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Admissible memory kernels for random unitary qubit evolution Filip A. Wudarski, Pawel Nalez˙yty, Gniewomir Sarbicki and Dariusz Chru´scin´ski Institute of Physics, Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University, Grudzia¸dzka 5/7, 87–100 Torun´, Poland We analyze random unitary evolution of a qubit within memory kernel approach. We provide sufficient conditions which guarantee that the corresponding memory kernel generates physically legitimate quantum evolution. Interestingly, we are able to recover several well-known examples and to generate new classes of nontrivial qubit evolution. Surprisingly, it turns out that a class of quantumevolutions with memory kernelgenerated byourapproach gives rise tothevanishingof a non-Markovianity measure based on the distinguishability of quantum states. 5 1 PACSnumbers: 03.65.Yz,03.65.Ta,42.50.Lc 0 2 t I. INTRODUCTION the standard GKSL form (8), then Λ =T exp( L du) r t 0 u p defines the so-called divisible dynamical map [12, 13] A Dynamics of open quantum systems plays an impor- whichisoftenconsideredasthe generalizationoRfMarko- tant role in the analysis of various phenomena like dis- vianity (see [14] for a generalization of the notion of di- 2 1 sipation, decoherence and dephasing [1, 2]. The usual visibility). Thesecondapproachisbasedonthenonlocal approach to the dynamics of an open quantum system Nakajima-Zwanzig equation [15] (see also [16]) ] consists of applying the Born-Markov approximation [1] h whichleadstoalocalmasterequationfortheMarkovian p t - semigroup ρ˙t = Kt−τρτdτ, (4) t n Z0 ρ˙ =L[ρ ] , (1) a t t u whereρ is the density matrixofthe investigatedsystem inwhichquantummemoryeffectsaretakenintoaccount q t [ andListhetime-independentgeneratorofthedynamical through the introduction of a memory kernel Kt. This semigroup defined as follows means that the rate of change of the state ρt at time t 4 depends on its history (starting at t = 0). The Marko- 9v L[ρ]=−i[H,ρ]+ 12 [Vα,ρVα†]+[Vαρ,Vα†] . (2) Tvihaentmimaest-derepEeqn.den(1t)kiesrnreelobistauinsueadllwyhreenferKretd=to2aδs(tt)hLe. 7 α X(cid:0) (cid:1) generator of the non-Markovianmaster equation. Equa- 1 5 Here H denotes the effective system Hamiltonian, and tion (4) applies to a variety of situations (see eg. [17]). 0 Vα represent noise operators [3, 4]. We call (2) the Becauseoftheconvolutionstructureof(4)thetime-local . GKSL form (Gorini-Kossakowski-Sudarshan-Lindblad). approach is often called time-convolutionless [1, 18, 19]. 1 0 The solution of (1) defines the Markoviansemigroup The structure and the properties of (4) were carefully 5 analyzed in [20–29]. In particular the generalization of ρ =Λ [ρ]=etLρ , (3) 1 t t Markovian evolution to the so-called semi-Markov was : investigated within the memory kernel approach by Bu- v where ρ is an initial state. The dynamical map Λ =etL t i iscompletelypositiveandtrace-preserving(CPTP)[1,3– dini[21]andBreuerandVacchini[23](seealsodiscussion X in [28]). 