doi: 10.1098/rsta.2011.0485, 5259-5269370 2012 Phil. Trans. R. Soc. A

Francesco Ticozzi and Lorenza Viola quantum dynamical semigroupsStabilizing entangled states with quasi-local

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Phil. Trans. R. Soc. A (2012) 370, 5259–5269doi:10.1098/rsta.2011.0485

Stabilizing entangled states with quasi-localquantum dynamical semigroupsBY FRANCESCO TICOZZI1,2 AND LORENZA VIOLA2,*

1Dipartimento di Ingegneria dell’Informazione, Università di Padova,via Gradenigo 6/B, 35131 Padova, Italy

2Department of Physics and Astronomy, Dartmouth College, 6127 WilderLaboratory, Hanover, NH 03755, USA

We provide a solution to the problem of determining whether a target pure state canbe asymptotically prepared using dissipative Markovian dynamics under fixed localityconstraints. Besides recovering existing results for a large class of physically relevantentangled states, our approach has the advantage of providing an explicit stabilizationtest solely based on the input state and constraints of the problem. Connections withthe formalism of frustration-free parent Hamiltonians are discussed, as well as controlimplementations in terms of a switching output-feedback law.

Keywords: quantum dynamical semigroups; entanglement generation; environment engineering

1. Introduction

While uncontrolled couplings between a quantum system of interest and itssurrounding environment are responsible for unwanted non-unitary evolution anddecoherence, it has also been long acknowledged that suitably engineering theaction of the environment may prove beneficial in a number of applications acrossquantum control and quantum information processing [1–3]. It is well known,in particular, that open-system dynamics are instrumental in control tasks suchas robust quantum state preparation and rapid purification, and both open-loop and quantum feedback methods have been extensively investigated in thiscontext [4–8], including recent extensions to engineered quantum memories [9]and ‘pointer states’ in the non-Markovian regime [10].Remarkably, it has also been recently shown that it is, in principle, possible

to design dissipative Markovian dynamics so that non-trivial strongly correlatedquantum phases of matter are prepared in the steady state [11,12] or the outputof a desired quantum algorithm is retrieved as the asymptotic equilibrium [13].From a practical standpoint, scalability of such protocols for multi-partite systemsof increasing size is a key issue, as experimental constraints on the availablecontrol operations may, in fact, limit the set of attainable states. Promisingresults have been obtained by Kraus et al. [14] and Verstraete et al. [13] for*Author for correspondence (lorenza.viola@dartmouth.edu).

One contribution of 15 to a Theo Murphy Meeting Issue ‘Principles and applications of quantumcontrol engineering’.

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5260 F. Ticozzi and L. Viola

a large class of entangled pure states, showing that Markovian dissipation actingnon-trivially only on a finite maximum number of subsystems is, under genericconditions, sufficient to generate the desired state as the unique ground stateof the resulting evolution. As proof-of-principle methodologies for engineeringdissipation are becoming an experimental reality [15,16], it is important to obtaina more complete theoretical characterization of the set of attainable states underconstrained control resources, as well as to explore schemes for synthesizing therequired dissipative evolution.Building on our previous analysis [6,17,18], in this work, we address the

problem of determining whether a target pure state of a finite-dimensionalquantum system can be prepared using ‘quasi-local’ dissipative resourceswith respect to a fixed locality notion (see also Yamamoto [19] for relatedresults on infinite-dimensional Markovian–Gaussian dissipation). We provide astabilizability analysis under locality-constrained Markovian control, including adirect test to verify whether a desired entangled pure state can be asymptoticallyprepared. In addition to recovering existing results within a system-theoreticframework, our approach has the important advantage of using only twoinputs: the desired state (control task) and a specified locality notion (controlconstraints), without requiring a representation of the state in the stabilizer,graph or matrix-product formalisms.

