EntanglementEntanglement CreationCreation inin OpenOpen Quantum SystemsQuantum Systems

Fabio BenattiFabio Benatti

Department of Theoretical PhysicsDepartment of Theoretical Physics

University of TriesteUniversity of Trieste

Milano, December 1, 2006Milano, December 1, 2006

In collaboration with In collaboration with R. FloreaniniR. Floreanini

OutlineOutline

OpenOpen quantum dynamics:quantum dynamics: dissipation and dissipation and decoherencedecoherence

Entanglement in open quantum systems: Entanglement in open quantum systems: creation and its persistencecreation and its persistence

F F. B., R. Floreanini, M. Piani: Phys. Rev. Lett. 91 (2003) 070402F F.B., R. Floreanini: Int. J . Mod. Phys. B 19 (2005) 3063F F.B., R. Floreanini: J . Opt. B 7 (2005) 5429F F.B., R. Floreanini: J . Phys. A 39 (2006) 2689F F. Benatti, R. Floreanini: Int. J . Quant. Inf. 4 (2006) 395

OpenOpen Quantum Dynamics Quantum Dynamics

qubits (S) qubits (S) in interaction with anin interaction with an environment (E): environment (E): heat bath,heat bath, external classical noiseexternal classical noise

weak coupling limit:weak coupling limit: Lindblad Lindblad equationequation

HHSS++EE == HH00SS ­­ 11EE ++ 11SS ­­ HHEE ++¸

PP®®VV®® ­­ BB®®

LLaammbb--sshhiifftteedd HH00SSNOISENOISEFRICTIONFRICTION

@@tt½½((tt)) == LL[[½½((tt))]]

== ¡¡ ii[[HHSS ;; ½½((tt))]]++ DD[[½½((tt))]]

DD[[½½((tt))]]==XX

ii jj

DDii jj££FFii ½½((tt))FFyy

jj ¡¡1122©©FFyy

ii FFjj ;; ½½((tt))ªª¤¤

Phenomenological CoefficientsPhenomenological Coefficients

Kossakowski matrixKossakowski matrix dissipative generatordissipative generator afterafter ergodic mean ergodic mean

DD ::== [[DDii jj ]] ¸ 00

VV®®((!! )) ==XX

EE aa ¡¡ EE bb==!!

PPaa VV®®PPbb HH00SS ==

XX

aa

EEaa PPaa

hh®® ((!! )) == ddtt eeii!! tt GG®® ((tt))R

VV®®((!! )) ==XX

ii

((TTrr((FFyyii VV®®((!! ))))FFii

DD[[½½]]==XX

®®;;¯

XX

!!

hh®® ((!! ))((VV®®((!! ))½½VVyy¯ ((!! )) ¡¡

1122ffVVyy

¯ ((!! ))VV®®((!! )) ;; ½½gg))

TTrr((FFyyii FFjj )) == ±±ii jj

GG®® ((tt)) == !! EE ((BB®®((tt))BB¯ ))

Physical ConsistencyPhysical ConsistencyComplete PositivityComplete Positivity

necessary necessary for thefor the physical consistency physical consistency of of

equivalent toequivalent to complete positivity complete positivity of of

for all entangled states of for all entangled states of

any n-level ancillaany n-level ancilla

DD ::== [[DD®® ]] ¸ 00

°°tt == eett LL ;; tt ¸ 00 ;; ½½77!! ½½((tt)) == °°tt[[½½]]

°°tt

°°tt ­­ iidd[[½½eenntt]] ¸ 00

SS ++ SSnn

SSnn

OneOne openopen 2-level atom2-level atom

SS(ystem)+(ystem)+EE(nvironment):(nvironment):

LindbladLindblad equation for equation for 1 qubit1 qubit density matrices: density matrices:

DD ==DD1111 DD1122 DD1133

DD2211 DD2222 DD1133

DD3311 DD3322 DD3333

¸ 00 ++

33XX

ii;;jj ==11

DDii jj [[¾¾ii½½((tt))¾¾jj ¡¡1122ff¾¾jj ¾¾ii ;; ½½((tt))gg]]

++¸33XX

ii==11

¾¾ii ­­ BBiiHHSS++EE == (! 0

2

3X

i=1

ni¾i)

| {z }H 0

S

­ 1E +1S ­ HE

@@tt½½((tt)) == ¡ i[HS ; ½(t)]

Two openTwo open 2-level atoms 2-level atoms

SS(1)+(1)+SS(2)+(2)+EE: : nono directdirect SS(1)(1)--S--S(2) (2) interactioninteraction

((!! 00

22

33XX

ii==11

nnii¾¾ii ))

|| {{zz }}HH 00

11

HHSS++EE ==

+1S1­ ­ 1E

­­ HHEE

((!! 00

22

33XX

ii==11

nnii¾¾ii

|| {{zz }}HH 00

22

))