5]. TheBorn-Markovapproximationassumesweakinter- r a action and a separation of time scales between the sys- Inapresentarticlewestudyrandomunitaryevolution temanditsenvironment. Suchapproachworksperfectly ofaqubitwithinthememorykernelapproach. Inpartic- well for many quantum optical systems [6–8]. When the ularweaddressthefollowingproblem: whatisthestruc- aboveassumptionisnolongervalidthedescriptionbased ture of the correspondingmemory kernel K which leads t on (1) is not satisfactory. Recent technological progress to the legitimate CPTP dynamical map Λ . The article t andmodernlaboratorytechniquescallforamorerefined has the following structure: in Section II we recall basic approach which takes into account memory effects com- factsaboutrandomunitaryevolutionsandinSectionIII pletely neglected in the description based on Markovian we formulatethe sufficientconditionfor K to guarantee t semigroups. In recent years we observed an intense re- legitimatephysicalevolutions. InSectionIVweexamine search activity in the field of non-Markovian quantum the issue of Markovianity. Surprisingly,it turns out that evolution (see the recent review [9], a collection of arti- a subclass of quantum evolutions with memory kernel cles in [10] and a recent comparative analysis in [11]). generated by our approach gives rise to the vanishing of There are basically two approaches which generalize a non-Markovianity measure based on the distinguisha- the standard Markovian master equation (1): time-local bility of quantum states [31]. Section V illustrates our approach replaces L by a time-dependent generator L . approach with several examples. Final conclusions are t Interestingly,ifforalltthetime-dependentgeneratorhas collected in Section VI. 2 II. RANDOM UNITARY QUBIT EVOLUTION III. CONSTRUCTION OF LEGITIMATE MEMORY KERNELS AquantumchannelE :B(H)→B(H)iscalledrandom unitary [32] if its Kraus representation is given by In this article we analyze the nonlocal description based on the following memory kernel equation E[X]= p U XU† , (5) k k k t k X Λ˙ = K Λ dτ , (15) where U is a collection of unitary operators and {p } t t−τ τ k k stands for a probability distribution. In this article we Z0 considera randomunitary dynamicalmapΛ defined by t with 3 3 Λ [ρ]= p (t)σ ρσ , (6) t α α α K [ρ]= k (t)(σ ρσ −ρ), (16) t i i i α=0 X i=1 where σ are Pauli matrices with σ = I [33]. Initial X α 0 2 where k (t) (i = 1,2,3) represent nontrivial memory ef- conditionΛ =1limpliesp (0)=δ . Recentlyatime- i t=0 α α0 fects. Note, that equation (15) considerably simplifies local description based on the following master equation after performing the Laplace transform was analyzed [34, 35] 1 Λ˙ =L Λ , (7) Λ˜ = , (17) t t t s s−K˜ s where L is a time-local generator defined by t whereΛ˜ := ∞e−stΛ dtandsimilarlyforK˜ . Theques- 3 s 0 t s tionweaddressis: whataretheconditionsfork (t)which L [ρ]= γ (t)(σ ρσ −ρ), (8) i