A continuous-time diffusion limit theorem for dynamical decoupling and intrinsic decoherence
AA Continuous-Time Diffusion Limit Theoremfor Dynamical Decoupling andIntrinsic Decoherence
Robin Hillier , Christian Arenz , Daniel Burgarth Department of Mathematics and StatisticsLancaster UniversityLancaster LA1 4YF, UK Department of MathematicsAberystwyth UniversityAberystwyth SY23 2BZ, UKOctober 16, 2018
Abstract
We discuss a few mathematical aspects of random dynamical decoupling, a keytool procedure in quantum information theory. In particular, we place it in the con-text of discrete stochastic processes, limit theorems and CPT semigroups on matrixalgebras. We obtain precise analytical expressions for expectation and variance of thedensity matrix and fidelity over time in the continuum-time limit depending on thesystem Lindbladian, which then lead to rough short-time estimates depending only oncertain coupling strengths. We prove that dynamical decoupling does not work in thecase of intrinsic (i.e., not environment-induced) decoherence, and together with theabove-mentioned estimates this yields a novel method of partially identifying intrinsicdecoherence.
The aim of this article is two-fold: first, to provide an analytical description of ran-dom dynamical decoupling because analytical expressions are often more manageable thancombinatoric-numerical ones; second, to use this description to propose a partial methodof detecting intrinsic decoherence of quantum systems.Dynamical decoupling is a method applied to stabilise states of quantum registersagainst undesired time-evolution. Originally invented in NMR technology, it has been gen-eralised to a wider context, in particular in quantum information theory [LB13, VKL99]. Itworks by application of repeated instantaneous unitary correction pulses on the quantumregister, perturbing the original time-evolution. The procedure is particularly interestingand effective when performed in a random way [VK05, SV05].While several general estimates and specialisations of the procedure have been proposedin the past (cf. [LB13] and the references therein), our focus here is on finding handyanalytical descriptions of the time-evolution of expectation and distribution of physicallyinteresting quantities like the density matrix process or the gate fidelity process arising1 a r X i v : . [ qu a n t - ph ] A p r rom this random time-evolution. We believe that such descriptions are a valuable tool infuture computations and enable predictions in experiments.We would like to provide now a rough overview of the content of this article. Let ( H , L )stand for a generic finite-dimensional quantum system, with H a finite-dimensional com-plex Hilbert space and L a possibly time-dependent Lindblad generator (cf. [BP02, Wol11]for general background information). We start in Section 2 by introducing dynamical de-coupling and show that the decoupling condition (4) can be satisfied only if L = i[ H, · ]for some Hamiltonian H . This might sound like a contradiction since dynamical decou-pling aims to eliminate decoherence (noise) arising from open systems. It can be resolvedby differentiating between intrinsic and extrinsic decoherence : the latter one is wheredecoherence arises from interaction with an actual quantum heat bath or environmentsuch that the total space time-evolution is unitary, the former one is where decoherenceis actually the time-evolution of a closed system [Adl04]. It is unclear whether intrinsicdecoherence may appear in nature, and it would, of course, contradict the axiom of uni-tary time-evolution. But in order to find out whether it may exist or whether the axiomof unitarity is always verified, one has to perform experiments and develop mathematicaltools.To this end, Sections 3 and 4 provide a probabilistic-analytical approach to dynam-ical decoupling, namely: we set up a probabilistic description of a random walk in thecompletely positive trace-preserving (CPT) maps of the quantum system arising from therandom correction pulses; then we study the continuum-limit of this random walk undera suitable scaling, which becomes a Gaussian (Markov) process in the CPT maps. Weuse this to determine the expectation and higher moments of the density matrix process ρ t . In the 4th section we then compute the expectation of the gate fidelity, which mightbe regarded as a mean fidelity when averaging over all states on B ( H ) in a suitable man-ner. We illustrate all constructions and considerations with an easy example that shallaccompany us through the paper.Up to this point, things were quite general, but this is where we can turn to oursecond aim: distinguishing between intrinsic and extrinsic decoherence (with boundedHamiltonian dilations, cf. [AHB14, App.]). We therefore specialise in the final Section5 on providing approximative bounds for the gate fidelity in these two extremal casestogether with a recipe which should enable the experimenter to determine the type ofdecoherence present in his setting. Ideally he should just know the pulse length τ , the totaltime t of evolution, and the coupling strength of the undesired decoherence. In some casesunfortunately some further input is needed. However, the overall moral is roughly speakingthe following: the rate of decoherence decreases to 0 when τ → τ → Acknowledgements.
We would like to thank Micha(cid:32)l Gnacik, Lorenza Viola and thereferee for useful discussions and/or comments on the manuscript. Moreover, RH wouldlike to thank Gernot Alber and Burkhard K¨ummerer for guidance in his master thesisseveral years ago, in which a special case of Section 3 had been developed.
Let us start with some notation used throughout the article. We shall denote our quan-tum system in question by ( H , L ), with H a finite-dimensional complex Hilbert space (of2imension d H ) and we denote the adjoint of linear maps on it by ‘ ∗ ’; L is the Lindblad op-erator on B ( H ), generating the completely positive trace-preserving (CPT) time evolutionmaps α t = e t L , t ∈ R + , of the quantum system (cf. [BP02, Wol11] for general backgroundinformation). Let us abbreviate A := B ( H ), which has dimension d = d H and whichbecomes a Hilbert space again with scalar product ( x, y ) ∈ A × A (cid:55)→ (cid:104) x, y (cid:105) := tr( x ∗ y ),and we denote adjoints of maps on this Hilbert space by ‘ † ’. We write Ad (or ad) for theadjoint representation of the unitary group (or its Lie algebra, respectively) on A , i.e.,Ad( v )( x ) = vxv ∗ and ad( H )( x ) = [ H, x ], for v, H, x ∈ A with v unitary and H selfadjoint.By a decoupling set in A we mean a finite group of unitaries V := ( v j ) j ∈ J ⊂ A suchthat 0 ∈ J and v = , (cid:88) j ∈ J Ad( v j )( x ) ∈ C , ∀ x ∈ A. (1)Notice that in this case we automatically have | J | (cid:80) j ∈ J v j xv ∗ j = d H tr( x ) . Example 1.
The standard illustrative example of a finite-dimensional quantum system tokeep in mind throughout this paper is an N -qubit quantum system, so H = ( C ) ⊗ N and d = 2 N ; there typically the decoupling set V consists of the 4 N different combinations ofPauli matrices { , σ , σ , σ } on the tensor factors. Here v ∗ j = v j ., for all j .Given the CPT semigroup of time evolution maps ( α t ) t ∈ R + of our system and a(“short”) time τ , consider the externally modified time evolution α ( τ )( n +1) τ = Ad( v ∗ j v j n ) ◦ α τ ◦ Ad( v ∗ j n v j n − ) ◦ α τ ◦ . . . ◦ α τ ◦ Ad( v ∗ j v j ) , (2)where n ∈ N , and ( j i ) i ∈ N forms a certain sequence in J with j = 0, meaning we apply instantaneous decoupling or correction pulses v ∗ j i v j i − at time iτ ; set α τt = α t − nτ ◦ α τnτ whenever t ∈ [ nτ, ( n + 1) τ ). The sequence j i can be fixed or random, leading to deter-ministic or random dynamical decoupling . It turns out that random decoupling has manyadvantages [VK05, SV05, KAS05] and moreover is mathematically more interesting, andthat is why we want to investigate it here.Our first goal is to find an analytical description of the externally modified time evolu-tion α ( τ ) t . In the random setting, ( α ( τ ) t ) t ∈ τ N becomes a stochastic process (a random walkwith steps lasting time τ ) induced by the process ( v j t ) t ∈ τ N with independent identicallydistributed (iid) and equidistributed [Shi96] increments in V , and we are interested in thelimit τ →
0, which would enable nice analytical expressions.Since α t = exp( t L ), we find α ( τ )( n +1) τ = exp( τ Ad( v j n ) ◦ L ◦ Ad( v ∗ j n )) ◦ α ( τ ) nτ , (3)so the increment during the time interval ( t − τ, t ) is given by exp( τ Ad( v j t ) ◦ L ◦ Ad( v ∗ j t )).We say that V satisfies the decoupling condition for ( H , L ) if (cid:88) j ∈ J Ad( v j ) ◦ L ◦ Ad( v ∗ j ) = 0 . (4)The idea behind this condition is that it ensures cancellation of interaction at first orderin τ (cid:107)L(cid:107) , i.e., for short time τ , and thus higher order terms contribute.We say (the time evolution of) our quantum system ( H , L ) is purely unitary if L =i ad( H ), with H ∈ A selfadjoint, because in this case α t = e t L is induced by a one-parameter family of unitary matrices; in this case L † = −L . The “opposite case”, namelywhere L † = L we call purely dephasing . 3 heorem 2. A decoupling set for ( H , L ) satisfies the decoupling condition (4) iff ( H , L ) is purely unitary. Proof.