­ (1S2­ 1E )

+1S1­ 1S2

++¸33XX

ii==11

¡¡¾¾ii ­­ 11SS22

)) ­­ BBii ++ ((11SS11­­ ¾¾ii )) ­­ BBii++11

LindbladLindblad forfor two open atoms two open atoms

++33XX

ii ;;jj ==11

BBii jj [[((¾¾ii ­­ 1122))½½((tt))((1111 ­­ ¾¾jj )) ¡¡1122ff¾¾ii ­­ ¾¾jj ;; ½½((tt))gg]]

++33XX

ii ;;jj ==11

BB¤¤jj ii[[((1111 ­­ ¾¾ii))½½((tt))((¾¾jj ­­ 1122)) ¡¡

1122ff¾¾jj ­­ ¾¾ii ;; ½½((tt))gg]]

++33XX

ii;;jj ==11

AAiijj [[((¾¾ii ­­ 1122))½½((tt))((¾¾jj ­­ 1122)) ¡¡1122ff¾¾jj ¾¾ii ­­ 1122 ;; ½½((tt))gg]]

++33XX

ii;;jj ==11

CCiijj [[((1111 ­­ ¾¾ii ))½½((tt))((1111 ­­ ¾¾jj )) ¡¡1122ff1111 ­­ ¾¾jj ¾¾ii ;; ½½((tt))gg]]

++DD[[½½((tt))]]

== ¡¡ ii[[HH11 ­­ 1122 ++ 1111 ­­ HH22 ++ HH1122 ;; ½½((tt))]]

DD[[½½((tt))]] ::

LLHH [[½½((tt))]]@@tt½½((tt)) == LL[[½½((tt))]]==

BByy == [[BB¤¤jj ii ]]

BB == [[BBii jj ]]

AA == [[AAiijj ]]

CC == [[CCii jj ]]

Environment Environment induced induced

interactioninteraction

¾(®) = ¾®­ 12 ®= 1;2;3

¾(®) = 11 ­ ¾®¡ 3 ®= 4;5;6

DD ==AA BBBByy CC

¸ 00CanCan

withoutwithout direct interactiondirect interaction generategenerate entanglemententanglement ? ?HH1122

¡¡ ii[[HH11 ­­ 1122 ++ 1111 ­­ HH22 ;; ½½((tt))]]++DD[[½½((tt))]]

@@tt½½((tt)) ==

DD[[½½((tt))]]==66XX

®®;;¯ ==11

DD®® [[¾¾((®®)) ½½((tt))¾¾((¯ )) ¡¡1122ff¾¾((¯ )) ¾¾((®®)) ;; ½½((tt))gg]]

Sufficient Condition Sufficient Condition ((F.B., R. Floreanini, M. Piani, PRL 2003)F.B., R. Floreanini, M. Piani, PRL 2003)

an initial an initial 2 qubit2 qubit separable separable statestate gets gets entangledentangled as soon as as soon as ifif

jÁ1ihÁ1j ­ jÂ1ihÂ1jtt >> 00

(Re(B))ij = 12(Bij +B¤

ij )

jjuuii ==hhÁÁ11jj¾¾11jjÁÁ22iihhÁÁ11jj¾¾22jjÁÁ22iihhÁÁ11jj¾¾33jjÁÁ22ii

;; ÁÁ11 ?? ÁÁ22 jjvvii ==hhÂÂ22jj¾¾11jjÂÂ11iihhÂÂ22jj¾¾22jjÂÂ11iihhÂÂ22jj¾¾33jjÂÂ11ii

;; ÂÂ11 ?? ÂÂ22

hujAjui hvjCT jvi < jhujRe(B)jvij2

Idea for proofIdea for proof

use use partial transposition partial transposition to check whetherto check whether

as well the the maps as well the the maps the mapsthe maps

form a form a semigroupsemigroup with generator with generator

id­ T

((iidd±±TT)) ±±°°tt[[jjÁÁ11iihhÁÁ11jj ­­ jjÂÂ11iihhÂÂ11jj]] ¸ 00

°°tt ;; tt ¸ 00 ;;

++RR[[½½((tt))]]¡¡ ii[[eeHH ;; ½½((tt))]]

RR[[½½]]==66XX

®®;;¯ ==11

QQ®® [[¾¾((®®)) ½½¾¾((¯ )) ¡¡1122ff¾¾((¯ )) ¾¾((®®)) ;; ½½gg]]

GG[[½½]]==

ggtt ::== ((iidd­­ TT)) ±±°°tt ±±((iidd­­ TT)) == eettGG

need need notnot be be positivepositive

need not preserve the positivity of

((iidd­­ TT))[[jjÁÁ11iihhÁÁ11jj ­­ jjÂÂ11iihhÂÂ11jj]]

need not be positive

((iidd­­ TT)) ±±°°tt[[jjÁÁ11iihhÁÁ11jj ­­ jjÂÂ11iihhÂÂ11jj]]