t k k k R guarantee that the solution Λ provides a legitimate dy- t k=1 X namical map? with time-dependent decoherence rates γ (t). One asks k Denoting by κ (t) the eigenvalues of K , α t the following question: what are the conditions for γ (t) k which guarantee that the solution Λt = exp( 0tLτdτ) Kt[σα]=κα(t)σα , (18) provides a legitimate dynamical map? Note, that the R equation(15) gives rise to the following set of equations: solution defines a random unitary evolution with p (t) α given by t λ˙ (t)= κ (t−τ)λ (τ)dτ, i=1,2,3. (19) i i i 3 1 Z0 p (t)= H λ (t) , (9) α αβ β 4 Note, that κ (t) = 0 and hence λ (t) = 1 = const. In 0 0 βX=0 termsoftheLaplacetransformsλ˜ (s)andκ˜ (s)onefinds i i where H is the Hadamard matrix αβ 1 1 1 1 1 λ˜i(s)= . (20) s−κ˜ (s) 1 1 −1 −1 i H = , (10) 1 −1 1 −1 In terms of λ˜ (s) conditions (13)–(14) may be equiva- i 1 −1 −1 1 lently reformulated as follows:   and λβ(t) are time-dependent eigenvalues of Λt 1 +λ˜ (s)+λ˜ (s)+λ˜ (s) is CM , (21) 1 1 2 Λ [σ ]=λ (t)σ , (11) s t α α α which read as follows: Λ (t)=1 and and 0 1 λ1(t) = exp(−2[Γ2(t)+Γ3(t)]), +λ˜3(s)−λ˜1(s)−λ˜2(s) is CM , s λ (t) = exp(−2[Γ (t)+Γ (t)]), (12) 2 1 3 1 λ3(t) = exp(−2[Γ1(t)+Γ2(t)]), s +λ˜2(s)−λ˜1(s)−λ˜3(s) is CM , (22) t 1 with Γk(t) = 0 γk(τ)dτ. Now, the map (6) is CP iff +λ˜1(s)−λ˜3(s)−λ˜2(s) is CM , p (t) ≥ 0, which is equivalent to the following set of s α R conditions for λs [34, 35]: where CM stands for a completely monotone function 1+λ (t)+λ (t)+λ (t)≥0 , (13) [37], i.e. a smooth function f : [0,∞) → R satisfying 1 2 3 the condition and dn λ (t)+λ (t) ≤ 1+λ (t), (−1)n f(s)≥0, s≥0, n=0,1,2,... (23) 1 2 3 dsn λ (t)+λ (t) ≤ 1+λ (t), (14) 3 1 2 The equivalence of (14) and (22) results fromthe follow- λ2(t)+λ3(t) ≤ 1+λ1(t). ing 3 Theorem 1 (Bernstein’s Theorem) A function and f: [0,∞) → R is completely monotone on [0,∞) if and 1 1 1 only if it is a Laplace transform of a finite non-negative + ≥ , a a a Borel measure µ on [0,∞), i.e. f is of the form 1 2 3 1 1 1 + ≥ , (30) ∞ a a a f(s)= e−stdµ(t). (24) 2 3 1 1 1 1 Z0 + ≥ , a a a Note that the initial condition p (0) = 1 and p (0) = 0 3 1 2 0 k for k =1,2,3 is equivalent to λ (0)=1 due to define a legitimate memory kernel k Theorem 2 (Initial Value Theorem) Let f˜(s) be the 1 3 Laplace transform of f(t). Then the following relation is K˜s[ρ]= κ˜α(s)σktr[σkρ] , (31) 2 true: k=1 X limf(t)= lim sf˜(s) (25) i.e. the corresponding λ˜k(s) satisfy (21)–(22) and (26). t→0 s→∞ Proof: note that formula (28) implies it is equivalent to 1 1 lim sλ˜k(s)=1 , (26) λ˜k(s)= s 1− a W(s) , (32) s→∞ (cid:18) k (cid:19) for k =1,2,3. This way we have proved and hence Theorem 3 The map Λ˜s represented by the following 1 +λ˜3(s)−λ˜1(s)−λ˜2(s) s spectral decomposition 1 1 1 1 1 = + − , (33) 3 1 sW(s) a a a Λ˜ [ρ]= λ˜ (s)σ tr[σ ρ] , (27) (cid:18) 1 2 3(cid:19) s α α α 2 α=0 which proves that 1 +λ˜ (s)−λ˜ (s)−λ˜ (s) is CM due X s 3 1 2 