Jian-Wei Pan

Transcription

1 Lecture Note

2 open system dynamics 0 E 0 U ( t) ( t) 0 E ( t) E U 1 E ( t) 1 1 System Environment U() t ( ) E 0 E ( t) + 1 E ( t) * 0 01 E1 E0 q() t = TrEq+ E = * E0 E1 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?

3 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

4 QC based on Decoherence- free Subspace Error-free Transfer in QC

5 e g + g e U (t ) e i e t e e i g t g + e i g t g e i e t e = e i ( e + g )t ( ) e g + g e

6 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 1997 [Phys. Rev. Lett. 79, 1953 (1997); Phys. Rev. Lett. 79, 3306 (1997); Phys. Rev. Lett. 81, 2594 (1998)] Symmetrization/Bang-bang methods [Phys. Rev. A 58, 2733 (1998); 1998 Phys.Lett. A 258, 77 (1999) ] Robustness to perturbing error processes [Phys. Rev. Lett. 81, 2594 (1998); Phys. Rev. A 60, 1944 (1999)] [Phys. Rev. Lett. 82, 4556 (1999)] [Phys. Rev. A (R) (1999)] [Phys. Rev. Lett. 84, 2525(2000)] How do we perform quantum communication in a DFS?

7 Collective Rotation NoiseNoise can be seen as some unitary transformation as U(,), if for all the channel, the unitary is the same, then it is called collective noise. If is 0, i.e., UU(), it is called collective rotation noise U (,): H Cos H + e i Sin V ( ) = 1 2 H V V H 1 2 V e i Sin H + Cos V (( Cos H + ei Sin V )( e i Sin H + Cos V ) ( e i Sin H + Cos V )( Cos H + e i Sin V )) 1 2 (( Cos2 + Sin 2 ) H V ( Cos 2 + Sin 2 ) V H ) = 1 ( 2 H V V H )

8 + = 1 ( 2 H H + V V ) 1 2 U ( ): H Cos H + Sin V V Sin H + Cos V (( Cos H + Sin V )( Cos H + Sin V )+ ( Cos V Sin H )( Cos V Sin H )) 1 2 (( Cos2 + Sin 2 ) H H + ( Cos 2 + Sin 2 ) V V ) ( ) = 1 2 H H + V V [P. G. Kwiat et al., Science 290, 498(2000); J. B. Altepeter, et al., Phys. Rev. Lett. 92, (2004)]

9 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 290, 498(2000);

10 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.

11 [Q. Zhang et al., PRA 73, (R) 2006]

12 QBER of DFS and traditional BB84 under the collective rotation noise. > /8, QBER BB84 >11%

13 HG01 HG10

14 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

15 + = 1 ( 2 H V + V H ) { ( Cos H + e i Sin V ) e i Sin H + Cos V ( ) + ( e i Sin H + Cos V )( Cos H + e i Sin V )} ( Cos 2 Sin 2 ) ( H V + V H )+ CosSin ( e i e )( i H H + V V ) CosSin ( e i + e )( i H H V V )} ( )+ 3 ( HH VV ) { = 1 ( HV + VH )+ 2 HH + VV U (,): H Cos H + e i Sin V V e i Sin H + Cos V ( = 1 )

16 First apply a time delay between H and V, the state will be HV + VH HV + V H After a collective noise T T HV T + V T H 2 { HV T V H T + ( 1 HV T + V H T )+ 2 { 2 V T 2 ( HH T + V V T )+ ( 3 HH T V V T )} + H H T V + ( 1 V T H + H T V )+ ( H H + V T T V )+ 3 H T H V T V ( )} Bob can measure in any direction (H /V ) which also can be considered as part of the collective noise.

17 Then again, Bob apply a time delay between H and V, the state will be 2 { + 2 H T 2 { 2 V T V H TT + ( 1 H T V T + V H TT )+ ( H H T TT + V V T )+ ( 3 H T H TT V V T )} V T H TT H T V + ( 1 V T H + H T TT V )+ ( H H TT T + V T V )+ 3 ( H TT H T V T V )} The last operation is to project the state onto the subspace in which the photons arrive exactly at the same time

18 We will get with a probability 1/3 by a random unitary transformation

19 Experimental Setup [T.-Y Chen et al., Phys. Rev. Lett (2006)]

20 4m fiber Experimental Result without random rotations with random rotations average QBER

21 1km fiber without random rotations Experimental Result with random rotations average QBER [T.-Y Chen et al., Phys. Rev. Lett (2006)]

22 QC based on Decoherence- free Subspace Error-free Transfer in QC

23

24 two bits flipping (p 2 ) can t be corrected CNot Operation Required!!! [D. Bouwmeester, PRA 63, (R) (2001).]

25 1234 = ( ) 1 ( ) ( ) Bell Measurement 00 Between 1 & ( 0 + 1) No coincedence ( ) ( 0 + 1)+ ( 0 1)

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

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

28 123 = 1 ( 2 H H + V V 2 ) ( H 3 + V 3 ) ( ) each photon in 1 the two arms of PBS 2 H H H + V 1' 2' 3' 1' V 2' V 3' = 1 ( 2 2 H H + V 1' 2' 1' V ) 2' + 3' + 1 ( 2 2 H H V 1' 2' 1' V ) 2' 3' ( ) + 3' Coincedence ( H 1" H 2" + V 1" V 2" ) + 3' ( ) + 2" + 3' 2 p 2 H 1' H 2' + V 1' V 2' between 2" and 3" H 1" + V 1" Through a noisy channel with bit-flip error rate pnew the remaining QBER will be ( ) 3 ( 1 p) 2 + p 2 o( ) p2

29 Experimental Set-up Trigged by D 4 possibility of two pair emission will be much lower

30 Bit-flip-error simulation By one HWP inside two QWP, any U-transmit can be implemented!

31 = cos 2 (2) u u + sin 2 (2) v v v = u f denotes bit-flip of u cos 2 if u = sin 2 ei we can get = 1 1+ cos(4)cos( ) sin( ) cos() icos(4) sin( ) 2 sin( ) cos() icos(4) sin( ) 1 cos(4)cos( )

32 Quantum Noisy Channel

33 Experimental Results [Y.-A. Chen et al., PRL 96, (2006)]

34 The phase-shift error rejection can be realized.

35 The higher order bit-flip error can be rejected. encoding unknown quantum states into higher multiphoton entanglement (N), the higher order (up to N-1) error can be rejected

36 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 92, (2004)]