We write out the generator L in Christensen-Evans form [CE79, Wol11] L ( x ) = Ψ( x ) + ax + xa ∗ , ∀ x ∈ A, with a certain a ∈ A and completely positive Ψ which is not a multiple of id A (w.l.o.g.,because adding 2 λ id A to Ψ has the same result on L as adding λ to a ). Suppose first that α is purely unitary; then Ψ = 0 and a ∗ = − a . From (1), with J indexing the decouplingset V = { v j : j ∈ J } as above, we obtain1 | J | (cid:88) j ∈ J Ad( v j ) ◦ L ◦ Ad( v ∗ j )( x ) = 1 | J | (cid:88) j ∈ J Ad( v j )( a Ad( v ∗ j )( x ) + Ad( v ∗ j )( x ) a ∗ )= 1 | J | (cid:88) j ∈ J (Ad( v j )( a ) x + x Ad( v j )( a ∗ ))= 1 d H tr( a + a ∗ ) x = 0 , x ∈ A, so V satisfies the decoupling condition. If instead α is not purely unitary, we have Ψ (cid:54) = 0and hence Φ := 1 | J | (cid:88) j ∈ J Ad( v j ) ◦ Ψ ◦ Ad( v ∗ j )is completely positive and nonzero. Suppose Φ( x ) equals − | J | (cid:88) j ∈ J (Ad( v j )( a Ad( v ∗ j )( x )) + Ad( v j )(Ad( v ∗ j )( x ) a ∗ )) = − d H tr( a + a ∗ ) x, x ∈ A. Then, for every rank-one projection p ∈ A , we have Φ( p ) = − d H tr( a + a ∗ ) p . But for every ξ ∈ H with pξ = 0, we have0 = − d H tr( a + a ∗ ) (cid:104) ξ, pξ (cid:105) = (cid:104) ξ, Φ( p ) ξ (cid:105) = 1 | J | (cid:88) j ∈ J (cid:104) ξ, Ad( v j ) ◦ Ψ ◦ Ad( v ∗ j )( p ) ξ (cid:105) , with each single term ≥
0, due to the positivity of Ψ and the scalar product, and thusactually = 0. In particular, since v = , we have (cid:104) ξ, Ψ( p ) ξ (cid:105) = 0, so Ψ( p ) ∈ R + p . Let uswrite Ψ in (minimal) Kraus form with rank(Ψ) its Kraus rank and certain b i ∈ A [Wol11]:Ψ( x ) = rank(Ψ) (cid:88) i =1 b i xb ∗ i , x ∈ A. This entails then, for any two mutually orthogonal vectors η, ξ ∈ H ,0 = (cid:104) ξ, Ψ( | η (cid:105)(cid:104) η | ) ξ (cid:105) = rank(Ψ) (cid:88) i =1 |(cid:104) ξ, b i η (cid:105)| . Hence, for every i , we see that b i η ∈ C η , or in other words, η must be an eigenvector of b i .This holds for every η ∈ H , so b i ∈ C , and thus Ψ is a multiple of id A , which contradictsour initial assumptions. Therefore, Φ( x ) (cid:54) = − d H tr( a + a ∗ ) x , and1 | J | (cid:88) j ∈ J Ad( v j ) ◦ L ◦ Ad( v ∗ j ) (cid:54) = 0 , i.e., V on ( H , L ) does not satisfy the decoupling condition. (cid:3) Despite this result, it will turn out in the course of this paper that dynamical decouplingis still interesting beyond the unitary case. 4
The continuous-time limit of random dynamical decou-pling
We continue with the notation and concepts introduced in the previous section. Let us,in particular, assume all increments v j i , with i ∈ N , in our random walk ( v j i ) i ∈ N ofdecoupling pulses to be iid and equidistributed in V as in the preceding section. Theinduced random walk ( α ( τ )( n +1) τ ) n ∈ N lies in the completely positive maps on A accordingto (2) and (3). Moreover, since completely positive maps are linear maps of the Hilbertspace A and since all the increments are invertible, the random walk actually lies in thegroup GL( A ) of invertible linear maps of A , and L ∈ gl ( A ), the Lie algebra of GL( A ).This induced random walk has again iid increments and is described by the measure µ ( τ ) := 1 | J | (cid:88) j ∈ J δ exp( τ Ad( v j ) ◦L◦ Ad( v ∗ j )) . (5)We would like to investigate it in the limit τ →
0. However, since τ is an actual physicalquantity in our set-up, we keep it and instead consider a fictitious limit, which should begood for small τ , as explained below. Considering simply µ ( τ ) and the limit of τ → Theorem 3.
Let G be an N -dimensional Lie group, with -chart ( U, x ) , Lie algebra basis ( X k ) ≤ k ≤ N and coordinate mappings x k : U → R extended to functions in C ∞ c ( G ) andhence to the one-point compactification G c . Let ( µ n ) n ∈ N be a family of probability measureson G converging to δ . Suppose there are numbers a k , a kl ∈ R such that ( a kl ) k,l =1 ...N ispositive semi-definite and, for all k, l = 1 , ..., N and n → ∞ : (i) (cid:82) G x k ( g ) dµ n ( g ) = a k /n + o (1 /n ) , (ii) (cid:82) G x k ( g ) x l ( g ) d µ n ( g ) = a kl /n + o (1 /n ) , (iii) µ n ( ˜ U c ) = o (1 /n ) for every -neighbourhood ˜ U ⊂ G c .Then the sequence (( µ n ) ∗ n ) n ∈ N converges *-weakly to a measure ν on G c which belongs tothe convolution semigroup ( ν t ) t ∈ R + whose corresponding operator semigroup ( T t ) t ∈ R + on C ( G c ) has infinitesimal generator L := dd t T t (cid:22) t =0 = N (cid:88) k =1 a k D X k + N (cid:88) k,l =1 a kl D X k D X l with dom( L ) = C ( G c ) . We would like to apply this theorem to our setting, namely where G = GL( A ) and g = gl ( A ) regarded as (real!) linear Lie group and algebra, respectively, and subspaces of5 ( A ). The plan is as follows: in a first step we shall construct a continuous-time stochasticprocess in G , and in a second step use this to obtain a description of the induced behaviourof the density matrix process ( ρ t ) t ∈ R + . In our setting this means we first have to definesuitable and physically realistic measures µ n to which to apply our limit procedure ofTheorem 3. The drift part should correspond to the original drift part resulting from (5).Putting ¯ L := 1 | J | (cid:88) j ∈ J Ad( v j ) ◦ L ◦ Ad( v ∗ j )and L j := Ad( v j ) ◦ ( L − ¯ L ) ◦ Ad( v ∗ j ) , let us define the measures µ n := 1 | J | (cid:88) j ∈ J δ exp (cid:0) τn / L j + τn ( ¯ L− τ L j ) (cid:1) , n ∈ N , (6)which conceptually imitate a diffusion part for the variation around the mean ¯ L and a driftpart for the mean movement. Apart from being mathematically clear and plausible fromthe classical Donsker invariance principle, the meaningfulness of this limit shall moreoverbe confirmed by numerical analysis carried out partially in the final section and mainly in[AHB14].Let us drop a quick side remark: as a rough first approximation for µ n we might alsostudy the purely drift-like µ (drift) n := 1 | J | (cid:88) j ∈ J δ exp( τn Ad( v j ) ◦L◦ Ad( v ∗ j )) , (7)similar to (5) but with scaling variable τ /n instead of τ as we now would like to keep τ fixed.In analogy to the law of large numbers this would lead to even nicer expressions and CPTdynamics but less faithful modelling; sometimes we will consider it briefly for comparisonreasons, cf. Remark 6. It can be shown that all other types of scaling (i.e., others than1 /n or 1 / √ n ) essential lead to either trivial or singular not well-defined expressions. Weshall therefore stick to µ n henceforth if not explicitly mentioned otherwise. Moreover, itis clear that Ad( v j ) ◦ ¯ L ◦
Ad( v ∗ j ) = ¯ L (as V is a group) and hence (cid:80) j ∈ J L j = 0. We have¯ L = 0 iff V satisfies the decoupling condition for ( H , L ).Using now the defining property of the coordinate maps in the limit n → ∞ , (i) isobtained with a series expansion of exp (cid:0) τn / L j + τn ( ¯ L − τ L j ) (cid:1) , namely: a k = lim n →∞ nτ (cid:90) G c x k ( g ) d µ n ( g )= lim n →∞ nτ | J | (cid:88) j ∈ J x k (cid:16) exp (cid:0) τn / L j + τn ( ¯ L − τ L j ) (cid:1)(cid:17) = lim n →∞ nτ | J | (cid:88) j ∈ J (cid:16) τn / (cid:104)L j , X k (cid:105) g + τn (cid:104) ¯ L − τ L j , X k (cid:105) g + τ n (cid:104)L j , X k (cid:105) g + O (cid:0) n / (cid:1)(cid:17) = (cid:104) ¯ L , X k (cid:105) g . a kl = lim n →∞ nτ (cid:90) G c x k ( g ) x l ( g ) d µ n ( g )= lim n →∞ nτ | J | (cid:88) j ∈ J x k (cid:16) exp (cid:0) τn / L j + τn ( ¯ L − τ L j ) (cid:1)(cid:17) x l (cid:16) exp (cid:0) τn / L j + τn ( ¯ L − τ L j ) (cid:1)(cid:17) = lim n →∞ nτ | J | (cid:88) j ∈ J (cid:16) τ n (cid:104)L j , X k (cid:105) g (cid:104)L j , X l (cid:105) g + O (cid:0) n / (cid:1)(cid:17) = τ | J | (cid:88) j ∈ J (cid:104)L j , X k (cid:105) g (cid:104)L j , X l (cid:105) g , for every k, l = 1 , . . . , N . Finally, it is easy to see that condition (iii) is satisfied as µ n hasdiscrete support in | J | points only which converge to 0 as n → ∞ . Thus we get L = N (cid:88) k =1 a k D X k + N (cid:88) k,l =1 a kl D X k D X l = D ¯ L + τ | J | (cid:88) j ∈ J D L j (8)for the generator of the limit convolution semigroup ( ν t ) t ∈ R + on G c , which can be inter-preted as a combination of drift and diffusion on G c . This has been the first big step in ourconstruction, namely the construction of the convolution semigroup of measures ( ν t ) t ∈ R + on G ; it implicitly describes a stochastic process ( α (cid:48) t ) t ∈ R + on G (according to Theorem 3)with α (cid:48) = id A .Our second step shall be to calculate the time evolution of the density matrix andrelated physically significant quantities out of the stochastic process ( α (cid:48) t ) t ∈ R + . This isslightly involved, but can be done using some tools which we are now going to derive.A general fact is that, for every f ∈ C ( G c ), we have E [ f ◦ α (cid:48) t ] = (cid:90) G c f ( g ) d ν t ( g ) = T t f ( ) . (9)We define the subsemigroup G [1] := { g ∈ G : (cid:107) g (cid:107) ≤ } ⊂ G ⊂ G c . Since α ( τ ) t are contrac-tions, the measures µ n must all be supported in G [1] . This implies that the convolutions µ ∗ mn of those measures are supported in G [1] (cf. Appendix A for a proof). For given t >
