ψ = α 0 + α 0 Decoherence System Environment 0 E 0 U ( t) ( t) 0 E ( t) E U E ( t) U() t ( ) α 0 + α E α 0 E ( t) + α E ( t) 0 0 0 * α0 α0α E E 0 ρq() t = TrEρq+ E = * αα 0 E0 E α 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?
Possible solutions to overcome decoherence in long-distance quantum communication (QC) 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
QC based on Decoherencefree Subspace Error-free Transfer in QC
Decoherence-free subspace (DFS) U() t iω t iωgt e + g e e e + e g e g + g e U() t iω t iω t iω t iω t + e e e e g g e g g e e e = i( ω ) e+ ωg t e e g + g e ( )
Decoherence Free Subspace General Definitions, Collective Decoherence 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 997 [Phys. Rev. Lett. 79, 953 (997); Phys. Rev. Lett. 79, 3306 (997); Phys. Rev. Lett. 8, 594 (998)] Symmetrization/Bang-bang methods [Phys. Rev. A 58, 733 (998); 998 Phys.Lett. A 58, 77 (999) ] Robustness to perturbing error processes 000 999 DFS History [Phys. Rev. Lett. 8, 594 (998); Phys. Rev. A 60, 944 (999)] [Phys. Rev. Lett. 8, 4556 (999)] [Phys. Rev. A 60 79(R) (999)] [Phys. Rev. Lett. 84, 55(000)] How do we perform quantum communication in a DFS?
DFS under Collective Noise 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 iφ U( θφ, ) : H Cosθ H + e SinθV iφ ψ = ( H V V H ) V e Sinθ H + Cosθ V ( iφ )( iφ Cosθ H + e Sinθ V e Sinθ H + Cosθ V ) ( iφ )( iφ e Sinθ H Cosθ V Cosθ H e Sinθ V ) + + (( ) ( Cos θ + Sin θ H V Cos θ + Sin θ) V H ) = ( H V V H )
+ φ ( ) : DFS under Collective Rotation Noise U θ H Cosθ H + Sinθ V V Sinθ H + Cosθ V = ( H H + V V ) ( Cosθ H + Sinθ V )( Cosθ H + Sinθ V ) + ( Cosθ V Sinθ H )( Cosθ V Sinθ H ) + + + = ( H H + V V ) (( ) ( Cos θ Sin θ H H Cos θ Sin θ) V V ) [P. G. Kwiat et al., Science 90, 498(000); J. B. Altepeter, et al., Phys. Rev. Lett. 9, 4790(004)]
DFS for Collective Rotation Noise ψ + φ = = ( H V ( H H + V V H V ) ) 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. [P. G. Kwiat et al., Science 90, 498(000);
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. Application in quantum key distribution using a DFS H + = φ = ( H H + V V ) V = ψ = ( H V V H ) + = ( H + V ) = ( H + V = ( H V ) = ( H + V [X.B.Wang, Phys. Rev. A 7, 050304(R) (005)] + ) )
Experimental Setup [Q. Zhang, PRA 73, 0030 (R) 006]
Experimental Result QBER of DFS and traditional BB84 under the collective rotation noise. θ > π/8, QBER BB84 >%
Drawback DFS only for Collective Rotation Noise Other noise Free space phase drifting caused by temperature difference Long distance in optical fibers will cause a redoubtable obstacle Noise not only in H/V basis!
( ) iφ U θφ, : H Cosθ H + e SinθV iφ V e Sinθ H + Cosθ V Collective Noise ψ + = + ( H V V H ) iφ iφ ( Cosθ H + e Sinθ V )( e Sinθ H + Cosθ V ) + iφ iφ ( e Sinθ H + Cosθ V )( Cosθ H + e Sinθ V ) iφ iφ ( Cos θ Sin θ)( H V + V H ) + CosθSinθ( e e )( H H + V V ) iφ iφ CosθSinθ( e + e )( H H V V ) = δ( HV + VH ) + δ( HH + VV ) + δ3 ( HH VV ) ( δ + δ + δ = ) 3
A new protocol First apply a time delay between H and V, the state will be α HV + β VH α HV + β V H T T After a collective noise α HV T + β V H T ( ) + β + ( ) ( ) α α HV + β VH = ψ + ψ + ψ + ψ α HV T VH T + δ HV T + VH T + + δ ( ) ( ) HHT VVT δ3 HHT VV + + T β VH ( ) T HV T + δ VH T + HV T + δ ( ) ( ) H TH + V TV + δ3 H TH V TV Bob can measure in any direction (H /V ) which also can be considered as part of the collective noise.