gwiittihmλa˜t0e(sm)a=p1Λ/s,ifdaenfidneosnltyheifLcaopnladciteiotnrasn(s2f1o)r,m(2o2f)aalned- rtoemthaienifnagctctohnadtiti1soWns1(s()14is).CM. Similarly one proves th(cid:3)e t (26) are satisfied. Note,thatsince 1sW1(s) isCM,hence,due tothe Bern- stein theorem, it is the Laplace transform of a positive Itisworthemphasisingthattherearefewanalyticaltools function. Hence for dealing with CM functions, which is due to the fact 1 that an infinite set of conditions (23) must be verified. W(s)= , (34) Nevertheless, we found an important class of CM func- f˜(s) tions giving rise to CPTP dynamics with a straightfor- where f˜(s) is the Laplace transform of f(t) satisfying ward interpretation. To present them, let us first ob- t serve that CM functions have the following two proper- f(τ)dτ ≥0 for all t≥0. One finds 0 ties, which will not be proved: R −sf˜(s) κ˜ (s)= . (35) Property 1 Let f and g be arbitrary completely mono- k a −f˜(s) k tone functions. Then Note, that condition (29) implies 1. f ·g is CM, 1 1 1 t 2. αf +βg is CM for any α,β >0, + + f(τ)dτ ≤4 . (36) a a a Property 2 If s ≥0 then 1 is CM. (cid:18) 1 2 3(cid:19)Z0 0 s+s0 Hencetosummarize: ourclassischaracterizedbyasingle We are now ready to prove our main result: function f(t) and three numbers a1,a2,a3 >0 such that t F(t)= f(τ)dτ ≥0 and conditions (30) and (36) hold. Theorem 4 Let W(s) be a function such that 1 1 is 0 sW(s) One finds for p (t): α CM. Then the functions R 1 1 1 1 s p (t) = + − F(t) , κ˜k(s)=−akW(s)−1, k =1,2,3, (28) 1 4(cid:18)a2 a3 a1(cid:19) 1 1 1 1 with a1,a2,a3 >0 such that p2(t) = 4 a + a − a F(t) , (37) (cid:18) 3 1 2(cid:19) 1 1 1 1 1 1 1 1 1 4− + + is CM , (29) p (t) = + − F(t) , 3 s W(s) a a a 4 a a a (cid:18) (cid:20) 1 2 3(cid:21)(cid:19) (cid:18) 1 2 3(cid:19) 4 andp (t)=1−p (t)−p (t)−p (t). Inparticular,taking where x :=tr[ρσ ] and the relaxationtimes are defined 0 1 2 3 k k a =a =a and a =∞ one finds via 1 2 3 −sf˜(s) T = 1 , T = 1 , T = 1 . (48) κ˜1(s)=κ˜2(s)= a−f˜(s) , κ˜3(s)=0 , (38) 1 γ2+γ3 2 γ3+γ1 3 γ1+γ2 Itiswellknown[5]thatcomplete positivityisequivalent and hence to the following set of conditions upon T : k 1 sf˜(s) k˜ (s)=k˜ (s)=0 , k˜ (s)= , (39) 1 1 1 1 2 3 2a−f˜(s) + ≥ , T T T 1 2 3 1 1 1 gives rise to the legitimate memory kernel + ≥ , (49) T T T 2 3 1 K [ρ]=k (t)(σ ρσ −ρ), (40) 1 1 1 t 3 3 3 + ≥ , T T T 3 1 2 with arbitrary f(t) and a > 0 satisfying additional condition It is therefore clear that condition (30) is an analogue of (49). Note that condition (30) means that there exist t b ,b ,b >0 such that 0≤F(t):= f(τ)dτ ≤2a, (41) 1 2 3 Z0 1 1 1 1 = + , for all t≥0. The corresponding solution reads 2a b b 1 2 3 1 1 1 1 1 = + , (50) p0(t) = 1− 2aF(t) , 2a2 b3 b1 1 1 1 1 p1(t) = p2(t)=0 , (42) = + . 2a b b 1 3 1 2 p (t) = F(t) . 