2. Problem definition and preliminary results

(a)Multi-partite systems and locality of quantum dynamical semigroups

We focus on quantum dynamical semigroups generated by a (time-independent)Markovian master equation (MME) [20–22] in Lindblad form (h ! 1),

r(t)= L(r(t))="i[H , r(t)] +!

k

"Lkr(t)L

†k "12{L†kLk , r(t)}

#, (2.1)

specified in terms of the Hamiltonian H =H † and a finite set of noise (orLindblad) operators {Lk}. We are interested in the asymptotic behaviour of MMEsin which the operators H , {Lk} satisfy locality constraints. More precisely, let usconsider a multi-partite system Q, composed of n (distinguishable) subsystems,labelled with index a = 1, . . . ,n, with associated da-dimensional Hilbert spacesHa . Thus,HQ = $n

a=1Ha . LetB(H) andD(H) denote the sets of linear operatorsand density operators on H, respectively. It is easy to show ([17], proof oftheorem 2) that the semigroup generated by equation (2.1) is factorized withrespect to the multi-partite structure, i.e. the dynamical propagator

Tt := eLt =n%

a=1Ta,t #t $ 0,

with Ta,t a completely positive, trace-preserving map on B(Ha), if and only if thefollowing conditions hold:

— each Lk acts as the identity on all subsystems except (at most) one and— H = &

a Ha , where each Ha acts as the identity on all subsystems except(at most) one.

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Quasi-local stabilization 5261

This motivates the following definitions: we say that a noise operator Lk is local,if it acts as the identity on all subsystems except (at most) one, and that aHamiltonian H is local, if it can be written as a sum of terms with the sameproperty. Note that in the language of quantum networks [23], this correspondsto the concatenation product (I ,L,H )= !na=1(Ia ,La ,Ha), where I and Ia denoteidentity operations. However, it is easy to verify that if a semigroup associatedto local operators admits a unique stationary pure state, the latter must be aproduct state. Thus, in order for the MME (2.1) to admit stationary entangledstates, it is necessary to weaken the locality constraints.We shall allow the semigroup dynamics to act in a non-local way only on certain

subsets of subsystems, which we call neighbourhoods. These can be generallyspecified as subsets of the set of indexes labelling the subsystems

Nj % {1, . . . ,n}, j = 1, . . . ,M .In analogy with the strictly local case, we say that a noise operator L is quasi-local(QL) if there exists a neighbourhood Nj such that

L= LNj & INj ,

where LNj accounts for the action of L on the subsystems included in Nj ,and INj :=

$a /'Nj Ia is the identity on the remaining subsystems. Similarly, a

Hamiltonian is QL, if it admits a decomposition into a sum of QL terms,

H =!

j

Hj , Hj =HNj & INj .

An MME will be called QL if both its Hamiltonian and noise operators are QL.It is well known that the decomposition into Hamiltonian and dissipative parts of(2.1) is not unique: nevertheless, the QL property remains well defined because thefreedom in the representation does not affect the tensor structure of H and {Lk}.The above way of introducing locality constraints is very general and encompassesa number of specific notions that have been used in the physical literature, notablyin situations where the neighbourhoods are associated with sets of nearest-neighbour sites on a graph or lattice, and/or one is forced to consider Hamiltonianand noise generators with a weight no larger than t (so-called t-body interactions),see also Kraus et al. [14] and Verstraete et al. [13].We are interested in states that can be prepared (or, more precisely, stabilized)

by means of MME dynamics with QL operators. Recall that an invariant stater for a system driven by (2.1) is said to be globally asymptotically stable (GAS)if for every initial condition r0, we have

limt()

eLt[r0] = r.

In particular, following Kraus et al. [14], the aim of this paper is to characterizepure states that can be rendered GAS by purely dissipative dynamics, for whichthe state is ‘dark’. More precisely, we have the following.

Definition 2.1. A pure state rd = |J*+J| 'D(HQ), is dissipatively quasi-locally stabilizable (DQLS) if there exist QL operators {Dk}k=1,...,K on HQ, with

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5262 F. Ticozzi and L. Viola

Dk |J*= 0, for all k and Dk acting non-trivially on (at most) one neighbourhood,such that rd is GAS for

r = LD[r] =!

k

"DkrD

†k "12{D†kDk , r}

#. (2.2)

We will provide a criterion for determining whether a state is DQLS, and indoing so, we will also show how assuming a single QL noise operator for eachneighbourhood does not restrict the class of stabilizable states. From now on, wethus let K !M and Dk !DNk & INk .We begin by noting that if a pure state is factorized, then we can realize its

tensor components ‘locally’ with respect to its subsystems (see Ticozzi and co-workers [17,18] for stabilization of arbitrary quantum states in a given system with‘simple’ generators, involving a single noise term). Thus, we can iteratively reducethe problem to subproblems on disjoint subsets of subsystems, until the statesto be stabilized are either entangled, or completely factorized. A preliminaryresult is that the DQLS property is preserved by arbitrary local unitary (LU)transformations, of the form U = $n

a=1Ua . In order to show this, lemma 2.2 isneeded.

