On the Role of Hamiltonians forDissipative Entanglement Engineering ?

Francesco Ticozzi

⇤Lorenza Viola

⇤⇤

⇤ Dipartimento di Ingegneria dell’Informazione, Universita di Padova,via Gradenigo 6/B, 35131 Padova, Italy and Department of Physics

and Astronomy, Dartmouth College, 6127 Wilder, 03755 Hanover, NH(USA) (e-mail:ticozzi@dei.unipd.it)

⇤⇤ Department of Physics and Astronomy, Dartmouth College, 6127Wilder, 03755 Hanover, NH (USA)

(e-mail:Lorenza.Viola@dartmouth.edu)

Abstract: Following the growing interest on using dissipative evolution for quantum in-formation engineering, the problem of determining whether a given entangled state can beasymptotically prepared by quasi-local quantum dynamical semigroups has been recently solvedfor purely-dissipative continuous-time dynamics. Here, we extend our previous analysis byaddressing the role of internal Hamiltonian dynamics as well as Hamiltonian control resources forthe same task. We show that frustration-inducing Hamiltonians can genuinely extend the classof stabilizable states. We present stabilizing Hamiltonians, along with necessary and su�cientconditions for their existence, for maximally entangled GHZ-states and translationally invariantW-states: both these classes cannot be generally stabilized by dissipation alone. We concludewith an outlook on existing open problems.

Keywords: Quantum information and control; Global stability; Distributed models

1. INTRODUCTION

Entanglement is one of the most intriguing features ofquantum theory, allowing for a composite system to existin a state exhibiting correlations that are not possible in aclassical probabilistic framework. Nonetheless, observingthe e↵ects of these peculiar quantum features typicallyrequires precise, state-of-the-art engineering of interactingexperimental systems. In particular, maximally entangledstates are fragile and hard to engineer: it is then natural toask which states with long-range quantum correlations canbe created and made stable by dynamical interactions thatare “local” in some suitable sense, as the available controlresources typically are. Additional motivation for lookinginto this problem comes from quantum information appli-cations: entanglement is not only the key of the advantageof certain quantum algorithms over their classical equiva-lent, but it has been shown that certain entangled states ofa (2D) register of quantum bits, graph states, are su�cientresources for performing quantum computation, which isthen implemented with a series of conditional measure-ment. In order to take advantage of this computationalmethod, these quantum states have to be generated, and“kept alive” until the sequence measurements is ready tobe performed.

A way of tackling this problem is engineering dissipation(see e.g. Poyatos et al. (1996); Beige et al. (2000); Carvalhoet al. (2001); Ticozzi and Viola (2008)) by controlled inter-action of the system of interest with auxiliary ones, or by

? F.T. acknowledges support by the QUINTET and QFutureprojects of the University of Padova.

implementing measurements and feedback control loops.In fact, in order to prepare a pure entangled state from anarbitrary one it is necessary to rely on open quantum dy-namics, and in particular considering Markov semigroupshas the advantages of time-invariant dynamical modelswhich, for stabilization purposes, are robust with respectto the initial condition. A number of theoretical results onMarkovian dynamics for entangled state preparation havebeen recently presented in Kraus et al. (2008); Verstraeteet al. (2009); Ticozzi and Viola (2009); Perez-Garcia et al.(2008); Ticozzi and Viola (2012), including methods forperforming quantum computation in a dissipative fash-ion (Verstraete et al. (2009)), for the preparation of newphases of matter (Diehl et al. (2008)), and experimentalresults for the digital simulation of continuous-time dy-namics in trapped ions (Barreiro et al. (2011)) and formacroscopic entangled-state engineering (Krauter et al.(2011)). In particular, in our earlier contribution (Ticozziand Viola (2012)) we provided necessary and su�cientconditions for an entangled state to be stabilizable bypurely-dissipative means in the presence of given localityconstraints. These conditions allow for a simple mathemat-ical test, and provide us with an explicit form for a set ofe↵ective noise actions to be engineered, among the viableones. In obtaining this result, we restricted our attention toa particular case: we assumed we could not engineer quasi-local Hamiltonians, but only noise operators (in a suitablesense), and that there was not a drift, uncontrolled partof the dynamical generator.

We here explore the general case, namely the possibil-ity of achieving global asymptotic stabilization of a given

entangled state for quasi-local, continuous-time Markovdynamics, by employing both Hamiltonian and dissipators,and possibly in the presence of drift dynamics. We developnecessary conditions on the Hamiltonian to make a stateasymptotically stable under quasi-local constraints, whenit was not stabilizable by dissipation alone. We provideexamples for Hamiltonians satisfying these properties ande↵ectively stabilizing physically relevant classes of entan-gled states under suitable locality constraints. In the lastSection, we discuss explore the role of underlying driftdynamics (both Hamiltonian and dissipative) , showinghow various cases can be recast into the former driftlessproblem, or can be solved by adapting our previous results.

2. PRELIMINARIES

2.1 Quantum dynamical semigroups and locality notions

We consider a multipartite system Q, composed of n(distinguishable) subsystems, labeled with index a =1, . . . , n, with associated da-dimensional Hilbert spacesHa. Thus,HQ =

Nna=1 Ha. In the statistical description of

quantum systems, a state for the system is associated to adensity operator ⇢, which is a trace-one, positive operatoron the system Hilbert space H. Let D(H) denote the setsof density operators on H.