0, choosing a sequence ( m n ) n ∈ N such that m n /n → t , one can check that µ ∗ m n n → ν t .Hence the limit semigroup ( ν t ) t ∈ R + is supported in G [1] , meaning the Gaussian process( α (cid:48) t ) t ∈ R + stays almost surely in G [1] . Then it follows that T t preserves the closed subspace C ,b ( G [1] c ) ⊂ C ( G c ) of bounded continuous functions on the complement G [1] c of G [1] vanishing at the boundary { g ∈ G : (cid:107) g (cid:107) = 1 } : T t f ( g ) = (cid:90) G [1] f ( hg ) d ν t ( h ) = 0 , for every g ∈ G [1] , as f ( hg ) = 0 for h, g ∈ G [1] , i.e., T t f has support in G [1] c and isbounded by (cid:107) f (cid:107) ∞ . The corresponding quotient Banach space C ( G c ) /C ,b ( G [1] c ) can beidentified with C b ( G [1] ): namely, f ∈ C ( G c ) induces a function f (cid:22) G [1] ∈ C b ( G [1] ) andv.v., two extensions f , f of a function f ∈ C b ( G [1] ) to G c lead to f − f ∈ C ,b ( G [1] c ),thus a unique element in C ( G c ) /C ,b ( G [1] c ). Write q for the corresponding quotient mapand f [1] := q ( f ), for every f ∈ C ( G c ), so that f [1] ( g ) = f ( g ) if g ∈ G [1] . Then weget the quotient semigroup ( T [1] t ) t ∈ R + as in Appendix A, with infinitesimal generator K = q ( Lq − ( · )) and dom( K ) = q (dom( L )) (cid:39) C b ( G [1] ), the twice differentiable functionson G [1] which and whose first and second order derivatives are all bounded.7n order to achieve a description of the time evolution ( ρ t ) t ∈ R + of the density matrix,the idea is to study every entry of ρ t in a certain orthonormal basis. To this end, let( e k ) k =1 ...d be an arbitrary fixed orthonormal basis of A . We consider, for every k, l , thefunction f kl : g ∈ G (cid:55)→ (cid:104) e l , g ( e k ) (cid:105) , which is ν t -integrable (because bounded by 1 on the support) and which lies in C ∞ ( G ) butnot in C c ( G ). Write f [ ∞ ] kl for an arbitrary but fixed function in C ∞ c ( G ) (and hence C ∞ ( G c ))coinciding with f kl on G [1] , which can always be achieved, e.g. by multiplying with asmoothed indicator function on G [1] (easy exercise); moreover, following the notation ofthe preceding paragraph we write f [ ∞ , kl := q ( f [ ∞ ] kl ). Then E [ f kl ( α (cid:48) t ( · ) g )] = E [ f [ ∞ ] kl ( α (cid:48) t ( · ) g )] = T t f [ ∞ ] kl ( g ) = T [1] t f [ ∞ , kl ( g ) , t ∈ R + , g ∈ G [1] . Noticing furthermore that f [ ∞ , kl ∈ C b ( G [1] ), we have, for every g ∈ G [1] , Kf [ ∞ , kl ( g ) = q ( Lf [ ∞ ] kl )( g ) = Lf [ ∞ ] kl ( g )= 1 | J | (cid:88) j ∈ J (cid:16) D ¯ L + τ D L j (cid:17) f [ ∞ ] kl ( g )= dd t (cid:104) e l , e t ¯ L g ( e k ) (cid:105) (cid:22) t =0 + τ | J | (cid:88) j ∈ J d d t (cid:104) e l , e t L j g ( e k ) (cid:105) (cid:22) t = s =0 = (cid:104) e l , (cid:16) ¯ L + τ | J | (cid:88) j ∈ J L j (cid:17) ( ge k ) (cid:105) = (cid:104) e l , ˆ L ( ge k ) (cid:105) (10)with ˆ L := ¯ L + τ | J | (cid:88) j ∈ J L j ∈ B ( A ) . Analogously K n f [ ∞ , kl ( g ) = (cid:104) e l , ˆ L n ( ge k ) (cid:105) , which is bounded by (cid:107) ˆ L (cid:107) n uniformly in g ∈ G [1] .Therefore, z ∈ C (cid:55)→ ∞ (cid:88) n =0 z n n ! K n f [ ∞ , kl = ∞ (cid:88) n =0 z n n ! (cid:104) e l , ˆ L n ( · e k ) (cid:105) = (cid:104) e l , e z ˆ L ( · e k ) (cid:105) ∈ C ∞ ( K )converges and is an analytic continuation of t (cid:55)→ T [1] t f [ ∞ , kl , so f [ ∞ , kl ∈ C ∞ ( K ) is an entireanalytic vector for T [1] t (cf. Appendix A).Recalling (9) and noticing that ∈ G [1] and that f kl is linear in its argument, thisenables us to compute the expectation value E [ (cid:104) e l , α (cid:48) t ( e k ) (cid:105) ] = E [ f kl ( α (cid:48) t ( · ))] = 2 E [ f kl ( 12 α (cid:48) t ( · ))] = 2 E [ f [ ∞ ] kl ( 12 α (cid:48) t ( · ))]=2 T t f [ ∞ ] kl (cid:16) (cid:17) = 2 T [1] t f [ ∞ , kl (cid:16) (cid:17) = (cid:104) e l , e t ˆ L ( e k ) (cid:105) , for every k, l , so E [ α (cid:48) t ( e k )] = e t ˆ L ( e k ). Since this holds for every basis vector e k , it holdsfor all elements in A . Applying it to the A -valued “density matrix stochastic process”( ρ t := α (cid:48) t ( ρ )) t ∈ R + , we find E [ ρ t ] = e t ˆ L ( ρ ) , concluding our second step, too.We summarize this all in 8 heorem 4. The continuous-time limit ( α (cid:48) t ) t ∈ R + of the above random walk determinedby a quantum system ( H , L ) and random dynamical decoupling with decoupling set V =( v j ) j ∈ J and (6) leads to a contraction semigroup with generator (8) . The density matrix ( ρ t ) t ∈ R + is then a stochastic process in A with expectation E [ ρ t ] = e t ˆ L ( ρ ) , ∀ t ≥ , where ˆ L = ¯ L + τ | J | (cid:88) j ∈ J L j . Remark 5.
If the intrinsic time evolution is not constant (but still continuously differen-tiable), then the continuous-time limit can be carried out in the same way, resulting in atime-dependent generatorˆ L ( t ) = ¯ L ( t ) + τ | J | (cid:88) j ∈ J Ad( v j ) ◦ ( L ( t ) − ¯ L ( t )) ◦ Ad( v ∗ j ) , ∀ t ∈ R + , and just a time-ordered integral [LB13] E [ ρ t ] = T e (cid:82) t ˆ L ( t (cid:48) ) d t (cid:48) ( ρ ) = ∞ (cid:88) n =0 (cid:90) t (cid:90) t (cid:48) n . . . (cid:90) t (cid:48) ˆ L ( t (cid:48) n ) . . . ˆ L ( t (cid:48) ) d t (cid:48) . . . d t (cid:48) n (11)instead of the semigroup. However, this analytic expression will be a good approximationof the original random walk usually only if τ is sufficiently small such that τ (cid:13)(cid:13)(cid:13) dd t (cid:48) L ( t ) (cid:13)(cid:13)(cid:13) (cid:28) (cid:107)L ( t (cid:48) ) (cid:107) , ∀ t (cid:48) ∈ [0 , t ] . For simplicity we shall only deal with the time-independent version here below.
Remark 6.
Let us write ˆ L (drift) for the generator and ( ν (drift) t ) t ∈ R + for the convolutionsemigroup of measures corresponding to the drift-like continuous-time limit of the randomwalk with µ (drift) n as in (7) instead of (6), and accordingly E (drift) and Var (drift) for expec-tation and variances with respect to ( ν (drift) t ) t ∈ R + . Then going through the constructionof Theorem 4, we see that the generator of ( T (drift) t ) t ∈ R + becomes L (drift) = D ¯ L . Henceˆ L (drift) = ¯ L , which vanishes iff the decoupling condition is fulfilled iff the original timeevolution α was unitary, according to Theorem 2. In this case T (drift) t = id, for all t ∈ R + ,hence E (drift) [ ρ t ] = ρ . Example 7. (1) We continue our Example 1 from the preceding section, the N -qubitsystem, with V the group of tensor products of N Pauli matrices. Suppose our timeevolution is unitary, so L = i Ad( H ) with H the system Hamiltonian. Then we findˆ L (drift) = ¯ L = 0, so L j = i[ v j Hv j , · ] andˆ L = − τ | J | (cid:88) j ∈ J [ v j Hv j , [ v j Hv j , · ]] . Now a variety of special cases may be investigated. If e.g. H acts only on the first qubit,i.e., it can be written as H = H ⊗ ⊗ ( n − , then so does ˆ L . If moreover ρ splits as aproduct state on the tensor factors, then so does E [ ρ t ], for all t >
0, with only the firsttensor factor changing over time. 92) Another example, which shall turn up in Figure 1 and which is treated in detailin [AHB14] is the amplitude-damping model. In this setting H is the one-qubit Hilbertspace C , A = M ( C ) and L ( x ) = − γ (cid:0) x − i σ x − i xσ − σ xσ − σ xσ − i σ xσ − i σ xσ (cid:1) , x ∈ A, with a certain coefficient γ ∈ R + . The Pauli matrices constitute the decoupling set V = { v = , v j = σ j : j = 1 , , } . In order to compute the generator ˆ L = ¯ L + τ (cid:80) j =0 L j ,one checks: ¯ L ( x ) = − γ (cid:0) x − σ xσ − σ xσ (cid:1) and L ( x ) = −L ( x ) = −L ( x ) = L ( x ) = − i γ (cid:0) σ x + xσ + σ xσ − σ xσ (cid:1) , x ∈ A. A computer can now easily calculate E [ ρ t ] = e t ˆ L ( ρ ), for any given t > ρ ∈ A . The result should be a good approximation for the actual random walk if τ (cid:28) / (cid:107)L(cid:107) .Important related quantities like gate fidelity shall be computed in the following sec-tion. Before concluding the present section let us derive here a tool that shall allow usto compute higher moments (including variance) of random variables, beyond the presentlinear ones (expectation value). Proposition 8.