Lemma 2.2. Let L denote the Lindblad generator associated to operatorsH , {Lk}. Then, for every unitary operator U , we have

UL[U †rU ]U † = L,[r], (2.3)

where L, is the generator associated to H , =UHU †, L,k =ULkU †, andU eLt[U †r0U ]U † = eL,t[r0], #t $ 0. (2.4)

Identity (2.3) is easily proved by direct computation, while (2.4) follows directlyfrom the properties of the (matrix) exponential. The desired invariance of the QLstabilizable set under LU transformation follows.

Proposition 2.3. If r is DQLS and U is LU, then r, =UrU † is also DQLS.

Proof. Assume that the generator L associated to QL operators {Dk} stabilizesr. Because r is GAS, for any initial condition r0, we may write

r, = limt(+)

U eLt[r0]U † = limt(+)

U (eLt[U †r0U ])U †. (2.5)

By applying lemma 2.2, it suffices to show that each D,k =UDkU † is QL.Because Dk =DNk & INk , we have D

,k =U (DNk & INk )U

† = (UNkDNkU†Nk)& INk ,

where UNk :=$

!'Nk U!. Hence, D,k is QL. "

(b)Quantum dynamical semigroups for unconstrained stabilization

We next collect some stabilization results that do not directly incorporate anylocality constraint, but will prove instrumental to our aim. Let HS := span{|J*}.Given corollary 1 in Ticozzi & Viola [17], rd is invariant if and only if

Lk ='LS ,k LP,k0 LR,k

(and iHP "

12

!

k

L†S ,kLP,k = 0, (2.6)

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Quasi-local stabilization 5263

where we have used the natural block representation induced by the partitionHQ = HS -H.

S and labelled the blocks as

X ='XS XPXQ XR

(.

Assume rd to be invariant. Then, |J* must be a common eigenvector of eachLk . Call the corresponding eigenvalue !k ! LS ,k . By lemma 2.4 in Ticozzi & Viola[17], the MME is invariant upon substituting Lk with Lk = Lk " !kI , and H withH =H + (i/2)&k(!

/kLk " !kL

†k). Using the series product of Gough & James [23],

this means (I , Lk , H )= (I , [lk ], 0) 0 (I , [Lk ],H ). In this way, we have LS ,k = 0 forall k, so that H P must be zero in order to fulfill the above condition. Thus, His block-diagonal, with |J* being an eigenvector with eigenvalue h ! H S . Usingthis representation for the generator, we can let LS ,k = !k = 0 and HP = 0=HQ .This further motivates the use of noise operators Dk such that Dk |J*= 0 in theDQLS definition.

Lemma 2.4. An invariant rd = |J*+J| is GAS for the MME (2.1), if there areno invariant common (proper) subspaces for {Lk} other than HS = span{|J*}.Proof. By lemma 8 and theorem 9 in Ticozzi & Viola [6], rd is GAS if and only

if there are no other invariant subspaces for the dynamics. Given the conditionson Lk for the invariance of a subspace, rd is GAS as long as there are no otherinvariant common subspaces for the matrices Lk . "On the basis of the above characterization, in order to ensure the DQLS

property, it suffices to find operators {Dk}Kk=1 1B(HQ) such that HS is the uniquecommon (proper) invariant subspace for the {Dk}.

3. Characterization of dissipatively quasi-locally stabilizable states

(a)Main result

The key elements in our approach are the reduced states that the target state rdinduces with respect to the given locality structure. Let us define

rNk = traceNk (rd), (3.1)

where traceNk indicates the partial trace over the tensor complement of theneighbourhood Nk , namely HNk = $

a /'Nk Ha . Lemma 3.1 follows from theproperties of the partial trace.

Lemma 3.1. supp(rd)%)k supp(rNk & INk ).

Proof. From the spectral decomposition rNk = &q pqPq , we can construct a

resolution of the identity {Pq} such that rNk & INk = &q pqPq & INk , where,

by definition of the partial trace, pq = trace(rdPq & INk ). If pq = 0 for some q,then it must be rd(Pq & INk )= 0 and therefore supp(rd). supp(Pq & INk ). Thus,supp(rd)%

*q supp(pqPq & INk )= supp(rNk ), for all k. "

Let us now focus on QL noise operators Dk =DNk & INk such that Dk |J*= 0.

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5264 F. Ticozzi and L. Viola

Lemma 3.2. Assume that a set {Dk} makes rd = |J*+J| DQLS. Then, for eachk, we have supp(rNk )% ker(DNk ).

