Stevens Innovation Expo
-
Upload
noah-valencia -
Category
Documents
-
view
22 -
download
2
description
Transcript of Stevens Innovation Expo
![Page 1: Stevens Innovation Expo](https://reader037.fdocuments.net/reader037/viewer/2022103100/56813361550346895d9a767f/html5/thumbnails/1.jpg)
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.
1
2
3
4
5
6