In the setting of Theorem 4, for all x , . . . , x n , y , . . . y n ∈ A , let f x ...x n ,y ...y n ( g ) := (cid:104) y , g ( x ) (cid:105) · · · (cid:104) y n , g ( x n ) (cid:105) = ( f x ,y · · · f x n ,y n )( g ) , g ∈ G, and define the linear operator ˆ L ( n ) on A ⊗ n by ˆ L ( n ) ( x ⊗ ... ⊗ x n ) := n (cid:88) l =1 x ⊗ ... ⊗ ¯ L ( x l ) ⊗ ... ⊗ x n + τ | J | (cid:88) j ∈ J (cid:16) n (cid:88) l =1 x ⊗ ... ⊗ L j ( x l ) ⊗ ... ⊗ x n + 2 n (cid:88) k =1 ,l>k x ⊗ ... ⊗ L j ( x k ) ⊗ ... ⊗ L j ( x l ) ⊗ ... ⊗ x n (cid:17) and linear extension. Then E [ f x ...x n ,y ...y n ◦ α (cid:48) t ] = (cid:104) ( y ⊗ ... ⊗ y n ) , e t ˆ L ( n ) ( x ⊗ ... ⊗ x n ) (cid:105) . Proof.
Following the notation and the truncation and quotient space procedure exactlyas in the case of f kl , we can define (not uniquely) a smooth function f [ ∞ ] x ...x n ,y ...y n ∈ C ∞ ( G )from f x ...x n ,y ...y n and hence a function f [ ∞ , x ...x n ,y ...y n ∈ C b ( G [1] ), which is analytic for K , i.e., in C ∞ ( K ); we can and do choose it such that f [ ∞ , x ...x n ,y ...y n = f [ ∞ ] x ,y · · · f [ ∞ ] x n ,y n .10xploiting then the product rule for differentiation, we obtain Kf [ ∞ , x ...x n ,y ...y n ( ) = Lf [ ∞ ] x ...x n ,y ...y n ( )= L (cid:0) f [ ∞ ] x ,y · · · f [ ∞ ] x n ,y n (cid:1) ( )= D ¯ L (cid:0) f [ ∞ ] y ,x · · · f [ ∞ ] y n ,x n (cid:1) ( ) + τ | J | (cid:88) j ∈ J D L j (cid:0) f [ ∞ ] y ,x · · · f [ ∞ ] y n ,x n (cid:1) ( )= n (cid:88) l =1 (cid:104) y , x (cid:105) · · · (cid:104) y l , ¯ L ( x l ) (cid:105) · · · (cid:104) y n , x n (cid:105) + τ | J | (cid:88) j ∈ J (cid:16) n (cid:88) l =1 (cid:104) y , x (cid:105) · · · (cid:104) y l , L j ( x l ) (cid:105) · · · (cid:104) y n , x n (cid:105) + n (cid:88) k =1 ,l>k (cid:104) y , x (cid:105) · · · (cid:104) y k , L j ( x k ) (cid:105) · · · (cid:104) y l , L j ( x l ) (cid:105) · · · (cid:104) y n , x n (cid:105) (cid:17) = (cid:104) ( y ⊗ ... ⊗ y n ) , ˆ L ( n ) ( x ⊗ ... ⊗ x n ) (cid:105) . Analogously, for higher powers we have K k f [ ∞ , x ,...,x n ,y ,...y n ( ) = (cid:104) ( y ⊗ ... ⊗ y n ) , ( ˆ L ( n ) ) k ( x ⊗ ... ⊗ x n ) (cid:105) whence exp( t ˆ L ( n ) ) is well-defined on A ⊗ n . Thus we find E [ f x ,...,x n ,y ,...y n ◦ α (cid:48) t ] = T [1] t f [ ∞ , x ,...,x n ,y ,...y n ( ) = (cid:104) ( y ⊗ ... ⊗ y n ) , e t ˆ L ( n ) ( x ⊗ ... ⊗ x n ) (cid:105) . (cid:3) Analogously, one can prove
Proposition 9.
In the setting of Theorem 4, for all x, x i , y, y i ∈ A , let ˇ L (2) ( x ⊗ y ) := ¯ L ( x ) ⊗ y + x ⊗ ¯ L † ( y )+ τ | J | (cid:88) j ∈ J (cid:16) L j ( x ) ⊗ y + 2 L j ( x ) ⊗ L † j ( y ) + x ⊗ ( L † j ) ( y ) (cid:17) and ˇ L (4) ( x ⊗ y ⊗ x ⊗ y ) := ¯ L ( x ) ⊗ y ⊗ x ⊗ y + x ⊗ ¯ L ( y ) ⊗ x ⊗ y + x ⊗ y ⊗ ¯ L † ( x ) ⊗ y + x ⊗ y ⊗ x ⊗ ¯ L † ( y )+ τ | J | (cid:88) j ∈ J (cid:16) L j ( x ) ⊗ y ⊗ x ⊗ y + x ⊗ L j ( y ) ⊗ x ⊗ y + x ⊗ y ⊗ ( L † j ) ( x ) ⊗ y + x ⊗ y ⊗ x ⊗ ( L † j ) ( y )+ 2( L j ( x ) ⊗ y + x ⊗ L j ( y )) ⊗ ( L † j ( x ) ⊗ y + x ⊗ L † j ( y ))+ 2 L j ( x ) ⊗ L j ( y ) ⊗ x ⊗ y + 2 x ⊗ y ⊗ L † j ( x ) ⊗ L † j ( y ) (cid:17) . Then L ( | f [ ∞ ] x,y | )( ) = (cid:104) ( y ⊗ x ) , ˇ L (2) ( x ⊗ y ) (cid:105) and L ( | f [ ∞ ] x,y | )( ) = (cid:104) ( y ⊗ x ⊗ y ⊗ x ) , ˇ L (4) ( x ⊗ y ⊗ x ⊗ y ) (cid:105) . Distribution of the gate fidelity
The most interesting quantity in control theory of a quantum system is its fidelity; as wewant to decouple independently of the state, we consider the gate fidelity [LB13], whichis given by the random variable F t := 1 − d d (cid:88) k,l =1 |(cid:104) e l , (id − α (cid:48) t )( e k ) (cid:105)| , independent of the actual choice of the orthonormal basis ( e k ) k =1 ...d of A . Most otherversions of fidelity can be treated using similar ideas.We are interested in E [ F t ] and Var[ F t ]. Proposition 10.