Proof. Because |J* is by hypothesis in the kernel of each Dk , with respect tothe decomposition HQ = HS -H.

S every Dk must be of block form,

Dk ='0 DP,k0 DR,k

(,

which immediately implies DkrdD†k = 0. It then follows that traceNk (DkrdD

†k)=

0= traceNk (DNk & INkrdD†Nk& INk ). Therefore, it also follows that DNkrNkD

†Nk

=0. If we consider the spectral decomposition rNk !

&j qj |fj*+fj |, with qj > 0, the

latter condition implies that, for each j , DNk |fj*+fj |D†Nk

= 0. Thus, it must besupp(rNk )% ker(DNk ), as stated. "Theorem 3.3. A pure state rd = |J*+J| is DQLS if and only if

supp(rd)=+

k

supp(rNk & INk )!+

k

HNk . (3.2)

Proof. Given lemmas 3.1 and 3.2, for any set {Dk} that make rd DQLS, wehave

supp(rd)%+

k

supp(rNk & INk )%+

k

ker(DNk & INk ).

By negation, assume that supp(rd)!)k supp(rNk & INk ). Then, there would be

(at least) another invariant state in the intersection of the kernels of the noiseoperators, contradicting the fact that rd is DQLS. Thus, a necessary conditionfor rd to be GAS is that supp(rd)=

)k supp(rNk & INk ). Conversely, if the latter

condition is satisfied, then for each k we can construct operators DNk thatrender each supp(rNk ) GAS on HNk (see [6,17,18] for explicit constructions).Then,

)k ker(DNk & INk )= supp(rNk ), and there cannot be any other invariant

subspace. By lemma 2.4, rd is hence rendered GAS by QL noise operators. "

(b) An equivalent characterization: quasi-local parent Hamiltonians

Consider a QL Hamiltonian H = &k Hk , Hk =HNk & INk . A pure state rd =

|J*+J| is called a frustration-free ground state if+J|Hk |J*=min l(Hk), #k,

where l(·) denotes the spectrum of a matrix. A QL Hamiltonian is called a parentHamiltonian if it admits a unique frustration-free ground state [24].Suppose that a pure state admits a QL parent Hamiltonian H . Then, the QL

structure of H may be naturally used to derive a stabilizing semigroup: it sufficesto implement QL operators Lk that stabilize the eigenspace associated to theminimum eigenvalue of each Hk . In view of theorem 3.3, it is easy to show thatthe following condition is also necessary.

Corollary 3.4. A state rd = |J*+J| is DQLS if and only if it is the ground stateof a QL parent Hamiltonian.

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Quasi-local stabilization 5265

Proof. Without loss of generality, we can consider QL Hamiltonians H =&k Hk , where each Hk is a projection. Let rd be DQLS, and define Hk := P.

Nk&

INk , with P.Nkbeing the orthogonal projector onto the orthogonal complement of

the support of rNk , i.e. HNk 2 supp(rNk ). Given theorem 3.3, |J* is the uniquepure state in

)k supp(rNk & INk ), and thus the unique state in the kernel of

all the Hk . Conversely, if a QL parent Hamiltonian exists, to each Hk we canassociate an Lk that asymptotically stabilizes its kernel. A single operator perneighbourhood is, in principle, always sufficient—see Ticozzi & Viola [6,17] forexplicit constructions and examples of Lk stabilizing a desired subspace. "

The earlier-mentioned result directly relates our approach to the one pursuedin Kraus et al. [14] and Verstraete et al. [13] and a few remarks are in order.In these works, it has been shown that matrix product states (MPSs) areQL stabilizable, up to a condition (the so-called injectivity) that is believedto be generic [24]. MPSs that allow for a compact representation (i.e. in thecorresponding ‘valence-bond picture’, those with a small bond dimension) areof key interest in condensed matter as well as quantum information processing[24–27]. However, any pure state admits a (canonical) MPS representation ifsufficiently large bond dimensions are allowed, suggesting that arbitrary purestates would be DQLS. The problem with this reasoning is that the locality notionthat is needed in order to allow stabilization of a certain MPS is in general inducedby the state itself. The number of elements to be included in each neighbourhoodis finite but need not be small: while this is both adequate and sufficient foraddressing many relevant questions in many-body physics (where typically athermodynamically large number of subsystems is considered), engineering therequired dissipative process may entail interactions that are not easily availablein experimental settings. For this reason, our approach may be more suitablefor control-oriented applications. It is also worth noting that the injectivityproperty is sufficient but not necessary for the target state to admit a QL parentHamiltonian (an example on a two-dimensional lattice is provided in Perez-Garciaet al. [26]). Once the locality notion is fixed, our test for DQLS can be performedirrespective of the details of the MPS representation, and it is thus not affectedby whether the latter is injective or not (rather, our DQLS test may be used tooutput a QL parent Hamiltonian if so desired).

