Inhomogeneous One-Dimensional Spin Models For Solving Problems Of Quantum Information Theory E.B. Fel dman

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1 Istitute of Problems of hemical Physics of Russia Academy of Scieces Ihomogeeous Oe-imesioal Spi Models For Solvig Problems Of Quatum Iformatio Theory E.B. Fel dma

2 OUTLIE Applicatios of ihomogeeous spi models i quatum iformatio processig (QIP). iagoalizatio of the XY-Hamiltoia of the simple ihomogeeous ope spi chais. Eergetic spectra of homogeeous ad alteratig ope spi chais. The ideal trasfer of quatum states from oe ed of a chai to the other ed. The high probability state trasfer alog a ihomogeeous ope chai. Separate ad etagled states i quatum-mechaical systems. The etaglemet i alteratig ope spi chais ad log distace etaglemet. Fluctuatios of quatum etaglemet. oclusios.

3 Alteratig hais Homogeeous XY-models ) I I I (I I ω H y y x x z ) I I I (I ω I H y y x x z eve. ω odd ω ω ;... ;...

4 iagoalizatio of the Hamiltoia of homogeeous chais. The mai ideas.. Jorda-Wiger trasformatios / c c I c I Π ) ( ii I I c I Π ) ( ii I I z lz l l y x lz l l y x. The fermio represetatio of the Hamiltoia

5 3. The classical solutio for homogeeous chais u ( ) jasi jp j... Eigestates: u j / πj si Eigevalues: ε 0 cos ω π...

6 Solutio for Alteratig hais - odd u A si ( p ).... p π - eve u u A A si ( p ) si p. 0 E.B.Fel dma M.G.Rudavets JETP Letters 8 47(005) E.I.Kuzetsova E.B.Fel dma JETP0 88 (006)....

7 δ si p si p 0 δ sh q sh p / π iq q δ < ( )/ (δ /) δ > ( )/

8 - odd.... 5)/ ( 3)/ ( ) ω (ω ω ω )/ ( ω )/...( ) ω (ω ω ω λ cos π Eigestates Ad Eigevalues for Alteratig hais (-j)/ j()/ / B(-δ) j35... u 0 j4...- δ - B. δ j ) (j si ) (j δsi ω λ A j j A si j u π π π

9 - eve... / cosp ) ω (ω ω ω /... cosp ) ω (ω ω ω λ.../) ( p p

10 )/ ( δ > )p ( B si 4... p A si u )/ ( δ < / ω λ ω λ p si )p ( si - A... ω λ ω λ ) ( A B )q ( sh B (-) 4... q sh A (-) u )/ (- / / ω λ ω λ q sh )q ( sh A

11 Model of quatum register o the basis of periodic ope spi chais 3 ω ω ω3 ω ω ω ω ω d ωd XY-Hamiltoia of a geeral ope liear spi chai i the earest-eighbour approximatio is ( ) H ω I I I I I j jz j j x j x j y j y j j Jorda-Wiger trasformatio reduces the study of Hamiltoia H to the diagoalizatio of a three-diagoal matrix: H ω 0 L 0 ω O 0 0 ω O M M O O O ω 3

12 SPETRA FOR LIEAR HAIS ε ε < 0 ε < 0 ε > 0 the gap ε < 0 ε ε ε 3 < < 0 0 Homogeeous chais ε cos π ( >>) ε ± Alteratig chais ( > ) π cos

13 Perfect State Trasfer i Quatum Alteratig hais -itλ P(t) exp( iht) u u e j j j { { Perfect State Trasfer is possible for at the followig δ : 4 δ whole oegative umbers. (δ / )

14 High probability quatum state trasfer alog the ope alteratig spi chai with the XY-Hamiltoia E.I.Kuzetsova A.I.Zechu Physical Letters A (008)

15 ETAGLE STATES ψ ( ψ ( Spi-flip trasformatio: ψ spi flip ) - the separable state ) ( ) spi flip - the etagled state ~ ψ σ σ ψ ~ ψ ( y ( ψ ) 0 ( ψ ) y ). ocurrece of the pure states: ( ψ ) ψ ~ ψ

16 The measure of the etaglemet of pure states: Φ( ) E( ψ ) Φ( ( ψ )); h h( x) xlog x ( x)log( x). ; E( ρ) Φ( ( ρ)); Wooters criteria for mixig states: ( ρ) max{0 λ λ λ3 λ4}; R ~ ρ ; ρ ρ ~ ( σ y σ y ) ρ * ρ ( σ y σ y ); λ λ λ3 λ 4.

17 Ed to ed cocurrece for log ad δ δ

18 Ed to ed cocurrece for log ad δ δ

19 Ed to ed cocurrece for log ad δ δ

20 Quatum etaglemet ad its fluctuatios (E.B.Fel dma M.A.Yurishchev Pis ma v JETP 90 75(009)) Alice ψ Bob LO: m E Pure states: E S ρ ) S( ρ ) ( A B ρ A Tr ψ ψ ρ B B Tr A ψ ψ { ρs( ρ) } S( ρ) log ρ ( ρ { ρ ρ }) S( ρ) Tr A B

21 ( ) [ ] / S S E d c b a d c b a ψ Eigevalues: ( ) bc ad B A ± : λ ρ ρ Momets: log log ) ( S Etaglemet: ) ( )log ( log ) ( ) ) / (( x x x x x H H E

22 ( ) log E EE: ( ) ( ) f f f f f l l f At 0<< f the system is i the regio of the strog fluctuatios whe E>E.

25 Mixed states ρ i ψ i ψ i pi 0 pi E i p i i i i { p Ψ } mi p S( Ψ ) ε ε i i E [ { } ] / 0 0 S Ψ ) S( Ψ ) ( i i { λ λ λ 0} max 3 4 λ λ i - eigevalues λ λ λ λ 0) of the matrix ( 3 4 R ( ) ( ) σ ρ * σ σ ρ σ y y y y

26 The Heiseberg dimer σ σ r r J H ( ) T H Z T B / exp ) ( ρ < e e B T T T T T J T 0 ) / 3exp( ) ( B e J T l 3 E(T) E(T) at the temperature T f : ) 3( l / f f f B J T

29 OLUSIOS Applicatios of ihomogeeous spi models allows to solve the ew problems of quatum iformatio processig (QIP). The XY-Hamiltoia of the some ihomogeeous ope spi chais ca be diagoalized exactly. The groud state of some ihomogeeous systems is separated by a gap from excited states. The high probability state trasfer alog a ihomogeeous ope chai icreases possibilities of quatum iformatio chaels. Log distace etaglemet emerges i ope alteratig spi chais with eve umbers of spis. Fluctuatios of quatum etaglemet ca be very importat i order to predict possibilities of differet materials to be used as registers of quatum computers.

30 Acowledgemets. M.A.Yurishchev. A.I.Zechu 3. E.I.Kuzetsova

C191 - Lecture 2 - Quantum states and observables

C191 - Lecture 2 - Quantum states and observables C191 - Lecture - Quatum states ad observables I ENTANGLED STATES We saw last time that quatum mechaics allows for systems to be i superpositios of basis states May of these superpositios possess a uiquely