QQ ==AA RRee((BB))

RRee((BBTT )) CCTT

ggtt == ((iidd­­ TT)) ±±°°tt ±±((iidd­­ TT))

EÃ;Á1;Â1(t) := hÃjgt[jÁ1ihÁ1j ­ j¤

1ih¤1j]jÃi

TT[[jjÂÂ11iihhÂÂ11jj]]== jj¤¤11iihhÂÂ

¤¤11jj

<< 00@tEÃ;Á1;Â1(0) =

6X

®;¯ =1

Q® hÃj¾(®) (jÁ1ihÁ1j ­ j¤1ihÂ

¤1j)¾(¯ ) jÃi

EÃ;Á1;Â1(0) = jhÃjÁ1 ­ ¤

1ij2 = 0

and get and get entangledentangledjÁ1i ­ jÂ1i

EEÃÃ;;ÁÁ;;ÂÂ11((tt)) << 00 tt !! 00++

Particular Case:Particular Case:

choosechoose sufficient conditionsufficient condition becomes becomes

AA == BB == CC ¸ 00

ÁÁ11 == ÂÂ22 ==)) jjuuii == jjvvii

hhuujjAAjjuuiihhuujjAATT jjuuii << jjhhuujjRRee((AA))jjuuiijj22

((hhuujjII mm((AA))jjuuii))22 >> 00

AA == AAyy == RRee((AA)) ++ II mm((AA)) ;; <<((AA)) ==AA ++ AATT

22II mm((AA)) ::==

1122((AA ¡¡ AATT ))

AA ==aa11 ii bb 00¡¡ ii bb aa22 0000 00 aa33

;; aa11;;22;;33 ¸ 00 ;; aa11aa22 ¸ bb22 II mm((AA)) ==

00 ii bb 00¡¡ ii bb 00 0000 00 00

Example:Example:

jjÁÁ11ii == jjÂÂ22ii == jj¡¡ ii ;; ¾¾33jj§§ ii == §§ jj§§ ii jjuuii ==

11ii00

hhuujjIImm((AA))jjuuii))22 == 44bb22 >> 00

If ,

gets entangled for small times

bb66==00 jj¡¡ iihh¡¡ jj ­­ jj++iihh++jj

Two atomsTwo atoms in a in a scalar thermal field scalar thermal field in in equilibrium at inverse temperature equilibrium at inverse temperature

qubit 1qubit 1 and and qubit 2qubit 2 linearly coupled to linearly coupled to

full Hamiltonian:full Hamiltonian:

H01 = H0

2 =! 0

2

3X

i=1

ni ¾i

FF((xx))

hFi(x)Fj (y)i = ±ij G(x¡ y)

= ±ijd4k

(2¼)3µ(k0)±(k2)(

e¡ ik(x¡ y)

1¡ e¡ ¯ k0+

e+ik(x¡ y)

e k0 ¡ 1)

++¸33XX

ii==11

((((¾¾ii ­­ 1122)) ++ ((1111 ­­ ¾¾ii)))) ­­ FFii((ff ))

FFii ((ff )) ==RRRR 33 ddxxff ((xx))FFii ((xx)) ff ((xx)) ==

11¼¼22

""==22xx22 ++ ((""==22))22

HHSS++EE == H01 + H0

2 + HE

¯

KossakowskiKossakowski matrix matrix DD explicitly calculable explicitly calculable

A =a+cn2

1 cn1n2 ¡ i bn3 cn1n3 +ibn2cn1n2 +ibn3 a+cn2

2 cn2n3 ¡ ibn1cn1n3 ¡ i bn2 cn2n3 +ibn1 a+cn2

3

DD ==AA AAAA AA

cc==11

22¼¼¡¡

!!44¼¼

11++ ee¡¡ ¯ !!

11¡¡ ee¡¡ ¯ !!aa==!!44¼¼

11++ ee¡¡ ¯ !!

11¡¡ ee¡¡ ¯ !!bb==

!!44¼¼

Can Can entanglemententanglement created irreversibly created irreversibly survive survive decoherencedecoherence??