(c) Examples

(i)Greenberger–Horne–Zeilinger states and W states

Consider an n-qubit system and a target Greenberger–Horne–Zeilinger(GHZ) state rGHZ = |J*+J|, with |J* ! |JGHZ*= (|000 . . . 0*+ |111 . . . 1*)/

32.

Any reduced state on any (non-trivial) neighbourhood is an equiprobable mixtureof |000 . . . 0* and |111 . . . 1*. It is then immediate to see that

span{|000 . . . 0*, |111 . . . 1*}%+

k

supp(rNk & INk ),

and hence rGHZ is not DQLS. In a similar way, for any n, the W staterW = |J*+J|, with |J* ! |JW*= (|100 . . . 0*+ |010 . . . 0*+ . . . + |000 . . . 1*)/3n

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5266 F. Ticozzi and L. Viola

has reduced states that are statistical mixtures of |000 . . . 0* and a smaller Wstate |JW, *, of the dimension of the neighbourhood. Thus,

span{|000 . . . 0*, |JW, *}%+

k

supp(rNk & INk ),

and rW is not DQLS (except in trivial limits, see also below). Note that forarbitrary n, both rGHZ and rW are known to be (non-injective) MPSs with(optimal) bond dimension equal to two.

(ii) Stabilizer and graph states

A large class of states does admit a QL description, and in turn they are DQLS.Among these are stabilizer states, and general graph states. Here, the relevantneighbourhoods are those that include all the nodes connected to a given one byan edge of the graph. The details are worked out in Kraus et al. [14]. Notice thatGHZ states are indeed graph states, but associated only with ‘star’ (or completelyconnected) graphs. In such cases, relative to the locality notion naturally inducedby the graph, any central node has a neighbourhood that encompasses the wholegraph, rendering the constraints effectively trivial.

(iii)Dissipatively quasi-locally stabilizable states beyond graph states

Consider a four-qubit system arranged on a linear graph, with (up to)three-body interactions. The two neighbourhoods N1 = {1, 2, 3},N2 = {2, 3, 4} aresufficient to cover all the subsystems, and contain all the smaller ones. The staterT = |J*+J|, with

|J* ! |JT *=(|1100*+ |1010*+ |1001*+ |0110*+ |0101*+ |0011*)3

6,

is not a graph state because if we measure any qubit in the standard basis,we are left with W states on the remaining subsystems, which are known notto be graph states. In contrast, proposition 9 of Hein et al. [28] ensures that theconditional reduced states for a graph state would have to be graph states as well.Nonetheless, by constructing the reduced states and intersecting their supports,one can establish directly that |JT * is indeed DQLS.

4. Switched feedback implementation

From theorem 3.3, it follows that a DQLS state can be asymptotically preparedprovided we can engineer QL noise operators Dk =DNk & INk that stabilize thesupport of each reduced state rNk on each neighbourhood. Restricting to HNk , wemust have DNk =

,0 DP,k0 DR,k

-, with the blocks DP,k ,DR,k such that the support of rNk

is attractive, i.e. such that no invariant subspace is contained in its complement.Following the ideas of Ticozzi & Viola [6] and Ticozzi et al. [18], a natural explicitchoice is to consider noise operators with the following structure:

DP,k =

.

/0 0 · · · 0... 0 · · · 0!1 0 · · · 0

0

1 and DR,k =

.

2/

0 !2 0 0

0 0 !3. . .

.... . . . . .

