1From quantum random walks to quantum sto hasti o y lesAlexander BeltonDepartment of Mathemati s and Statisti sLan aster Universityhttp://www.maths.lan s.a .uk/~belton/Quantum Sto hasti s and Information SeminarS hool of Mathemati al S ien esUniversity of Nottingham22nd O tober 20081. A simple model of system-parti le intera tion2. Repeated intera tions and quantum random walks3. Quantum sto hasti o y les4. Parti les in a faithful normal state5. Bibliography

21. A simple modelA system (e.g., an atom) intera ts with a stream of identi al parti les (e.g., photons).The on�guration of the system is des ribed by an element of B(h).[More generally, B(h) ould be repla ed by any on rete von Neumann algebra, or even anoperator spa e.℄The on�guration of a parti le is des ribed by an element of B(k̂), where k̂ := C! � k.Let e0 := ! and let (ej) be an orthonormal basis for k.[B(h) and B(k̂) are the algebras of observables for the system and a parti le; we work in theHeisenberg pi ture (observables evolve over time) and the parti les are in the ve tor state !.℄Intera tionAll the parti les are assumed to be in the same on�guration before they intera t with thesystem. Hen e the intera tion an be des ribed by a linear map� : B(h) ! B(h k̂) = B(h)B(k̂):[Combining quantum systems orresponds to forming tensor produ ts.℄

31. A simple modelExampleLet � : a 7! U��(a I

k̂

)U� [Heisenberg pi ture℄where � is the duration of the intera tion and U� = exp(�i�Htot) is the unitary operator givenby the self-adjoint HamiltonianHtot = Hsys Ik̂ + Ih Hpar + Hint:[The self-adjoint operators Hsys 2 B(h) and Hpar 2 B(k̂) represent how the system and parti leevolve individually, whereas Hint 2 B(h k̂) des ribes the intera tion between them.℄Dipole intera tionTaking N := dim k <1, the Hamiltonian for dipole intera tion isHint = ∑Nj=1(Vj jejih!j + V �j j!ihej j) = [0 V �V 0 ] ;where Vj 2 B(h) and V 2 B(h; h k) is su h that u 7! ∑Nj=1(Vju) ej .[The parti le is an N+1-level atom with ground state ! and ex ited states e1; : : : ; eN; intera tiono urs when the parti le moves from an ex ited state to the ground state or vi e versa.℄

42. Repeated intera tionsThe system now intera ts periodi ally with a stream of parti les, as above; after intera ting,ea h parti le `moves o�' and does not interfere with its su essor.Having met n parti les, the ombined system is des ribed by�(n) : B(h) ! B(h)B(k̂) � � �B(k̂) = B(h k̂

n);a 7! (� IB(k̂n�1)) Æ�(n�1)(a):(The parti les move o� to the right in the tensor produ t B(k̂) � � �B(k̂).)[Boundedness onditions must be imposed on � (whi h hold if � is �-homomorphi , as above).℄This gives a quantum random walk, the family of mapsB(h) ! B(h � ); a 7! �(n)(a) I�[n (n > 1);where � := 1⊗m=0 k̂(m) = n�1⊗m=0 k̂(m) 1⊗m=n k̂(m) = �n) �[n(with k̂(m) := k̂ for all m) is toy Fo k spa e.[This tensor produ t is taken with respe t to (!(m))1m=0 (with !(m) := ! for all m).℄

52. Repeated intera tionsQuestionSubje t to some kind of s aling, does the quantum random walk onverge to a limit?Consider the analogous situation where a lassi al random walk onverges to Brownian motion.Theorem [Donsker (?)℄Let (�k)1k=1 be a sequen e of independent, identi ally distributed random variables, withP(�k = �1) = P(�k = 1) = 1=2 8 k > 1:If S0 = 0, Sm := ∑mk=1 �k andX(n)t := 1pn(Sm + (nt �m)�m+1) 8 t 2 [m=n; (m + 1)=n[then X(n) onverges in distribution to a standard Brownian motion. (The onstru tion of X(n)embeds the points of the random walk S at fm=pn : m 2 Zg and linearly interpolates betweenthem.)The natural state spa e for the random walk S is Z, the set of integers, but linear interpolationand Brownian motion require ontinuous, real-valued paths. In our set-up, Z orresponds to � ;what orresponds to R?