3 2a Now,ittermsofb ,b ,b ourresultmaybereformulated 1 2 3 as follows This approach resembles very much the semi-Markov construction [23, 28]: for any f(t) ≥ 0 satisfying ∞f(t)dt≤1 the memory kernel (40) with Corollary 1 For any b1,b2,b3 >0 and the function f(t) 0 satisfying R k˜ (s)= sf˜(s) , (43) t 1 1 1 −1 3 1−f˜(s) 0≤F(t):= f(τ)dτ ≤ + + , (51) b b b Z0 (cid:18) 1 2 3(cid:19) gives rise to CPTP evolution. In this case one finds and 1 p (t) = [1+λ (t)] , lim f˜(s)=0, (52) 0 1 2 s→∞ p (t) = p (t)=0 , (44) 1 2 the memory kernel defined by 1 p (t) = [1−λ (t)] , 3 2 1 sf˜(s) κ˜ (s)=− , (53) where k ak−f˜(s) f˜(s)+1 defines legitimate quantum evolution. Moreover one has λ˜ (s)=λ˜ (s)= . (45) 1 2 f˜(s)−1 1 p (t)= F(t) , (54) k b It is therefore clear that our approach goes beyond the k semi-Markov construction. and p (1)=1−p (t)−p (t)−p (t). 0 1 2 3 Let us recall that Markoviansemigroup generated by Letusobservethatitisveryhard,ingeneral,toinvert 3 1 formula (35) to the time domain. Now, we provide a L[ρ]= γ [σ ρσ −ρ], (46) k k k 2 familyofW(s)whichenablesonetoeasilycomputeκ (t) i k=1 X and have the memory kernel in time domain. the corresponding Bloch equation reads Theorem 5 Let W(s) be a polynomial 2 x˙ (t)=− x (t) , (47) k Tk k W(s)=(s+z1)...(s+zn), (55) 5 with z >0. If a ,a ,a satisfy (30) and where the integral is evaluated over the region where i 1 2 2 d||Λ [ρ − ρ ]|| > 0. Now, it has been proved [34] n dt t 1 2 tr 1 1 1 1 thatforrandomunitaryqubitevolutionifalleigenvalues z ≥ + + , (56) i 4 a a a λ (t)≥0, then (61) is equivalent to i=1 (cid:18) 1 2 3(cid:19) k Y then κ (t) defined via (28) define a legitimate memory d i λ (t)≤0 ; k =1,2,3. (63) k kernel. dt Proof: It is clear that it is enough to prove (21). Proposition 1 For a1,a2,a3 satisfying (30) and W(s)= 1 , where f˜(s) is CM and Lemma 1 One has the following decomposition f˜(s) 1 =A 1 − n ij−=11zj , (57) tf(τ)dτ ≤amin, amin =min{a1,a2,a3}, (64) s ni=1(s+zi) s i=1 ijQ=1(s+zj)! Z0 X thecorrespondingmemorykernelgivesrisetothedynam- whereQ Q ical map Λ such that N [Λ ]=0. t BLP t 1 A= . (58) n z Proof: Let us observe that condition (64) implies (29). i=1 i Indeed, from (64) one has For the proof see the AppeQndix. Now we show that condition (21) holds. According to (57) one has 1 1 1 t + + f(τ)dτ ≤3, (65) a a a 1 +λ˜ (s)+λ˜ (s)+λ˜ (s) (cid:18) 1 2 3(cid:19)Z0 1 2 3 s and hence the condition (29) follows. Now, observe that 1 1 1 1 1 = 4− + + s a a a W(s) 1 t (cid:18) (cid:20) 1 2 3(cid:21) (cid:19) λ (t)=1− f(τ)dτ ≥0, k = 1 4− 1 + 1 + 1 1 (59) ak Z0 s a a a n z (cid:18) (cid:20) 1 2 3(cid:21) i=1 i(cid:19) due to (64). Hence itis sufficient to showthat dλ (t)≤ + (cid:20)a11 + a12 + a13(cid:21) ni=11ziQj=n1 jiQ=1ij(=−s11+zizi). 0a.ndIthiesnccleeatrakdditnλgki(ntt)o≤ac0coifuanntd(2o0n)lyitiifs1eq−usivλãkl(edsnt)tkitsoCthMe X requirement that −κ˜ (s)λ˜ (s) is CM. One has therefore A second term in (59) is CQM due to thQe fact that it is a k k sumofCMfunctions. Hence,ifcondition(56)issatisfied f˜(s) then (21) holds. (cid:3) −κ˜k(s)λ˜k(s)= , (66) a k Note, that which ends the proof since f˜(s) is CM and a >0 (cid:3) s 1 s k κ˜ (s)=− =− , i akW(s)−1 ak(s−s1)...(s−sm) Remark 2 It W(s) = (s+z1)...