Entanglement Purification
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1 Lecture Note Entnglement Purifiction Jin-Wei Pn 6.5.
2 Introduction( Both long distnce quntum teleporttion or glol quntum key distriution need to distriute certin supply of pirs of prticles in mximlly entngled stte to two distnt users.
3 Introduction( owever, distriuted quits will interct with the environment nd decoherence will hppen. E U ( t ( t E ( t E U E ( t ere, represents the quit stte nd E represents the environment initil stte, U (t is the joint unitry time evolution opertor. or ritrry quit stte: ( U ( t α α E α E( t α E ( t ρ * α αα E E q( t TrEρq E * αα E E α The off-digonl element of the quit density mtrix will drop down with the Γ(t Γt rte E ( t E( t e, depends on the coupling etween quit nd environment. The mximlly entngled stte will e in some mixed stte with certin entnglement fidelity due to the process.
4 Introduction( Solution to the decoherence Quntum Error Correction for Quntum computtion Quntum Entnglement Purifiction for Quntum Communiction Quntum Communiction sed on Decoherence free Suspce The sic ide of entnglement purifiction is to extrct from lrge ensemle of lowfidelity EPR pirs smll su-ensemle with sufficiently high fidelity EPR pir. Entnglement Purifiction----improve purify of ny kind of unknown mixed stte Locl filtering----improve entnglement degree for known stte Entnglement Concentrtion---- improve entnglement degree for unknown stte
5 Principle of Entnglement Purifiction Model: Suppose Alice wnt to shre n ensemles of -quit mximlly entngled sttes with Bo vi noise chnnel. After the trnsmission, the stte hs een chnged into generl mixed stte M, the purity of M cn e expressed s. Severl ingredients in the Entnglement Purifiction: ( Bell sttes:, ; ( Werner stte: ; ( Puli rottion:,, ; ( CNOT gte:, / ( M x σ z σ σ y ( ± ± ( ± ± W y x x y x
6 Principle of Entnglement Purifiction Steps of Entnglement Purifiction: ( Rndom Bilterl Puli Rottion on ech photon in the sttes. This step cn chnge ritrry mixed stte into Werner stte: W σ y ( A Unilterl Rottions converting the sttes from mostly Werner sttes to the nlogous mostly sttes, ( σ mps ±, ( Bilterl CNOT opertions on two photon pirs in the Werner stte. y [C. Bennett, et l., PRL 76, 77 (996]
7 Bilterl CNOT opertions will convert the Bell sttes s the form: or exmple: (-/ Proility, we hve CNOT ( S T CNOT[( SA CNOT ( SA ( SA TA SB S T SB TA TB SB S SA TB SA SB T ( TA SA TA SB TA TB TB SB TA TB SA TB TA ] SA SB TA TB SB TB SA TA SA SB TA TB SB TB
8 Principle of Entnglement Purifiction ( Mesuring the trget pir in z sis, if the result is prllel, keep the source pir, if not, discrd the source pirs. After the protocol, the purity of the source pir hs een improved: If >/, then > i severl this kind processes, we cn purify generl mixed stte into highly entngled stte. [C. Bennett, et l., PRL 76, 77 (996]
9 Experiment of Entnglement Purifiction The theoreticl proposl needs C-Not gte which requires high non-linerities. Unluckily, no efficient C-Not gte exists t the moment A etter solution for experiments [J.-W. Pn et l., Nture, 67 (]
10 ( ( ( t s t s t s t s Initil Stte: Purified Stte: ( ρ ( ρ / (if ( > >
11 ( ( ( ( > > > > > our-fold events No four-fold events or the first cse, 5% / proility of
12 Similrly, ( 5% No four-fold events proility of ( / ( In this wy ρ ( > ( if > / (
13 Experimentl Reliztion [J.-W. Pn et l., Nture, 7 (]
14 contriution contriution our-fold contriution from doule pirs emission ( > > > > > [J.-W. Pn et l., Nture, 7 (]
15 To keep the phse stle, we use the polriztion-sptil entnglement ( ( ( Two-fold coincidence per second Time dely (μm Two-fold coincidence per 5 seconds Piezo position (nm