In the setting of Theorem 4, the expectation and variance of the gatefidelity of the quantum system ( H , L ) with decoupling set V are given by − E [ F t ] = 1 d d (cid:88) k,l =1 (cid:16) δ k,l − δ k,l (cid:104) e l , (e t ˆ L + e t ˆ L † )( e k ) (cid:105) + (cid:104) e l ⊗ e k , e t ˇ L (2) ( e k ⊗ e l ) (cid:105) (cid:17) and Var[ F t ] = 1 d d (cid:88) i,j,k,l =1 (cid:16) δ k,l δ i,j − δ i,j δ k,l (cid:104) e l , (e t ˆ L + e t ˆ L † )( e k ) (cid:105) + 2 δ i,j (cid:104) e l ⊗ e k , e t ˇ L (2) ( e k ⊗ e l ) (cid:105) + δ i,j δ k,l (cid:104) e i ⊗ e k , (e t ˇ L (1 , + e t ˇ L (1 , + e t ˇ L (1 , † + e t ˇ L (1 , † )( e i ⊗ e k ) (cid:105)− δ i,j (cid:104) e i ⊗ e l ⊗ e k , (e t ˇ L (3 , + e t ˇ L (3 , )( e i ⊗ e k ⊗ e l ) (cid:105) + (cid:104) e j ⊗ e i ⊗ e l ⊗ e k , e t ˇ L (4) ( e i ⊗ e j ⊗ e k ⊗ e l ) (cid:105) (cid:17) − d (cid:16) d (cid:88) k,l =1 (cid:16) δ k,l − δ k,l (cid:104) e l , (e t ˆ L + e t ˆ L † )( e k ) (cid:105) + (cid:104) e l ⊗ e k , e t ˇ L (2) ( e k ⊗ e l ) (cid:105) (cid:17)(cid:17) . with ˇ L (1 , ( x ⊗ y ) = ¯ L ( x ) ⊗ y + x ⊗ ¯ L ( y )+ τ | J | (cid:88) j ∈ J L j ( x ) ⊗ y + 2 L j ( x ) ⊗ L j ( y ) + x ⊗ L j ( y )ˇ L (1 , ( x ⊗ y ) = ¯ L † ( x ) ⊗ y + x ⊗ ¯ L ( y )+ τ | J | (cid:88) j ∈ J ( L † j ) ( x ) ⊗ y + 2 L † j ( x ) ⊗ L j ( y ) + x ⊗ L j ( y )ˇ L (3 , ( x ⊗ y ⊗ z ) = ¯ L ( x ) ⊗ y ⊗ z + x ⊗ ¯ L ( y ) ⊗ z + x ⊗ y ⊗ ¯ L † ( z )+ τ | J | (cid:88) j ∈ J (cid:16) L j ( x ) ⊗ y ⊗ z + 2 L j ( x ) ⊗ L j ( y ) ⊗ z + 2 L j ( x ) ⊗ y ⊗ L † j ( z )+ 2 x ⊗ L j ( y ) ⊗ L † j ( z ) + x ⊗ L j ( y ) ⊗ z + x ⊗ y ⊗ ( L † j ) ( z ) (cid:17) ˇ L (3 , ( x ⊗ y ⊗ z ) = ¯ L † ( x ) ⊗ y ⊗ z + x ⊗ ¯ L ( y ) ⊗ z + x ⊗ y ⊗ ¯ L † ( z )+ τ | J | (cid:88) j ∈ J (cid:16) ( L † j ) ( x ) ⊗ y ⊗ z + 2 L † j ( x ) ⊗ L j ( y ) ⊗ z + 2 L † j ( x ) ⊗ y ⊗ L † j ( z )+ 2 x ⊗ L j ( y ) ⊗ L † j ( z ) + x ⊗ L j ( y ) ⊗ z + x ⊗ y ⊗ ( L † j ) ( z ) (cid:17) . roof. Since we know α (cid:48) t , we find:1 − E [ F t ] = 1 d d (cid:88) k,l =1 E [ |(cid:104) e l , (id − α (cid:48) t )( e k ) (cid:105)| ]= 1 d d (cid:88) k,l =1 E [ δ k,l − δ k,l (cid:60)(cid:104) e l , α (cid:48) t ( e k ) (cid:105) + |(cid:104) e l , α (cid:48) t ( e k ) (cid:105)| ]= 1 d d (cid:88) k,l =1 (cid:16) δ k,l − δ k,l T t (cid:16) (cid:60) f [ ∞ , kl ( ) (cid:17) + T t | f [ ∞ , kl | ( ) (cid:17) = 1 d d (cid:88) k,l =1 (cid:16) δ k,l − δ k,l ∞ (cid:88) n =0 t n n ! K n (cid:16) f [ ∞ , kk + f [ ∞ , kk (cid:17) ( ) + ∞ (cid:88) n =0 t n n ! K n (cid:16) | f [ ∞ , kl | (cid:17) ( ) (cid:17) = 1 d d (cid:88) k,l =1 (cid:16) δ k,l − δ k,l ∞ (cid:88) n =0 t n n ! ( (cid:104) e k , ˆ L n ( e k ) (cid:105) + (cid:104) e k , ( ˆ L † ) n ( e k ) (cid:105) + ∞ (cid:88) n =0 t n n ! (cid:104) e l ⊗ e k , ( ˇ L (2) ) m ( e k ⊗ e l ) (cid:105) (cid:17) = 1 d d (cid:88) k,l =1 (cid:16) δ k,l − δ k,l (cid:104) e k , (e t ˆ L + e t ˆ L † )( e k ) (cid:105) + (cid:104) e l ⊗ e k , e t ˇ L (2) ( e k ⊗ e l ) (cid:105) (cid:17) . (12)Here the third equality follows from (9) and the quotient procedure; the fifth from the Leib-niz rule and Proposition 9, noticing that f kk ( g ) = (cid:104) ge k , e k (cid:105) and L n f kk ( ) = (cid:104) ˆ L n ( e k ) , e k (cid:105) = (cid:104) e k , ( ˆ L † ) n ( e k ) (cid:105) .The variance is obtained analogously:Var[ F t ] = E [ F t ] − E [ F t ] = E [(1 − F t ) ] − E [1 − F t ] = 1 d d (cid:88) i,j,k,l =1 E [ |(cid:104) e i , (id − α (cid:48) t )( e j ) (cid:105)| |(cid:104) e l , (id − α (cid:48) t )( e k ) (cid:105)| ] − (cid:16) d d (cid:88) k,l =1 E [ |(cid:104) e l , (id − α (cid:48) t )( e k ) (cid:105)| ] (cid:17) = 1 d d (cid:88) i,j,k,l =1 E (cid:104) δ k,l δ i,j − δ i,j δ k,l (cid:104) e l , ( α (cid:48) t + α (cid:48)† t )( e k ) (cid:105) + 2 δ i,j (cid:104) e l ⊗ e k , ( α (cid:48) t ⊗ α (cid:48)† t )( e k ⊗ e l ) (cid:105) + δ i,j δ k,l (cid:104) e i ⊗ e k , ( α (cid:48) t ⊗ α (cid:48) t + α (cid:48)† t ⊗ α (cid:48) t + α (cid:48) t ⊗ α (cid:48)† t + α (cid:48)† t ⊗ α (cid:48)† t )( e i ⊗ e k ) (cid:105)− δ i,j (cid:104) e i ⊗ e l ⊗ e k , (( α (cid:48) t + α (cid:48)† t ) ⊗ α (cid:48) t ⊗ α (cid:48)† t )( e i ⊗ e k ⊗ e l ) (cid:105) + (cid:104) e j ⊗ e i ⊗ e l ⊗ e k , ( α (cid:48) t ⊗ α (cid:48)† t ⊗ α (cid:48) t ⊗ α (cid:48)† t )( e i ⊗ e j ⊗ e k ⊗ e l ) (cid:105) (cid:105) − d (cid:16) d (cid:88) k,l =1 (cid:16) δ k,l − δ k,l (cid:104) e l , (e t ˆ L + e t ˆ L † )( e k ) (cid:105) + (cid:104) e l ⊗ e k , e t ˇ L (2) ( e k ⊗ e l ) (cid:105) (cid:17)(cid:17) . The terms in the first sum are all 0 , , , , E [ (cid:104) e i ⊗ e k , ( α (cid:48)† t ⊗ α (cid:48) t )( e i ⊗ e k ) (cid:105) ] = E [ f [ ∞ ] ii f [ ∞ ] kk ◦ α (cid:48) t ]= ∞ (cid:88) n =0 t n n ! L n (cid:0) f [ ∞ ] ii f [ ∞ ] kk (cid:1) ( )= ∞ (cid:88) n =0 t n n ! (cid:104) e i ⊗ e k , ( ˇ L (1 , ) n ( e i ⊗ e k ) (cid:105) with ˇ L (1 , ( x ⊗ y ) = ¯ L † ( x ) ⊗ y + x ⊗ ¯ L ( y )+ τ | J | (cid:88) j ∈ J ( L † j ) ( x ) ⊗ y + 2 L † j ( x ) ⊗ L j ( y ) + x ⊗ L j ( y )because L (cid:0) f [ ∞ ] ii f [ ∞ ] kk (cid:1) ( ) =( D ¯ L f [ ∞ ] ii ) f [ ∞ ] kk ( ) + f [ ∞ ] ii ( D ¯ L f [ ∞ ] kk )( )+ τ | J | (cid:88) j ∈ J ( D L j f [ ∞ ] ii ) f [ ∞ ] kk ( ) + 2( D L j f [ ∞ ] ii )( D L j f [ ∞ ] ii )( ) + f [ ∞ ] kk ( D L j f [ ∞ ] kk )( )= dd t f [ ∞ ] ii (e t ¯ L ) f [ ∞ ] kk ( ) + f [ ∞ ] ii ( ) dd t f [ ∞ ] kk (e t ¯ L )+ τ | J | (cid:88) j ∈ J d d t d s f [ ∞ ] ii (e s L j e t L j ) f [ ∞ ] kk ( )+ 2 d d t d s f [ ∞ ] ii (e s L j ) f [ ∞ ] ii (e t L j ) + f [ ∞ ] kk ( ) d d t d s f [ ∞ ] kk (e s L j e t L j ) (cid:22) s = t =0 = (cid:104) e i ⊗ e k , ¯ L † ( e i ) ⊗ e k + e i ⊗ ¯ L ( e k ) (cid:105) + τ | J | (cid:88) j ∈ J (cid:104) e i ⊗ e k , ( L † j ) ( e i ) ⊗ e k + 2 L † j ( e i ) ⊗ L j ( e k ) + e i ⊗ L j ( e k ) (cid:105) = (cid:104) e i ⊗ e k , ˇ L (1 , ( e i ⊗ e k ) (cid:105) . For the other 2-(anti-)linear expressions we obtain similar results but with operatorsˇ L (1 , , ˇ L (1 , † , ˇ L (1 , † instead. The remaining terms are treated analogously, by letting L act on the corresponding m -(anti-)linear functions, e.g. the 3-(anti-)linear case is obtainedwriting E [( f [ ∞ ] ii + f [ ∞ ] ii ) f [ ∞ ] kl f [ ∞ ] kl ◦ α (cid:48) t ] = ∞ (cid:88) n =0 t n n ! L n (( f [ ∞ ] ii + f [ ∞ ] ii ) f [ ∞ ] kl f [ ∞ ] kl )= ∞ (cid:88) n =0 t n n ! (cid:104) e i ⊗ e l ⊗ e k , (( ˇ L (3 , ) n + ( ˇ L (3 , ) n )( e i ⊗ e k ⊗ e l (cid:105) . Putting together all of this and expressing the power series back again as exponential14unctions, we finally obtain the statement in the proposition:Var[ F t ] = 1 d d (cid:88) i,j,k,l =1 E (cid:104) δ k,l δ i,j − δ i,j δ k,l (cid:104) e l , ( α (cid:48) t + α (cid:48)† t )( e k ) (cid:105) + 2 δ i,j (cid:104) e l ⊗ e k , ( α (cid:48) t ⊗ α (cid:48)† t )( e k ⊗ e l ) (cid:105) + δ i,j δ k,l (cid:104) e i ⊗ e k , ( α (cid:48) t ⊗ α (cid:48) t + α (cid:48)† t ⊗ α (cid:48) t + α (cid:48) t ⊗ α (cid:48)† t + α (cid:48)† t ⊗ α (cid:48)† t )( e i ⊗ e k ) (cid:105)− δ i,j (cid:104) e i ⊗ e l ⊗ e k , (( α (cid:48) t + α (cid:48)† t ) ⊗ α (cid:48) t ⊗ α (cid:48)† t )( e i ⊗ e k ⊗ e l ) (cid:105) + (cid:104) e j ⊗ e i ⊗ e l ⊗ e k , ( α (cid:48) t ⊗ α (cid:48)† t ⊗ α (cid:48) t ⊗ α (cid:48)† t )( e i ⊗ e j ⊗ e k ⊗ e l ) (cid:105) (cid:105) − d (cid:16) d (cid:88) k,l =1 (cid:16) δ k,l − δ k,l (cid:104) e l , (e t ˆ L + e t ˆ L † )( e k ) (cid:105) + (cid:104) e l ⊗ e k , e t ˇ L (2) ( e k ⊗ e l ) (cid:105) (cid:17)(cid:17) = 1 d d (cid:88) i,j,k,l =1 (cid:16) δ k,l δ i,j − δ i,j δ k,l (cid:104) e l , (e t ˆ L + e t ˆ L † )( e k ) (cid:105) + 2 δ i,j (cid:104) e l ⊗ e k , e t ˇ L (2) ( e k ⊗ e l ) (cid:105) + δ i,j δ k,l (cid:104) e i ⊗ e k , (e t ˇ L (1 , + e t ˇ L (1 , + e t ˇ L (1 , † + e t ˇ L (1 , † )( e i ⊗ e k ) (cid:105)− δ i,j (cid:104) e i ⊗ e l ⊗ e k , (e t ˇ L (3 , + e t ˇ L (3 , )( e i ⊗ e k ⊗ e l ) (cid:105) + (cid:104) e j ⊗ e i ⊗ e l ⊗ e k , e t ˇ L (4) ( e i ⊗ e j ⊗ e k ⊗ e l ) (cid:105) (cid:17) − d (cid:16) d (cid:88) k,l =1 (cid:16) δ k,l − δ k,l (cid:104) e l , (e t ˆ L + e t ˆ L † )( e k ) (cid:105) + (cid:104) e l ⊗ e k , e t ˇ L (2) ( e k ⊗ e l ) (cid:105) (cid:17)(cid:17) . (cid:3) For comparison reasons and some applications in [AHB14], we would like to statethe analogous formulae for the case of the drift-like continuous-time limit in the sense ofRemark 6. Since L (drift) can be regarded as a special case of L with vanishing L j , theexpressions in Proposition 10 simplify significantly and we obtain: Proposition 11.