YESYES

Moreover, Moreover, entangledentangled states can remain states can remain entangledentangled asymptoticallyasymptotically and even become and even become more entangledmore entangled

F.B., R. Floreanini (Int. J. Quant. Inf. 2006)F.B., R. Floreanini (Int. J. Quant. Inf. 2006)

QuantifyingQuantifying EntanglementEntanglement::ConcurrenceConcurrence

2 qubits 2 qubits entanglement contententanglement content (Wootters 1998) (Wootters 1998) ::

spectrum(spectrum(RR) = ) =

concurrence:concurrence:

¸2211 ¸ ¸22

22 ¸ ¸2233 ¸ ¸22

44

CC((½½)) ::== mmaaxxff00;;¸11 ¡¡ ¸22 ¡¡ ¸33 ¡¡ ¸44gg

½½77!! ^½½::== ((¾¾22 ­­ ¾¾22))½½¤¤ ((¾¾22 ­­ ¾¾22)) 77!! RR == ½½½½

½1 =14(11 ­ 12

+3X

i=1

¿ ¡ R2(1¡ (¿ +3)n2i )

2(3+R2)¾i ­ ¾i +

X

i6=j

R2(¿ +3)ni nj

2(3+R2)¾i ­ ¾j )

00·· RR ==bbaa ==

11¡¡ ee¡¡ ¯ !!

11++ ee¡¡ ¯ !!·· 11

½=14(11 ­ 12 +

3X

i=1

½0i 11 ­ ¾i +3X

i=1

½i0¾i ­ 12 +3X

i;j =1

½ij ¾i ­ ¾j )

Any initial stateAny initial state

goes intogoes into

¿¿ ==33XX

ii==11

TTrr((½½¾¾ii ­­ ¾¾ii ))

¡3X

i=1

Rni(¿ +3)3+R2

(11 ­ ¾i +¾i ­ 12)

RR((¯ == 00)) == 00

Asymptotic Concurrence:Asymptotic Concurrence:

initial state:initial state:

concurrence:concurrence:

asymptotic gain:asymptotic gain:

CC((½½)) == 11¡¡33ss22

;; ss <<2233

CC((½½11 )) ¡¡ CC((½½)) ==33RR22 ss33++ RR22

½½==ss44

11SS11­­ 11SS22

++ ((11¡¡ ss)) jjªª 0011iihhªª 0011jj

CC((½½11 )) ==33¡¡ RR22

22((33++ RR22))[[55RR22 ¡¡ 3333¡¡ RR22

¡¡ ¿¿]]

Two atomsTwo atoms separated by a separated by a distance Ldistance L

interaction Hamiltonian:interaction Hamiltonian:

smearing functions:smearing functions:

ff11((xx)) ==11¼¼22

""==22xx22 ++ ((""==22))22

ff22((xx)) == ff ((xx ++ LL))

HHiinntt ==33XX

ii==11

((¾¾((11))ii ­­ FFii((ff11)) ++ ¾¾((22))

ii ­­ FFii((ff22)) ))

Kossakowski MatrixKossakowski Matrix

bb==!!44¼¼

cc==11

22¼¼¡¡

!!44¼¼

11++ ee¡¡ ¯ !!

11¡¡ ee¡¡ ¯ !!

A =a+cn2

1 cn1n2 ¡ i bn3 cn1n3 +ibn2cn1n2 +ibn3 a+cn2

2 cn2n3 ¡ ibn1cn1n3 ¡ i bn2 cn2n3 +ibn1 a+cn2

3

aa==!!44¼¼

11++ ee¡¡ ¯ !!

11¡¡ ee¡¡ ¯ !!

DD==AA AA00

AA00 AA

AA00==aa00++ ccnn22

11 cc00nn11nn22 ¡¡ ii bb00nn33 cc00nn11nn33 ++ ii bb00nn22

cc00nn11nn22 ++ ii bb00nn33 aa00++ ccnn2222 cc00nn22nn33 ¡¡ ii bb00nn11

cc00nn11nn33 ¡¡ ii bb00nn22 cc00nn22nn33 ++ ii bb00nn11 aa00++ cc00nn2233

bb00==!!44¼¼

ssiinn((!! LL))!! LL

cc00==11

22¼¼¡¡

!!44¼¼

11++ ee¡¡ ¯ !!

11¡¡ ee¡¡ ¯ !!

ssiinn((LL))!! LLaa00==

!!44¼¼

11++ ee¡¡ ¯ !!

11¡¡ ee¡¡ ¯ !!

ssiinn!! LL!! LL

ControllingControlling Entanglement Creation Entanglement Creation

separable initial state:separable initial state: sufficient condition:sufficient condition:

with it becomeswith it becomes

separableseparable if if

½½== jj¡¡ ii ¡¡ jj ­­ jj++iihh++jj

jjuuii == jjvvii == ((11;;¡¡ ii;;00))

RR22 ++ SS22 >> 11

SS ==ssiinn((!! LL))

!! LL

RR ==bbaa ==

11¡¡ ee¡¡ ¯ !!

11++ ee¡¡ ¯ !!

hhuujjAAjjuuiihhvvjjAATT jjvvii << jjhhuujjRRee((AA00))jjvviijj22

½½11 LL >> 00 TT ==11¯ == 11