16 Experimentl Result Before purifiction, /.5.5. >> >>. 5 o >5 o > -5 o >-5 o > rction. rction... >> >> 5 o >-5 o > -5 o >5 o >.... After purifiction, /.5 >> >>.5 5 o >5 o > -5 o >-5 o >.. rction. rction.... >> >>. 5 o >-5 o > -5 o >5 o >.. [J.-W. Pn et l., Nture, 7 (]
17 Locl filtering The purifiction protocol is such wste for the photon pir sources. When we hve known some informtion of the stte, there is some more efficient method to improve the purity. ε or exmple, ( ε / ε ( is non-mximlly ε entngled stte, cn e converted into the mximlly entngled stte ( / y sujecting one of the quits to generlized mesurement filtering process: ε,. This process is clled locl filtering. It cn only e used in known stte nd cn only improve the entnglement degree. [P. Kwit et l., Nture 9, (]
18 Locl filtering Any inseprle two spin-/ system Mtrices Cn Be Distilled to Singlet orm with locl filtering nd entnglement purifiction A quntum system is clled inseprle if its density mtrix cnnot e written s mixture of product sttes: [M. orodecki, et l., PRL 78, 57 (997]
19 [P. Kwit et l., Nture 9, (]
20 Locl iltering nd idden Non-loclity or the stte ρε, λ λ ε ε ( sometimes it cn not violte Bell inequlity. After the Locl iltering process, the stte is chnged into the stte ( λ ε λ ε which cn violte the Bell inequlity. ε ( λ ( [P. Kwit et l., Nture 9, (]
21 Entnglement Concentrtion---Scheme ψ T ( α β ( α β R9 ( α β ( α β αβ. α PBS ψ ψ ( αβ. ( β [C.. Bennett et l., Phys. Rev. A 5, 6 (996] [Z. Zho et l., Phys. Rev. A6, (] [T. Ymmoto et l., Phys. Rev. A6, (]
22 Experiment Reliztion [Z. Zho et l, Phys. Rev. Lett. 9, 79(.] [T. Ymmoto et l., Nture, (]
23 Difficulties in Long-distnce Quntum Communiction Due to the noisy quntum chnnel ( sorption photon loss ( decoherence degrding entnglement qulity Solution to prolem (: Entnglement swpping! [N. Gisin et l., Rev. Mod. Phys. 7, 5 (] Solution to prolem (: Entnglement purifiction! [C.. Bennett et l., Phys. Rev. Lett. 76, 7 (996] [D. Deutsch et l., Phys. Rev. Lett. 77, 88 (996]
24 The Kernel Device for Long Distnce Quntum Communiction Quntum repeters: Require [. Briegel et l., Phys. Rev. Lett. 8, 59(998] entnglement swpping with high precision entnglement purifiction with high precision quntum memory
25 A proof-in-principle demonstrtion of quntum repeter [Z. Zho et l, Phys. Rev. Lett. 9, 79(.]
26 Results for Repeter S.58 ± ±. Stndrd Devition S ±.5 ±.8 Stndrd Devition S 5..8 ±.. ±.8 Stndrd Devition idelity.96 ±..9±..9±. [Z. Zho et l, Phys. Rev. Lett. 9, 79(.]
27 Drwck in ormer Experiments Asence of quntum memory Proilistic entngled photon source Proilistic entnglement purifiction Quntum memory uge photon loss in fier ree-spce entnglement distriution - we re working on km nd 5km scle Synchroniztion of independent lsers - we re working on entnglement swpping [T. Yng et l., PRL 96, 5 (6]
28 Drwck in ormer Experiments P / P In multi-stge experiments, the cost of resource is proportionl to N N/ P thus not sclle If one knows when the photon pir is creted nd the entngled pir cn e stored s demnded, the totl cost is then N/P P
29 Another Solution---- Quntum Communiction sed on Decoherence free Suspce or specil noise, we cn utilize some entnglement suspce to directly implement quntum communiction. or exmple, the phse flip error chnnel:, iφ e The Bell stte ψ, ψ re immune to the noise: ( / i iφ ( e e / i e ( ( / i iφ ( e e / i e ( / / We cn tke nd, Ech stte comined y the two sis cn e used for decoherence free communiction. The suspce is clled decoherence free suspce. [Q. Zhng et l., PRA 7. (R (6] [T.-Y. Chen et l., PRL 96, 55 (6]
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Clssicl Verifiction of Quntum Proofs rxiv:1505.0743v1 [qunt-ph] 7 My 015 Zhengfeng Ji Institute for Quntum Computing nd School of Computer Science, University of Wterloo, Wterloo, Ontrio, Cnd Stte Key
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