Preserving Entanglement via Quantum Error CorrectionXinyu Zhao, Samuel R. Hedemann, and Ting Yu

Center for Controlled Quantum Systems and the Department of Physics and Engineering Physics,Stevens Institute of Technology, Hoboken, New Jersey 07030, USA Stevens Innovation Expo

IntroductionQuantum coherence and quantum entanglement will deterioratewhen the system is coupled to its environment. In this project,we propose a new error-correction scheme based on the randomunitary (RU) decomposition of the quantum channel. We show thatquantum coherence and entanglement can be recovered bymeasuring the environment.

Advantages of our scheme:1. Deterministic process, (ideally 100% successful probability).2. High fidelity (ideally, the fidelity is 1).3. No extra quantum resources needed.4. Restoration operations are unitary.

Our project: Environmental assisted quantum error correction and entanglement preservation

1-qubit dephasing bathHamiltonian:Two types of Kraus operators:1. If m is odd2. If m is even

RU decomposition:

† †( )2 z k k k k k k k k kH b b g b b

| | 0 oddm zm U F

| | 0 evenmm U F I

21 | |odd

m m zK F 2

2 | |m mevenK F I

System initial state(0)S

Env. initial state vacuum | 0

Total evolved state

†0 (0)| 0 |S UU

Dephasing channel

†( )I k k z k kH g b b

Parity is odd or even?

Measure the parity of photon numbers

1 ||odd

mmM m

Measurement outcome

odd

System collapse into†

1 1(0)SK K

System collapse into†

2 2(0)SK K

even

Final state† †

1 1 1 1

† †2 2 2 2(0)

)

(

(0

0)

S

S

S

R K K

R K

R

K R

1 zR

2R I

Restoration

†2

1( ) (0)S i S iit K K

Error correction procedure

Merits of our scheme:High fidelity (ideally 1), high successful probability (ideally 100%), unitary restoration operation.

2-qubit common bathHamiltonian: , Non-RU decomposition:

Entanglement restoration:Unknown initially entangled state:Using the corresponding restoration operations:

The initial state can be fully recovered as

RU decomposition:

Measurement basis:Starting from a known set of basis (Fock basis)

The measurement basis for can be constructed by unitary transformation. can be treated as zero approximately. (see Fig. 2)

† †( )2 z k k k k k k k k kS SH b b g b b 1/ 2( )A B

z z zS

,

3 †

1 ,( ) (0)S C i S Ci iLt L

,1 1 1{ ( ),1,1, ( )}CL diag l t l t

,2 2 ( ) {1,0,0, 1}CL l t diag

,3 3( ) {1,0,0,1}CL l t diag

01

'

0( ) exp[ ( , )' ]'

t tt s dl t dt s

22 ( ) | |odd

mml t F

23( ) | |m

even

ml t F

|11| | 00

1,1 1 {1,1,1,1}C l diagR 1

2,2 {1,1,1, 1}C l diagR 13,3 {1,1,1,1}C l diagR

4 3

1 , 1

† †, , ,( ) (0) (0)S C i S C i C i Si i C iLKt K L

,1 1 {1,1,1,1}CK x diag

,2 2 { 1,1,1,1}CK x diag

,3 2 {1,1,1, 1}CK x diag

,4 4 {1, 1, 1,1}CK x diag

|' | 0m mL U

,C nK

N-qubit common bath – RU decompositionRandomly choose basis, like

The RU-type Kraus operators can be chosen as:

According to the relation,

we can determine the coefficients .

1 ( 1) / 2kN N N

1 {1,1,...,1}B diag {1, 1,..., 1}kN

B diag 1 { 1,1,..., 1}B diag

†

1(0) ( )* (0)k

i S i S

NK K C N

( 1 )i i i kiK c B to N

ic

Conclusion• RU decompositions for N-qubit systems are explicitly constructed.• Quantum coherence and entanglement can be recovered by

measuring Fock basis and performing a unitary operation.• Experimental realizations can be made with the existing

technologies.

References1. Xinyu Zhao, Samuel R. Hedemann, and Ting Yu, to be submitted.

2. M.Gregoratti and R. F. Werner, J. Mod. Opt. 50, 915 (2003).

† †, , , ,| | | |, ( 1, 2,3)C i C i C i C iR L L R i

2 ||even

mmM m

N-qubit separable bathsKraus operators are the tensor products of single qubit sbu-systems.

where are the Kraus operators for the total system in the case of separable baths, are the Kraus operators of the sub system.

, 2 )( 1 NS nK n to

, ii jK ij th i th

, 1, 1 2, 2 ,...S n j j N jNK K K K

, 'C n mm n mVK L

Fig. 1

Fig. 2

40 1

1| |n mnV m

4' ( 0)m mL

Arbitrary initial state can be fully recovered.

AcknowledgementWe acknowledge the grant support  from the NSF PHY-0925174 and

the AFOSR No. FA9550-12-1-0001.

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