(s+zn) with zk > 0 (60) and a ,a ,a satisfying (30) together with 1 2 3 where {s ,...,s } are the roots of the polynomial 1 m (akW(s)−1). Itisthereforeclearthatformula(60)may n 1 be easily inverted to the time domain. zi ≥ , k =1,2,3, (67) a k i=1 Y Remark 1 Note, that W(s) defined in (55) implies that 1 is CM and hence 1 1 is CM as well. then the corresponding dynamical map Λt satisfies W(s) sW(s) N [Λ ]=0. BLP t Remark 3 It was shown [14, 36] that BLP condition IV. CHECKING FOR NON-MARKOVIANITY (61) is equivalent to so-called P-divisibility. This means that Let us recall that according to [31] the evolution represented by Λt is non-Markovianif the condition Λt =Vt,sΛs, (68) d ||Λt[ρ1−ρ2]||tr ≤0 , (61) and for any t>s the propagator Vt,s is positive (but not dt necessarily completely positive). is violated for some initial states ρ and ρ . One defines 1 2 Interestingly, our construction provides a class of legiti- [31] a well-known non-Markovianity measure mate random unitary qubit evolutions generated by the d nontrivial memory kernel but still satisfying BLP condi- N [Λ ]= sup ||Λ [ρ −ρ ]|| dt , (62) BLP t dt t 1 2 tr tion (61), (cf. also [30]). It is clear that to violate (61) ρ1,ρ2Z 6 one needs a more refined construction such that 1 is with z >0. One finds W(s) not CM but 1 1 is already CM. It deserves further s sW(s) κ˜ (s)=− , (74) analysis. k a (s+z)−1 k Consider now the question of CP-divisibility which is fullycontrolledbythelocaldecoherenceratesin(8). One and the inverse Laplace transform gives may easily compute them in terms of f(t): κk(t)=−1 δ(t)− z− 1 e−[z−a1k]t . (75) z a (cid:18) (cid:20) k(cid:21) (cid:19) f(t) −1 1 1 γ (t) = + + , 1 Note, that if a =1/z, then the dynamics is purely local. 4 a −F(t) a −F(t) a −F(t) k (cid:18) 1 2 3 (cid:19) One easily finds f(t) 1 1 1 γ (t) = − + , 2 4 a −F(t) a −F(t) a −F(t) 1 (cid:18) 1 2 3 (cid:19) λk(t)=1− (1−e−zt), (76) f(t) 1 1 1 zak γ (t) = + − . 3 4 a −F(t) a −F(t) a −F(t) (cid:18) 1 2 3 (cid:19) and finally the solution for pk(t) is defined by (37) with The dynamical map Λt is CP-divisible iff γk(t) ≥ 0 for F(t)= 1(1−e−zt) . (77) k =1,2,3. Let us assume that z a ≤a ≤a . (69) Note, that condition (56) implies the following relation 1 2 3 between z and a ,a ,a : 1 2 3 Proposition 2 If a ,a ,a and f(t) ≥ 0 satisfy condi- 1 2 3 1 1 1 tions (29) and (30) the corresponding memory kernel 4z ≥ + + , (78) a a a 1 2 3 sf˜(s) κ˜ (s)=− , (70) which guarantees that p (t) ≥ 0. In the symmetric case k a −f˜(s) 0 k a =a = a =a one finds p (t) =p (t)= p (t)=: p(t) 1 2 3 1 2 3 with leads to a CP-divisible dynamical map iff 1 p(t)= [1−e−zt], (79) F(t)≤a1− (a2−a1)(a3−a1). (71) 4za Proof: Due to (69) it ispsufficient to show that γ1(t)≥0 and p0(t) = 1 − 3p(t) with 4za ≥ 3. One finds that which, for f(t)≥0, is equivalent to asymptotically −1 1 1 3 + + ≥0. (72) p0(t)→1− . (80) a −F(t) a −F(t) a −F(t) 4za 1 2 3 Note that for za > 1 one has asymptotically p (∞) < Let us assume that F(t) < a , which