In the setting of Proposition 10 but with L (drift) instead of L , we obtain E (drift) [ F t ] = 1 − d d (cid:88) k,l =1 |(cid:104) e l , (id − e t ˆ L (drift) )( e k ) (cid:105)| and Var (drift) [ F t ] = 0 . Some readers might find the vanishing variance intuitively expected, given that thelimiting procedure corresponds somehow to the classical law of large numbers where con-vergence is almost surely to the (non-constant but time-dependent) expectation value.
Proof.
The expression for E (drift) follows immediately from that of E in the precedingproof, specialising to ˆ L (drift) : since (cid:104) e l ⊗ e k , e t ( ¯ L⊗ id + id ⊗ ¯ L † ) ( e k ⊗ e l ) (cid:105) = (cid:104) e l ⊗ e t ¯ L ( e k ) , e t ¯ L ( e k ) ⊗ e l (cid:105) , the last line in (12) becomes simply (cid:104) e l ⊗ (id + e t ¯ L )( e k ) , (id + e t ¯ L )( e k ) ⊗ e l ) (cid:105) = |(cid:104) e l ⊗ (id + e t ¯ L )( e k ) | . (drift) , we analogously compute:Var (drift) [ F t ] = E (drift) [ F t ] − E (drift) [ F t ] = E (drift) [(1 − F t ) ] − E (drift) [1 − F t ] = 1 d d (cid:88) i,j,k,l =1 E (drift) [ |(cid:104) e i , (id − α (cid:48) t )( e j ) (cid:105)| |(cid:104) e l , (id − α (cid:48) t )( e k ) (cid:105)| ] − (cid:16) d d (cid:88) k,l =1 E (drift) [ |(cid:104) e l , (id − α (cid:48) t )( e k ) (cid:105)| ] (cid:17) = 1 d d (cid:88) i,j,k,l =1 |(cid:104) e i , (id − e t ˆ L (drift) )( e j ) (cid:105)| |(cid:104) e l , (id − e t ˆ L (drift) )( e k ) (cid:105)| − (cid:16) d d (cid:88) k,l =1 |(cid:104) e l , (id − e t ˆ L (drift) )( e k ) (cid:105)| (cid:17) =0 . (cid:3) Example 12.
We return to our former illustrative Example 1. Since L = i Ad( H ), wefind that L † = −L and hence ˆ L † = ˆ L, ˇ L (2) = ˆ L (2) , as follows immediately from the respective definition in Propositions 8 and 9. For shorttimes t the results of Proposition 10 become:1 − E [ F t ] ≈ − N N (cid:88) k t (cid:104) e k , ˆ L ( e k ) (cid:105) + 14 N N (cid:88) k,l t (cid:104) e k ⊗ e l , ˆ L (2) ( e l ⊗ e k ) (cid:105) = 2 tτ N | J | (cid:88) j ∈ J N (cid:88) k,l |(cid:104) e k , [ v j Hv j , e l ] (cid:105)| = 2 tτ N | J | (cid:88) j ∈ J N (cid:88) k | [ v j Hv j , e k ] | , which for special cases of H can be further simplified, but in general will be used in thisform for a computer and is of order O ( τ t (cid:107) H (cid:107) ). A similar procedure may be applied tovariance.In contrast, in the case of the drift-like limit, we would simply get E (drift) [ F t ] = 1 , Var (drift) [ F t ] = 0 , t ∈ R + , which is obviously less realistic than the diffusion-like limit, but on the other hand con-firms that for unitary time-evolution ( α t ) t ∈ R + dynamical decoupling works (i.e., decouples)optimally, in contrast to other types of α , cf. also Theorem 2! Remark 13 (Errors) . Theorem 4 gives us the expectation of our quantities in the con-tinuum limit, but we must ask two questions:-(1) How big is the difference between continuum limit and original discrete randompaths? 162) What is the distribution of the actual (continuum limit) paths around the expecta-tion value?These two errors add up to give the total maximal error, which we have to estimate now.Concerning (1), one has to work with a kind of Berry-Esseen theorem [Shi96] on theapproximation of random walks by Brownian motion. This is quite complicated, but wesatisfy ourselves here with the fact that this error tends to 0 as τ → Suppose a given quantum system ( H , L ) undergoes decoherence caused by interaction withan external quantum heat bath described by another quantum system ( H , i ad( H )).Then according to standard axioms of quantum mechanics, time evolution of the total(closed) system is unitary, thus described by a one-parameter automorphism family onthe operators of the total Hilbert space H (cid:48) = H ⊗ H , namely t ∈ R + (cid:55)→ T e i (cid:82) t ad( H (cid:48) ( t (cid:48) )) d t (cid:48) , and H (cid:48) is the (possibly time-dependent) Hamiltonian of the total system on H (cid:48) and thetime-ordered integral is defined in analogy to (11). The heat bath may be infinite-dimensional separable, but the involved Hamiltonian H (cid:48) is henceforth supposed to beuniformly bounded on compact intervals. It is unclear whether dynamical decouplingworks without this assumption, and maybe alternative requirements would have to bemade in case of unboundedness, cf. [AHB14, App.] for further discussion.The actual dynamics perceived on the subsystem H is given by t ∈ R + (cid:55)→ α t := E ◦ T e i (cid:82) t ad( H (cid:48) ( t (cid:48) )) d t (cid:48) ( · ⊗ ρ θ ) , where E : B ( H (cid:48) ) → B ( H ) is the partial trace (conditional expectation) onto the sub-system and ρ θ the initial state of the heat bath [LB13, BP02]. The resulting perceiveddynamics ( α t ) t ∈ R + then becomes a family of CPT maps. Under special assumptions on H (cid:48) , it actually produces the CPT semigroup with infinitesimal generator L , the Lindbladoperator, but usually α t is no longer an automorphism. We call this phenomenon, wherea CPT semigroup time evolution arises from interaction with an external quantum heatbath and unitary time evolution on the total system, extrinsic decoherence because thenon-unitarity of time evolution of the original system is caused by interaction with theexternal heat bath.In contrast to this, intrinsic decoherence we call the situation where time evolution ofa closed system ( H , L ) is no longer unitary and the non-unitarity is intrinsic to the system,i.e., does not arise from (unitary) interaction with a heat bath. It is a fundamental questionwhether this actually occurs in nature or whether the axiom of unitarity is always fulfilled– on a sufficiently large total system. Mathematically the two cases are described in thesame way (by CPT semigroups with unitary dilations), and also physically with usualobservations they seem to be indistinguishable.However, applying dynamical decoupling in the case of the above type of extrinsic de-coherence, the time evolution of the total system is unitary, and so the perceived evolution17n the subsystem is given by the discrete stochastic process α ( τ ) nτ = E ◦ n (cid:89) i =1 Ad( v j i ⊗ ) ◦ T e i (cid:82) iτ ( i − τ ad( H (cid:48) ( t (cid:48) )) d t (cid:48) ◦ Ad( v ∗ j i ⊗ )( · ⊗ ρ θ ) . Now we notice that, if (1) is satisfied for all x ∈ B ( H ), then it is also satisfied forall x ∈ B ( H (cid:48) ) modulo ⊗ B ( H ). In fact, x ∈ B ( H (cid:48) ) can be written as a finite sum (cid:80) k y k ⊗ z k + ⊗ ˜ z , with certain traceless y k ∈ B ( H ) and with z k , ˜ z ∈ B ( H ), and