63. QS o y lesToy Fo k spa e � := ⊗1m=0 k̂(m) is a dis rete-time version of F = F(R+), the Boson Fo kspa e over L2(R+; k).If I � R+ then Boson Fo k spa e F(I) is the ompletion of the linear span E of the linearlyindependent exponential ve tors f"(f ) : f 2 L2(I; k)g, with h"(f ); "(g)i := exphf ; gi.Boson Fo k spa e has the dire t-sum de ompositionF(I) = CI � 1⊕n=1 L2(I; k)symn;where I = "(0) is a distinguished unit ve tor (the va uum).The splitting � = �n) �[n := n�1⊗m=0 k̂(m) 1⊗m=n k̂(m)is the dis rete analogue of the ontinuous tensor-produ t stru tureF(R+) �= F([0; t[) F([t;1[);"(f ) 7! "(f j[0;t[) "(f j[t;1[):(The spa es split into past, up to time n or t, and future, after time n or t.)

73. QS o y lesGiven � > 0, there is a natural way to embed � into F . First split the Fo k spa e using the ontinuous tensor-produ t property: sin e R+ = [0; � [ [ [�; 2� [ [ � � � ,F(R+) �= 1⊗m=0F([�m; �(m + 1)[):Then, for all m > 0, isometri ally embed the mth opy of k̂ from � into F([�m; �(m + 1)[) inthe only natural manner: if I = [�m; (� + 1)m[ then let

k̂ 3 (�x) 7! �I + x ˜1I 2 CI � L2(I; k) � CI � 1⊕n=1 L2(I; k)symn = F(I);where ˜1I := 1I=k1IkL2(I;k) is the normalised indi ator fun tion of the interval I.TheoremIf D�� : � ! F is this isometri embedding then D��D� ! IF as � ! 0+.ProofShow this holds weakly for exponential ve tors and then noti e that D��D� is a proje tion. �

83. QS o y lesIn the limit, the maps jejihekj on B(k̂) give in�nitesimal in rements of the quantum noises �jkon F : h"(f );�jk(t)"(g)i = ∫ t0 f(j)(s)g(k)(s) dsh"(f ); "(g)i 8 t > 0;where f (0) = g(0) = 1, f(j) = hf ; eji and g(k) = hek; gi for j , k 6= 0.Relationship with lassi al sto hasti pro essesNote that �00 is the deterministi time pro ess, sin e �00(t) = tIF .Let X = (Xt)t>0 be an RN-valued pro ess, a ting by multipli ation on the appropriate L2 spa e.1. If X is lassi al Brownian motion, there exists an isometri isomorphism UW : L2(W ) ! Fsu h that U�W (�j0 + �0j )(t)UW = X(j)t (j = 1; : : : ; N):2. If X is a standard ompensated Poisson pro ess (so that P(X(j)t + t = n) = e�ntn=n!)there exists an isometri isomorphism UP : L2(P ) ! F su h thatU�P (�j0 + �jj + �0j )(t)UP = X(j)t (j = 1; : : : ; N):

93. QS o y lesQuantum sto hasti integralsIf (Ft)t>0 is a pro ess of operators in hF (with domains ontaining h�E) whi h is(i) adapted:Ftu"(f j[0;t[) 2 hF([0; t[) and Ftu"(f ) = [Ftu"(f j[0;t[)℄ "(f j[t;1[)for all t > 0; and(ii) lo ally square-integrable:∫ t0 kFsu"(f )k2 ds <1for all t > 0then the quantum sto hasti integral (∫ t0 Fs d�jk(s))t>0 exists, has these properties andsatis�es hu"(f ); ∫ t0 Fs d�jk(s)v"(g)i = ∫ t0 f(j)(s)g(k)(s)hu"(f ); Fsv"(g)i dsfor all t > 0.[Note: even if Fs is bounded for all s 2 [0; t℄, the operator ∫ t0 Fs d�jk(s) need not be.℄