means, that γ (t) 0 1 1 1/4. This property cannot be reproduced within the local is not singular. Inequality (72) is satisfied iff approach with regular generators L . Indeed, it follows t from (9) (see also [34] for more details) that F(t)∈(−∞,F ]∪[F ,+∞) − + 1 with p (t)= [1+λ (t)+λ (t)+λ (t)], (81) 0 1 2 3 4 F± =a1± (a2−a1)(a3−a1). and hence, using (14), one finds p Now, taking into account that F(t) < a one finally 1 1 proves (71). (cid:3) p0(t)≥ 4. (82) This Proposition shows that positivity of the function This example shows that local and memory kernel ap- f(t) is not sufficient for CP-divisibility. One needs an proaches may lead to essentially different evolutions. extra condition (71) which involves not only f(t) but {a ,a ,a } as well. 1 2 3 Example 2 Consider now the same polynomial W(s)= s+z but let z =2c>0. Moreover V. EXAMPLES 1 1 a =a = , a = . (83) 1 2 3 c 2c Example 1 Consider the simplest case with a polyno- One finds mial of degree one sc κ˜ (s)=κ˜ (s)=− , κ˜ (s)=−2c, W(s)=s+z, (73) 1 2 3 s+c 7 and hence Example 4 Let κ1(t)=κ2(t)=−cδ(t)+c2e−ct , κ3(t)=−2cδ(t). W(s)=s2+ω2. (92) Finally, one finds the following formula for the memory Note that 1 1 is CM since sW(s) kernel c 1 1 1 1 ω K [ρ] = δ(t)[σ ρσ +σ ρσ −2ρ] = , t 2 1 1 2 2 sW(s) ωs s2+ω2 (cid:18) (cid:19) c2 − e−ct[σ ρσ −ρ]. (84) t 3 3 is theLaplace transform of sin(ωτ)dτ which is positive 2 0 for all t≥0. Condition (29) implies One has R 1 1 1 λ1(t)=λ2(t)= 21 1+e−2ct , λ3(t)=e−2ct. a1 + a2 + a3 ≤2ω2 . (93) Interestingly,thisevolut(cid:0)ionreprod(cid:1)ucestime-localdescrip- Thecorrespondingeigenvalues ofthememorykernelread tion with 1 1 c c κ (t)=− cos ω2− t , (94) γ (t)=γ (t)= , γ (t)=− tanh(ct). (85) i a a 1 2 2 3 2 i (cid:18)r i (cid:19) asdiscussedin[35]. Itwasshown[36]thatΛt isaconvex for ω2 ≥1/ai, and combination of two Markovian semigroups Λ(1) and Λ(2) t t 1 1 generated by κ (t)=− cosh −ω2 t , (95) i a a c i (cid:18)r i (cid:19) L [ρ]= [σ ρσ −ρ] ; k =1,2, (86) k 2 k k for ω2 <1/ai. Moreover one finds that is, 1 λ (t)=1+ [cos(ωt)−1] , (96) 1 k akω2 Λ = etL1 +etL2 . (87) t 2 and hence (cid:0) (cid:1) This simple example shows that a convex combination 1 1 1 1 of Markovian semigroups leads to a quantum evolution p (t) = + − [1−cos(ωt)] , 1 4ω2 a a a displaying essential memory effects. (cid:18) 2 3 1(cid:19) 1 1 1 1 p (t) = + − [1−cos(ωt)] , (97) Example 3 Consider now a polynomial of degree two 2 4ω2 a a a (cid:18) 3 1 2(cid:19) W(s)=(s+c )(s+c ), (88) 1 1 1 1 1 2 p (t) = + − [1−cos(ωt)] , 3 4ω2 a a a (cid:18) 1 2 3(cid:19) with c >c >0. Our construction gives rise to a legiti- 2 1 mate memory kernel if condition (30) holds and togetherwithp (t)=1−p (t)−p (t)−p (t). Inparticular 0 1 2 3 taking 1 1 1 4c c ≥ + + . (89) 1 2 a a a 1 1 2 3 a =a = , a =∞, (98) 1 2 ω2 3 One finds one finds 1 s κ˜ (s) = − k ak(s+c1)(s+c2)− a1k κ1(t)=κ2(t)=−ω2 , κ3(t)=0 , (99) 1 s = − , (90) and hence a (s+s )(s+s ) k 1 2 ω2 with k1(t)=k2(t)=0 , κ3(t)= , (100) 