then1 | J | (cid:88) j ∈ J (cid:88) k ( v j ⊗ )( y k ⊗ z k )( v j ⊗ ) ∗ + ⊗ ˜ z = 1 | J | (cid:88) k (cid:16) (cid:88) j ∈ J v j y k v ∗ j (cid:17) ⊗ z k + ⊗ ˜ z = ⊗ ˜ z. Consider now for x the (possibly time-dependent) Hamiltonian H (cid:48) = (cid:80) k H ,k ⊗ H ,k + ⊗ H . The heat bath is by definition in a thermal equilibrium state ρ θ independent oftime, i.e., ad( H )( ρ θ ) = 0. Let L (cid:48) := i ad( H (cid:48) ) be the (purely unitary) Lindbladian of thetotal system and thus ¯ L (cid:48) = i ad( ⊗ H ), so ¯ L ( x ⊗ ρ θ ) = 0, for all x ∈ A . Then we obtainˆ L (cid:48) = ¯ L (cid:48) + τ | J | (cid:88) j ∈ J Ad( v j ⊗ ) ◦ ( L (cid:48) − ¯ L (cid:48) ) ◦ Ad( v ∗ j ⊗ ) , and hence E [ ρ t ] = E ◦ T e (cid:82) t ˆ L (cid:48) ( t (cid:48) ) d t (cid:48) ( ρ ⊗ ρ θ ) . The main dynamics comes from ¯ L (cid:48) , which leaves ρ ⊗ ρ θ invariant, but L (cid:48) j changes it, sothat higher-order terms disturb the invariance of the state.We can conclude: if the system dynamics is determined by extrinsic decoherence thenthe decoupling condition is satisfied in first-order approximation and the total time evolu-tion under decoupling in first-order approximation in τ and t is described as in the unitarycase; ⊗ ρ θ will in general not be invariant under L (cid:48) j , but those effects are of higher orderin τ .We would like to have an estimate of E [ F t ] that depends only on t and the couplingstrength, distinguishing between the two extremal cases of purely extrinsic decoherence(i.e., purely unitary on the dilation: Ψ (cid:48) = 0 and a (cid:48) = − a (cid:48)∗ in the notation of Theorem 2)and purely intrinsic decoherence ( Ψ (cid:48) (cid:54) = 0 and a = a ∗ ). Theorem 14.
Given a quantum system ( H , L ) with decoupling set V and the previousnotation, write Γ := max {(cid:107)L(cid:107) , (cid:107)L (cid:48) (cid:107) , (cid:107) ¯ L(cid:107)} . Then in the drift-like limit of Remark 6, forpurely extrinsic decoherence we have E (drift) [ F ( extr ) t ] = 1 . An approximate upper bound for the expectation of the fidelity in the case of purely intrinsicdecoherence, in the limit of τ (cid:28) t (cid:28) / Γ , is asymptotically given by E (drift) [ F ( intr ) t ] (cid:46) − d t (cid:107) ˆ L (drift) (cid:107) If in addition L = L † (so-called purely intrinsic dephasing) this can be made more precise: E (drift) [ F ( intr ) t ] ≤ − d (1 − e − t (cid:107)L(cid:107) / | J | ) . In the (physically more realistic) diffusion-like limit of Theorem 4, a lower bound for thefidelity of purely extrinsic decoherence is asymptotically given by E [ F ( extr ) t ] (cid:38) − d | J | τ (cid:90) t (cid:107)L (cid:48) ( t (cid:48) ) (cid:107) d t (cid:48) , n terms of the (possibly time-dependent) Lindbladian L (cid:48) on the dilated system, while anupper bound in the case of purely intrinsic decoherence is asymptotically given by E [ F ( intr ) t ] (cid:46) − d | J | τ t (cid:107)L − ¯ L(cid:107) − d t (cid:107) ¯ L(cid:107) . Proof.
We appeal to Proposition 10 for notation and the exact formulae underlying ourestimates here.For the case of pure dephasing (i.e., L † = L ), we first notice that ˆ L (drift) , being a sumof double commutators, is a selfadjoint operator on the Hilbert space A . Moreover, itmust be negative since e t ˆ L (drift) is a contraction. We may suppose that the orthonormalbasis ( e k ) k =1 ...d consists of the eigenvectors of ˆ L (drift) in decreasing order of eigenvalues.The smallest eigenvalue of ˆ L (drift) is less than −(cid:107)L(cid:107) / | J | : in fact, the smallest eigenvalueof L is −(cid:107)L(cid:107) , and ˆ L (drift) ≤ | J | L . Then we find1 − E (drift) [ F t ] = 1 d d (cid:88) k,l =1 (cid:104) e l , (id − e t ˆ L (drift) )( e k ) (cid:105) ≥ d (cid:104) e , (id − e t ˆ L (drift) )( e ) (cid:105) = 1 d (1 − e − t (cid:107)L(cid:107) / | J | ) . For general L , we can say at least1 − E (drift) [ F t ] = 1 d d (cid:88) k =1 (cid:104) (id − e t ˆ L (drift) )( e k ) , (id − e t ˆ L (drift) )( e k ) (cid:105)≥ d (cid:107) id − e t ˆ L (drift) (cid:107) ≈ d t (cid:107) ˆ L (drift) (cid:107) . Let us come to the actual diffusion-like limit. Using the power series of the exponentialfunction and neglecting higher order terms, we obtain1 − E [ F t ] = 1 d d (cid:88) k,l =1 (cid:16) δ k,l − δ k,l (cid:104) e l , (e t ˆ L + e t ˆ L † )( e k ) (cid:105) + (cid:104) e l ⊗ e k , e t ˇ L (2) ( e k ⊗ e l ) (cid:105) (cid:17) ≈ t (cid:16) τd | J | d (cid:88) k,l =1 (cid:88) j ∈ J (cid:104) e l ⊗ e k , ( L j ⊗ L † j + L † j ⊗ L j )( e k ⊗ e l ) (cid:105) (cid:17) + t (cid:16) d d (cid:88) k,l (cid:104) e l ⊗ e k , ( ¯ L ⊗ ¯ L † )( e k ⊗ e l ) (cid:105) (cid:17) = t (cid:16) τd | J | (cid:88) j ∈ J A ⊗ A ( L j ⊗ L † j ◦ φ ) (cid:17) + t (cid:16) d tr A ⊗ A ( ¯ L ⊗ ¯ L † ◦ φ ) (cid:17) ≥ d | J | τ t (cid:107)L j (cid:107) + 1 d t (cid:107) ¯ L(cid:107) , where j ∈ J is arbitrary, φ denotes the flip unitary on A ⊗ A , and ⊗ A ⊗ A the standard(non-normalised) trace on B ( A ⊗ A ).For the case of extrinsic decoherence, we have to work in the dilation algebra A ⊗ B ( H ).Let L (cid:48) be the corresponding (possibly time-dependent) Lindbladian on that algebra, cor-responding to unitary time evolution. Then after decoupling we obtainˆ L (cid:48) = ¯ L (cid:48) + τ | J | (cid:88) j ∈ J ( L (cid:48) j ) , L (cid:48) j := Ad( v j ⊗ ) ◦ ( L (cid:48) − ¯ L (cid:48) ) ◦ Ad( v ∗ j ⊗ ) , F extt (dotted blue line) coinciding with thelower bound for E (drift) [ F extt ] ≡
1, average F intt (solid orange line), and the upper boundfor E (drift) [ F intt ] from Theorem 14 (dashed red line) as a function of t , for Γ τ = 10 − andthe so-called amplitude-damping channel model with coupling strength γ (cf. Example7(2) and [AHB14] for further explanation). The average was taken over 25 paths, one ofthem illustrated for F extt in the inset plot.with the commutator ¯ L (cid:48) vanishing on A ⊗ ρ θ . As in the preceding case, we find1 − E [ F t ] = 1 d d (cid:88) k,l =1 T (cid:16) δ k,l − δ k,l (cid:104) e l ⊗ , (e (cid:82) t ˆ L (cid:48) ( t (cid:48) ) d t (cid:48) + e (cid:82) t ˆ L (cid:48)† ( t (cid:48) ) d t (cid:48) )( e k ⊗ ρ θ ) (cid:105) + (cid:104) e l ⊗ ⊗ e k ⊗ , e (cid:82) t ˇ L (cid:48) (2) ( t (cid:48) ) d t (cid:48) ( e k ⊗ ρ θ ⊗ e l ⊗ ρ θ ) (cid:105) (cid:17) ≈ τd | J | d (cid:88) k,l =1 (cid:88) j ∈ J (cid:68) e l ⊗ ⊗ e k ⊗ , (cid:90) t ( L (cid:48) j ⊗ L (cid:48)† j + L (cid:48)† j ⊗ L (cid:48) j )( t (cid:48) ) d t (cid:48) ( e k ⊗ ρ θ ⊗ e l ⊗ ρ θ ) (cid:69) = τd | J | (cid:88) j ∈ J A ⊗ A ⊗ B ( H ) ⊗ B ( H ) (cid:16) (cid:90) t ( L (cid:48) j ⊗ L (cid:48)† j )( t (cid:48) ) d t (cid:48) ◦ φ ⊗ ( ⊗ ρ θ ⊗ ⊗ ρ θ ) (cid:17) ≤ dτ (cid:90) t (cid:107)L (cid:48) ( t (cid:48) ) (cid:107) d t (cid:48) , because tr A ⊗ B ( H ) ( ⊗ ρ θ ) = d and ¯ L (cid:48) ( x ⊗ ρ θ ) = 0 for all x ∈ A . (cid:3) Notice that these bounds are probably not sharp at all, but they should rather serveas an inspiration and starting point for finding more specific and sharper bounds. Theinteresting fact, in any case, is that they separate the fidelity of intrinsic and extrinsicdecoherence dynamics, respectively, in the region τ (cid:28) t (cid:28) / Γ, cf. Figure 1.Moreover, under further assumptions on L like e.g. L † = L we can try to use a similarprocedure in order to achieve better bounds involving directly (cid:107)L(cid:107) . Conclusion 15 (Application to experiment – sketch) . If τ is sufficiently small, then forsuitable t the last two bounds in Theorem 14 provide a separation into two disjoint ranges20f the fidelity in the two intrinsic interaction cases, which allows the experimenter toidentify the type of decoherence. He would have to proceed as follows:(1) Given the intrinsic or extrinsic coupling strength Γ := max {(cid:107)L(cid:107) , (cid:107)L (cid:48) (cid:107) , (cid:107) ¯ L(cid:107)} , choose t (cid:28) / Γ.