103. QS o y lesA limit theoremIf the linear (and suitably bounded) maps� = �(�) = [�00 �0���0 ���] : B(h) ! B(h k̂) = [ B(h) B(h k; h)B(h; h k) B(h k) ]

and : B(h) ! B(h k̂) are su h that, as � ! 0+,

[(�00(a) � a)=� �0�(a)=p���0 (a)=p� ���(a) � a Ik] ! (a)then, as � ! 0+,K�t (a) := (Ih D�)�(�(m)(a) I�[m)(Ih D�) if t 2 [m�; (m + 1)� [ onverges to k t (a) strongly on h�E , where the limit o y le k satis�es the QSDEk t (a) = a IF + ∑j;k ∫ t0 k s ( (a)kj ) d�jk(s) (?)and (a) = ∑j;k (a)kj jejihekj.[If � is �-homomorphi then so is a 7! k t (a), for all t > 0.℄

113. QS o y lesChoi e of s alingKnowing the lassi al interpretation of the noises �jk makes the s aling in the previous theoremplausible: if B and P are Brownian motion and a ompensated Poisson pro ess then(dBt)2 = dt and (dPt)2 = dPt;so dPt = dBt=pdt = dt=dt and, in some sense,

[ dt d�k0d�0j d�kj ] � [ dt p dtp dt 1 ] dPt:PropagandaIn the same way that studying lassi al random walks leads naturally to Brownian motion (e.g.,Donsker's invarian e prin iple) studying quantum random walks leads to quantum sto hasti s.RemarkIt is not surprising that the limit o y le may involve unbounded operators even though theoperators whi h make up the walk are bounded: the same happens with lassi al walks andBrownian motion.

123. QS o y lesExample: s aled dipole intera tionIf � : a 7! exp(i�Htot)(a I

k̂

) exp(�i�Htot);where Htot = Hsys Ik̂ + Ih Hpar + Hint,Hpar = ∑Nj=0 �j jejihej j =

�0 0 00 . . . 00 0 �N and Hint = 1p� [0 V �V 0 ] ;then K� ! k , where (a) := [�i [a;Hsys℄ + V �(a Ik)V � 12fa; V �V g �iaV � + iV �(a Ik)�i(a Ik)V + iV a 0 ] :If Tt : B(h) ! B(h) gives the redu ed dynami s of the system, so thathu; k t (a)v ihF = hu; Tt(a)vihthen the QSDE (?) shows that Tt = exp(tL), whereL : a 7! �i [a;Hsys℄ + V �(a Ik)V � 12fa; V �V g: [Lindbladian℄These dynami s are irreversible: the generator ontains se ond-order dissipative terms, omingfrom the system-parti le intera tion.

133. QS o y lesExample: random-walk approximation of sto hasti exponentialsSuppose h = k = C and note that the olle tion of linear maps from B(h) to B(h k̂) is linearlyisomorphi to the 2 � 2 matri es over C via � 7! �(Ih).Fix 2 R and use this identi� ation to de�ne � and by setting�(Ih) = [ 1 �1=2�1=2 1 + ] (� > 0) and (Ih) = [0 11 ] :Then K� ! k and X = k (Ih) satis�es the QSDEX0 = IhF ; dXt = Xt(d�10 + d�01 + d�11) = Xt dYt ;where Y is (unitarily equivalent to) a lassi al pro ess: if = 0 then Y is Brownian motion;otherwise, Y is a ompensated Poisson pro ess with jump size and intensity �2. Hen e X isthe Dol�eans-Dade exponential of Y .Furthermore, �(m)(Ih) = �m(Ih) for all m > 0, soh; K�m�(Ih)ni = h!m; �(m)(Ih)n!mi = h!; �(Ih)n!im 8 n > 0:Therefore, in the va uum state, Zm := K�m�(Ih) is the produ t of m independent identi allydistributed random variables: if � is one su h thenP(� = 1 + �p 2 + 4�2 ) = 12 � 2p 2 + 4� :

144. Normal statesGibbs statesThe onstru tion of our model relies upon the distinguished ve tor !, whi h orresponds to theparti les being in a ve tor state.A more realisti model might have parti les in the Gibbs state�� := 1tr(e��Hpar)e��Hpar;where � > 0 is the inverse temperature.[If Hpar = ∑Nj=0 �j jejihej j, where �0 < �j for j = 1; : : : ; N, then �� ! je0ihe0j as � !1.℄More generally, we may onsider any faithful normal state � 2 B(K)� with orresponding densitymatrix % 2 B(K)+. Let (fj) be an orthonormal basis for the omplex Hilbert spa e K su h that% = ∑j �j jfjihfj j;where �j > 0 for all j and ∑j �j = 1.