2 1 s +s =c +c , s s =c c − . which proves that the constant (time independent) 1 2 1 2 1 2 1 2 a k k Hence the solution has the form (37) with the function K [ρ]= (σ ρσ −ρ), (101) t 3 3 2 F(t) given by provides a legitimate memory kernel for arbitrary k = 1 1 1 F(t)= [1−e−c1t]− [1−e−c2t] . (91) ω2 > 0. Moreover one finds for the local decoherence c2−c1 (cid:18)c1 c2 (cid:19) rates 8 CM. We stress that Theorem4 providesonly a sufficient condition and further analysis is needed to cover physi- ωsin(ωt) −1 γ (t)= callyinterestingcaseswhichdonotfittheassumptionsof 1 4 a1ω2−1+cos(ωt) the Theorem. Interestingly, it turns out that the quan- (cid:16) 1 1 tum evolution with a memory kernel generated by our + + , a ω2−1+cos(ωt) a ω2−1+cos(ωt) approach gives rise to vanishing non-Markovianity mea- 2 3 (cid:17) sure based on the distinguishability of quantum states andsimilarlyforγ2(t)andγ3(t). Note,thatiffor somek [31]. We also have shown when the corresponding dy- one has akω2 <1 then local decoherence rates are singu- namicalmapis CP-divisible. Itshowsthatthe evolution lar and hence in this case the non-local approach is more satisfying nonlocal master equation does not necessarily suitable. lead to a non-Markovian evolution. It would be also in- teresting to analyze the relation between semi-Markov evolutionandthe one governedby our approachin more VI. CONCLUSIONS detail. We analyzed random unitary evolution of a qubit withinmemorykernelapproach. Ourmainresultformu- Acknowledgements lated in Theorem 4 allows to construct legitimate mem- orykernelsleadingtoCPTPdynamicalmaps. Thepower We thank anonymous referee for valuable comments of this method is based on the fact that 1) it allows to and Karol Z˙yczkowski for pointing out [33]. This article reconstructwellknownexamplesoflegitimatequbitevo- was partially supported by the National Science Center lution,2)thestructureofpolynomialsW (s)enablesone k project DEC- 2011/03/B/ST2/00136. to perform the inverse Laplace transform and to find a formula for the kernel in the time domain. The math- ematical analysis heavily uses the notion of completely Appendix: proof of Lemma 1 monotone functions. These functions are not commonly used in theoretical physics and knowledge of their properties is rather limited. There are no known effective Let us observe that (57) may be represented in the methods allowing to check whether a given function is following form n (s+z )−s n (s+z )+z n (s+z )+... n−2z (s+z )+ n−1z 1 i=1 i i=2 i 1 i=3 i j=1 j n j=1 j =A , (102) s ni=1(s+zi) Q (cid:16)Q s Qni=1(s+zi) Q Q (cid:17) Q Q therefore, to prove the Lemma it suffices to show that n n n n n−2 n−1 z = (s+z )−s (s+z )+z (s+z )+... z (s+z )+ z . (103) i i i 1 i j n j   i=1 i=1 i=2 i=3 j=1 j=1 Y Y Y Y Y Y   Wewillprovethisbyinduction. Forn=1itisclearthat while RHS reads LHS=RHS=z . We assume that (103) is true for n and 1 prove it is also true for (n+1). 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