(2) Choose and vary τ (cid:28) t in that range.(3) Compute the fidelity of many decoupling pulse sample paths for these given valuesof Γ and varying τ, t , then average and extrapolate to get his averaged fidelity ¯ F t asa function of τ and t .(4) Compare it with the bounds for these given values of Γ , τ : then in the above limit,he will find either¯ F t (cid:38) − dτ (cid:90) t (cid:107)L (cid:48) ( t (cid:48) ) (cid:107) d t (cid:48) or ¯ F t (cid:46) − d | J | τ t (cid:107)L − ¯ L(cid:107) − d t (cid:107) ¯ L(cid:107) , t ∈ R + . the first case corresponding to extrinsic, the second to intrinsic dephasing.If he cannot carry out many runs, then it would also be necessary to take into accountthe quantiles from above in order to understand how well his experimental mean value¯ F t describes the analytical E [ F t ]. This can be done by considering higher moments E [ F nt ]and Var[ F t ] as in Proposition 10 For an arbitrarily large number of runs, however, this isnot necessary. Figure 1 illustrates the bounds with an average of concrete sample paths.Although the precise relation between (cid:107)L(cid:107) , (cid:107)L (cid:107) , (cid:107) ¯ L(cid:107) , (cid:107)L (cid:48) (cid:107) etc. is not clear and itis therefore difficult to compare the above bounds quantitatively, these values dependsomehow monotonically on one another, i.e., increase or decrease synchronically. In anycase, when τ → H (cid:48) is needed; otherwise alternative assumptions would have to be made that lead to futurework, cf. also [AHB14]. Moreover, it follows from the above bounds then that ¯ F t → F t converges to some function f ( t ) (cid:46) − t (cid:107) ¯ L(cid:107) /d in theintrinsic case.If extrinsic and intrinsic decoherence appear together: since the respective couplingstrengths will not be known, it is impossible to compute the above bounds; yet, for fixed t , letting τ →
0, the experimenter can check whether or not ¯ F t goes to 1, meaning pureextrinsic or intrinsic/mixed decoherence, respectively. (cid:50) A Lie groups and convolution semigroups
The aim of this appendix section is to sketch the necessary definitions and facts aboutLie groups and convolution semigroups necessary to understand the third and fourthsection. For a comprehensive study and the notation used in Section 3 we refer to anysuitable textbook: e.g. [FH91] for (linear) Lie groups and algebras, [Dav80, EN00] for one-parameter semigroups, [Gre08, Hey79, Ch.4] for probability and convolution measures onLie groups.An N -dimensional (real) Lie group G is an N -dimensional smooth manifold whichhas a group structure with neutral element such that multiplication and inversion aresmooth maps. In this paper G is always a linear algebraic Lie group , i.e., a group oflinear mappings on a finite-dimensional real vector space. The tangent space T G of G in forms a Lie algebra and is called the Lie algebra of G , denoted g with scalar product(nondegenerate bilinear form) (cid:104)· , ·(cid:105) g . 21here is a canonical diffeomorphism exp from a 0-neighbourhood in g to some -neighbourhood U ⊂ G mapping 0 to and called the exponential map, which in thepresent case can be identified with the standard exponential function of matrices. Given anorthonormal basis ( X k ) k =1 ...N of g , the corresponding (coordinate) -chart is the smoothfunction x : U → R N such that g = exp (cid:16) N (cid:88) k =1 x k ( g ) X k (cid:17) , g ∈ U, and x k : U → R is called the k -th coordinate map . One may extend the functions x k ∈ C ∞ ( U ) to functions in C ∞ c ( G ) denoted again by x k ; write G c for the one-pointcompactification of G if G is noncompact, otherwise we take G c = G , and every function f ∈ C c ( G ) is extended by f ( ∞ ) := 0 to G c ; this is needed in section 3 for technical reasons.One notes that dd t x k (e tY ) (cid:22) t =0 = (cid:104) Y, X k (cid:105) g , for every k = 1 , . . . , N and Y ∈ g . The directional derivative D Y for Y ∈ g is defined by D Y f ( g ) := dd t f (e tY g ) (cid:22) t =0 , f ∈ C c ( G ) , g ∈ G, and one has D X + λY = D X + λD Y , for X, Y ∈ g and λ ∈ R .The convolution of two probability measures µ , µ on G (with usual Borel σ -algebra B ( G )) is defined by µ ∗ µ ( A ) := ( µ × µ ) { ( g, h ) ∈ G × G : gh ∈ A } , A ∈ B ( G ) . Suppose that µ and µ are supported in a subsemigroup H ⊂ G . Then for every A ∈ B ( G ), we have µ ∗ µ ( A ) =( µ × µ ) { ( g, h ) ∈ G × G : gh ∈ A } =( µ × µ ) { ( g, h ) ∈ H × H : gh ∈ A } =( µ × µ ) { ( g, h ) ∈ H × H : gh ∈ A ∩ H } = µ ∗ µ ( A ∩ H ) , so µ ∗ µ is supported in H , too.Measures and convolution on G can be trivially extended to G c by setting g ∞ := ∞ g := ∞ , for all g ∈ G c . The set of probability measures on G c , equipped with the *-weak topology and convolution as multiplication, constitutes a topological monoid, wherethe Dirac measure δ serves as the neutral element. Here the *-weak topology on G c isdefined as follows: a net of measures ( µ i ) i ∈ I converges to a limit measure µ if for all f ∈ C ( G c ) (the continuous R -valued functions on G c ) the condition (cid:82) G c f d µ i → (cid:82) G c f d µ holds. A continuous convolution semigroup of probability measures on G is a set ( µ t ) t ∈ R + of probability measures on G (trivially extended to G c ) such that µ s ∗ µ t = µ s + t for every s, t ∈ R + and lim t → µ t = µ = δ *-weakly.Let ( µ t ) t ∈ R + be a continuous convolution semigroup of probability measures on G . For t ∈ R + define the operator T t : C ( G c ) → C ( G c ) , ( T t f )( g ) := (cid:90) G c f ( gh ) dµ t ( h ) , g ∈ G c . Then ( T t ) t ∈ R + forms a strongly continuous one-parameter contraction semigroup on C ( G c ).To ( T t ) t ∈ R + there corresponds an infinitesimal generator L := lim t → T t − id t T t -invariant dense domain dom( L ) ⊂ C ( G c ).Given a strongly continuous one-parameter semigroup ( T t ) t ∈ R + with generator ( L, dom( L ))on a Banach space E , we write C ∞ ( L ) := (cid:84) n ∈ N dom( L n ). A vector f ∈ C ∞ ( L ) is called entire analytic for L if z ∈ C (cid:55)→ ∞ (cid:88) n =0 z n n ! L n f ∈ C ∞ ( L )is analytic, in which case it extends t ∈ R + (cid:55)→ T t f ∈ E to an entire analytic function.Nonzero analytic vectors need not exist for one-parameter semigroups, whereas for one-parameter groups they do.If ( T t ) t ∈ R + is a strongly continuous one-parameter semigroup on a Banach space E with generator ( L, dom( L )) which leaves invariant a closed subspace E ⊂ E , then itinduces a strongly continuous one-parameter semigroup ( S t ) t ∈ R + on the quotient Banachspace E/E with infinitesimal generator ( K, dom( K )) as follows: denote the quotient mapby q : E → E/E , then S t q ( f ) := q ( T t f ) , f ∈ E, and Kq ( f ) = q ( Lf ) with dense dom( K ) = q (dom( L )) ⊂ E/E .Given a continuous convolution semigroup of probability measures ( µ t ) t ∈ R + on G ,there exists a probability space and a G -valued Markov process on this space such thatits transition probabilities from ( g, ∈ G c × R + to ( A, t ) ∈ B ( G c ) × R + are given by( µ t ∗ δ g )( A ). 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