0

31. (4.1)

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Quasi-local stabilization 5267

If the above QL Lindblad operators are not directly available for open-loopimplementation, a well-studied strategy for synthesizing attractive Markoviandynamics is provided by continuous measurements and output feedback. In theabsence of additional dissipative channels, and assuming perfect detection, therelevant feedback master equation takes the form [5]

r(t)="i[H +Hc + 12(FM +M †F), r(t)] + Lf r(t)L†f " 1

2{L†f Lf , r(t)},

where Hc is a time-independent control Hamiltonian, F =F † and M denote,respectively, the feedback Hamiltonian and the measurement operator, and Lf :=M " iF . Necessary and sufficient conditions for the existence of open- and closed-loop Hamiltonian control that stabilizes a desired subspace have been providedby Ticozzi and co-workers [6,17].In order to exploit the existing techniques in the current multi-partite

setting, it would be necessary to implement measurements and feedback in eachneighbourhood. If the measurement operators do not commute, however, onewould have to carefully scrutinize the validity of the model and the consequencesof ‘conflicting’ stochastic back-actions when acting simultaneously on overlappingneighbourhoods. These difficulties can be bypassed by resorting to a cyclicswitching of the control laws. Consider a DQLS state rd and the family ofgenerators {Lk}Mk=1, Lk [r] =DkrD†k " 1

2{D†kDk , r}, with Dk such that supp(rNk &

INk ) is the unique invariant subspace for Lk . Define a switching interval t > 0 andthe cyclic switching law j(t)= 4t/tM 5+ 1. We can then establish the following.Theorem 4.1. There exists QL {Dk} such that rd is GAS for the switched

evolution Lj(t).

Proof. Consider the trace-preserving, completely positive maps Tj(r)= eLjt[r].It is easy to see that rd is invariant for each Tj : as a corollary of theorem 1 inBolognani & Ticozzi [29], it follows that rd is GAS if it is the only invariantstate for T = TM 6 · · · 6 T1. Assume that r is invariant for T : then either it isfixed for all Tk , which means that necessarily r = rd , or there exists a periodiccycle. Because each Tj is a trace-distance contraction [22], this means that eachmap preserves the trace distance, i.e. 7Tj(rd " r)71 = 7rd " r71. This would inturn imply that each Tj admits eigenvalues on the unit circle, and hence eachLk would have imaginary ones. However, if we choose Dk as in equation (4.1), invectorized form, the Liouvillian generator reads

Lk = (D†k)T &Dk "12I &D†kDk "

12(D†kDk)

T & I ,

which is an upper triangular matrix with eigenvalues either equal to zero,"|!j |2/2,or "(|!j |2 + |!i|2)/2. Therefore, for this choice, rd is the only invariant pure statefor T and hence it is GAS. "

5. Concluding remarks

We have presented a characterization of DQLS pure states for fixed localityconstraints, from a system-theoretic perspective. As a by-product of our analysis,an easily automated algorithm for checking DQLS states is obtained. The

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5268 F. Ticozzi and L. Viola

necessary steps entail: (i) calculating the reduced states on all the neighbourhoodsspecifying the relevant QL notion; (ii) computing their tensor product withthe identity on the remaining subsystems, and the corresponding supports; and(iii) finding the intersection of these subspaces. If such intersection coincideswith the support of the target state alone, the latter is DQLS. If so, we haveadditionally showed that the required Markovian dynamics can, in principle, berealized by switching output-feedback control. While we considered homodyne-type continuous-time feedback MME, the study of discrete-time strategies isalso possible along similar lines, see also Bolognani & Ticozzi [29] and Barreiroet al. [15].Our present results have been derived under two main assumptions: the

absence of underlying free dynamics, and the use of purely dissipative control(no Hamiltonian control involved). In case a drift internal dynamics is present, asimilar approach can, in principle, be adapted to determine what can be attainedby dissipative control. When we additionally allow for Hamiltonian control, onemay use the algorithm described in §III.B of Ticozzi et al. [30] to search for aviable QL Hamiltonian when dissipation alone fails. Nonetheless, in the presenceof locality constraints, a more efficient design strategy may be available: an in-depth analysis of the role of additional internal dynamics as well as of combinedcoherent and dissipative control will be presented elsewhere.It is also worth noting that in various experimental situations, the available

dissipative state preparation procedures involve two steps: first, enact local noiseoperators that prepare a known pure state that is factorized; next, use open-loopcoherent control to steer the system on the desired entangled target. The approachwe discussed here is expected to have an intrinsic advantage in terms of theoverall robustness against initialization errors and finite-time perturbations of thedynamics [13,16]. While establishing rigorous robustness results require furtherstudy, the actual answer will crucially depend on the physical implementation andits characteristic time scales. Lastly, the estimation of the speed of convergencestill presents numerous challenges, most importantly its optimization and acharacterization of its scaling with the number of subsystems involved.F.T. acknowledges support by the QFuture project of the University of Padova.

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