A new protocol Then again, Bob apply a time delay between H and V, the state will be ( ) α HV T T VH TT + δ HV T T + VH TT + δ T TT T 3 T TT T β VH ( ) T T H TTV + δ VH T T + H TTV + + δ TT T T 3 TT T T ( H H + VV ) + δ ( H H VV ) ( H H + V V ) + δ ( H H V V ) The last operation is to project the state onto the subspace in which the photons arrive exactly at the same time + δ δ ( ) α HV ( ) T T + β VH T T + α V H TT + β H TTV + δ + δ3 δ δ3 T TT TT T T T ( α H H + β H H ) + ( α VV + β V V )
A new protocol We will get probability α H V + β V H T T T T ( + δ ) / with a /3 by a random unitary transformation
Experimental Setup [T.-Y Chen et al., Phys. Rev. Lett. 96 50504 (006)]
4m fiber Experimental Result without random rotations with random rotations average QBER
km fiber without random rotations Experimental Result with random rotations average QBER [T.-Y Chen et al., Phys. Rev. Lett. 96 50504 (006)]
QC based on Decoherencefree Subspace Error-free Transfer in QC
Bit-flip Error Correction α 0 + β CNot ψ3 = α 3 + β 3 3 0 0 α 0 + β α 000 + β ( 000 ) with a probability poccurs a bit-flip error ( α β ) U ( α β ) ( ) ( ) α00 + β 0 + 0 0 + CNot α 00 + β 0 α 0+ β 0 α 00 β0 + α 0 + β 0 two bits flipping (p ) can t be corrected CNot Operation Required!!! [D. Bouwmeester, PRA 63, 04030(R) (00).]
Error-free transfer ψ = α + β + ( 0 ) ( 000 ) 34 34 34 Bell Measurement Between & ( α 00 34 β 34 ) ( ) ( ) + α 00 34 + β 34 + α 0 + β + α 0 β α 0 34 + β 0 34 No coincedence α 0 34 + β 0 34 α 34 β 00 34 α β + + ( 0 + ) + ( α 0 β )
Problem in Experimental Realization Possibility of two pair emission is in the same order and will cause four-fold coincidence!
Error-free transfer [X.-B. Wang, PRA 69, 030 (004)]
ψ ( ) ( ) 3 = HH + VV α H3 + β V3 each photonin ( α HHH ) the two arms of PBS ' ' 3' + βvvv ' ' 3' = ( α H H + βvv ) + + ( α H H βvv ) ( α HH βvv ) Coincedence between "and3" ' ' ' ' 3' ' ' ' ' 3' + + ' ' ' ' 3' ( α HH βvv ) + + ( α H βv ) + + + " " " 3' " " " " 3' Through a noisy channel with bit-flip error rate pnew the remaining QBER will be ( ) p ( ) p + p 3 ~ o ( p )
Experimental Set-up Trigged by D 4 possibility of two pair emission will be much lower [Y.-A. Chen et al., PRL 96, 0504 (006)]
Bit-flip-error simulation HWP : QWP : cos( β) sin( β) i sin( β) cos( β) π + i 4 sin( α) cos( α) e 0 sin( α) cos( α) π cos( α) sin( α) = i cos( α) sin( α) 4 0 e + i cos( α) i cos( α)sin( α) i cos( α) i cos( α)sin( α) By one HWP inside two QWP, any U-transmit can be implemented!
= u u + v v v = u ρ cos ( β) sin ( β) if u ρ = γ cos = we can get γ iϕ sin e π π cos( β) i sin( β) QWP( ) HWP( β ) QWP( ) = i sin( β) cos( β) Now set the angel of the HWP to β and β. γ cos π π if ψ( β) = QWP( ) HWP( β) QWP( ) γ iϕ sin e we can get ψ( β) ψ( β) + ψ( β) ψ( β) = + cos(4 β )cos( γ) sin( γ) [ cos( ϕ) i cos(4 β) ] sin( γ) sin( γ) [ cos( ϕ) i cos(4 β) ] sin( γ) cos(4 β)cos( γ) [ i β ] + cos(4 β)cos( γ) sin( γ) cos( ϕ) cos(4 ) sin( γ ) sin( γ )[ cos( ϕ) i cos(4 β) ] sin( γ) cos(4 β) cos( γ) +
Quantum Noisy Channel
Experimental Results [Y.-A. Chen et al., PRL 96, 0504 (006)]
The phase-shift error rejection can be realized. 0 ( 0 + ), ( 0 ) phase shift error can be changed to bit flip error α H + β V = α+ β H + V + α β H V ( )( ) ( )( ) Phase shift α+ β H V + α β H + V = α H β V ( )( ) ( )( ) ( α + β)( H + V ) ( α β)( H V ) + ( α β) ( α β) + H + V ( α + β)( H + V ) ( α β)( H V ) +
The higher order bit-flip error can be rejected. encoding unknown quantum states into higher multiphoton entanglement (N), the higher order (up to N-) error can be rejected α H + β V α HH... H + β VV... V
Applied to the quantum key distribution the threshold of tolerable error rate over the quantum noisy channel can be greatly improved. [X.-B. Wang, PRL 9, 07790 (004)]