154. Normal statesLet k̂ be the Hilbert spa e given by ompleting B(K), with the embedding of B(K) into k̂ denotedby X 7! [X℄, su h that h[X℄; [Y ℄i

k̂

:= �(X�Y ) 8X; Y 2 B(K):Let ! := [IK℄ 2 k̂ and let �(A) 2 B(k̂) be su h that �(A)[X℄ = [AX℄ for all A, X 2 B(K).The mapping � : B(K) ! B(k̂) is a normal inje tive unital �-homomorphism andh!; �(X)!ik̂

= �(X) 8X 2 B(K):[The triple (�; k̂; !) is the Gelfand-Naimark-Segal representation given by �.℄The unital �-homomorphism˜� := IB(h)� : B(h K) ! B(h k̂)is normal and inje tive.If � : B(h) ! B(h K) is (suitably bounded and) linear then ˜� Æ� : B(h) ! B(h k̂) is thegenerator of the quantum random walk with generator � and parti le state �.[The ve tor state on B(k̂) given by ! orresponds to the state � on B(K).℄

164. Normal statesA onditional expe tationLet D � B(K) be the von Neumann algebra generated by fjfjihfj jg.There exists a unique onditional expe tationd : B(K) ! B(K)onto D su h that � Æ d = �. [Expli itly, d : X 7! ∑jhfj ; XfjiKjfjihfj j.℄[More generally, one an take any onditional expe tation d on B(K) whi h preserves �.℄The diagonal map Æ := IB(h) d is a onditional expe tation from B(h K) onto B(h)D.If S 2 B(K; K K) is the S hur isometry su h that Sfj = fj fj for all j thenÆ(T ) = (Ih S)�(T IK)(Ih S)for all T 2 B(h K).Note that Æ is ompletely positive, Æ Æ Æ = Æ,Æ(a IK) = a IK and Æ(T1Æ(T2)) = Æ(T1)Æ(T2) = Æ(Æ(T1)T2)for all a 2 B(h) and T1, T2 2 B(h K).

174. Normal statesS alingIf � > 0 and � : B(h) ! B(h K) is linear (and suitably bounded) then de�nes(�; �) : B(h) ! B(h K); a 7! ��1Æ(�(a) � a IK) + ��1=2Æ?(�(a));where Æ? := IB(hK) � Æ is the o�-diagonal map.TheoremIf � = �(�) and are su h that s(�; �) ! as � ! 0+ then K˜� Æ� ! k, where : B(h) ! B(h k̂); a 7! [ (˜� Æ )(a) E�!(˜� Æ Æ? Æ )(a)E�E��(˜� Æ Æ? Æ )(a)E! 0 ] ;with the sli e map

˜� := IB(h) � : B(h K) ! B(h); T 7! trK

((Ih %)T )and the embedding maps E! : h ! h k̂; u 7! u !and E� : h k ,! h k̂ the anoni al embedding.

184. Normal statesExampleLet Hd and Ho be self-adjoint elements of B(h K) su h that Hd = Æ(Hd) and Ho = Æ?(Ho).If � > 0, the self-adjoint operator Htot := Hd + ��1=2Ho and� : B(h) ! B(h K); a 7! exp(i�Htot)(a IK) exp(�i�Htot)then s(�; �) ! as � ! 0+, where : B(h) ! B(h K);a 7! �i [a IK;Hd + Ho℄ + Æ(Ho(a IK)Ho) � 12fa IK; Æ(H2o)g:Hen e K˜� Æ� ! k, where(a) =

�i [a; ˜�(Hd)℄ + ˜�(Ho(a IK)Ho) � 12fa; ˜�(H2o)g �iE�!˜�([a IK;Ho℄)E��iE�˜�([a IK;Ho℄)E! 0 :

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