The dynamical models we are interested in are associatedto Lindblad-GKS Master Equations (ME):

⇢(t) =L(⇢(t)) (1)

=�i[H, ⇢(t)] +X

k

⇣Lk⇢(t)L

†k � 1

2{L†

kLk, ⇢(t)}⌘,

that represent the general form of generators of completely-positive, trace-preserving continuous Markov semigroups{eLt, t � 0} acting on D(H). H is the Hermitian operatorassociated to the Hamiltonian of the systems, while {Lk}are called noise operators and specify the non-Hamiltonianpart of the generator leading to irreversible dynamics (the“noise action”).

The locality constraints on the semigroup dynamics areintroduced by limiting the action of each operator Lk

to a↵ect only certain subsets of subsystems, which wecall neighborhoods, and an equivalent condition will beimposed on H. The neighborhoods can be specified, infull generality, as subsets of the set of indexes labeling thesubsystems:

Nj ✓ {1, . . . , n}, j = 1, . . . ,M.

We say that a noise operator L is Quasi-Local (QL) if thereexists a neighborhood Nj such that:

L = LNj ⌦ INj,

where LNj accounts for the action of L on the subsystemsincluded in Nj , and INj

:=N

a/2NjIa is the identity on

the remaining subsystems. Similarly, a Hamiltonian is QLif it admits a decomposition into a sum of QL terms:

H =X

j

Hj , Hj = HNj ⌦ INj.

A ME will be called QL if both its Hamiltonian andnoise operators are QL 1 . The introduction of locality1 While it is well known that the decomposition into Hamiltonianand dissipative part of (1) is not unique, it can be proven that the QL

constraints through the neighborhoods allows us to includein a single definition a number of specific notions that havebeen used in the physical literature, where the locality no-tions are typically associated with sets of nearest-neighborsites on a graph or lattice, or the Hamiltonian and noisegenerators are forced to act on a given maximum numbert of subsystems (so-called t-body interactions), see alsoKraus et al. (2008), Verstraete et al. (2009).

2.2 Standard form for the ME and stabilizing QL dynamics

An state ⇢d for a system driven by (1) is said to be GloballyAsymptotically Stable (GAS) if for every initial condition⇢0 we have

limt!1

eLt[⇢0] = ⇢d.

A necessary condition for a state ⇢d = | ih | to be GASis that the state is invariant for the dynamics, namelyL(⇢d) = 0: the following proposition, that has been provenin Ticozzi and Viola (2012), provides us with a simple wayto check its invariance.

Proposition 1. Let the system dynamics be driven by L.Then ⇢d = | ih | is invariant for the dynamics if and onlyif

Lk| i= `k| i, `k 2 C 8k,H| i= h| i, h 2 R, (2)

with H = H + iP

k(`⇤kLk � `kL

†k).

Building on this result, in the Lemma below we introduce aconvenient standard form, or more precisely, an equivalentrepresentation of a stabilizing generator in terms of newHamiltonian and noise operators with some useful proper-ties.

Lemma 1. If a generator L associated to H, {Lk} makes⇢d = | ih | GAS, then the same generator can beassociated to H, {Dk} such that H| i = h| i and 8k,Dk| i = 0.

Proof. Assume ⇢d = | ih | is GAS for (3), by Proposition1 it follows that

Lk| i = `k| i, `k 2 C 8k,and H| i = h| i, H = i

Pk(`

⇤kLk�`kL

†k). Then, if `k 6= 0,

use Lemma 2 in Ticozzi and Viola (2008) to substituteLk with Dk = Lk � `kI. In this way we can always,without loss of generality, consider `k = 0, 8k, and Han Hamiltonian that has | i as an eigenstate. 2

In other words, we can always transform the semigroupoperators in a standard form, so that `k = 0 for all ks.This is similar to what is done in the preliminary analysisof Kraus et al. (2008), when an equivalent (possibly non-physical) process is built with the desired state as theunique dark state.

Our main task is to find ways to determine if a target stateis stabilizable with QL resources:

property remains well defined since the freedom in the representationdoes not a↵ect the tensor structure of H and {Lk}.

Definition 1. A pure state ⇢d = | ih | 2 D(HQ), isQuasi-Locally Stabilizable (QLS) if there exist QL opera-tors H, {Dk}k=1,...,K as in Lemma 1 such that ⇢d is GASfor the associated generator L.In Ticozzi and Viola (2012), we restricted our attention tothe case in which H = 0, and the stabilization is enactedby the dissipators alone:

Definition 2. A pure state ⇢d = | ih | 2 D(HQ), isDissipatively Quasi-Locally Stabilizable (DQLS) if thereexist QL operators {Dk}k=1,...,K on HQ, with Dk| i = 0,such that ⇢d is GAS for

⇢ =X

k

⇣Dk⇢D

†k � 1

2{D†

kDk, ⇢}⌘. (3)

It can be easily shown, given Theorem 1 below, that forDQLS states we can restrict to the case of a single noiseoperator for neighborhood without loss of generality. Acharacterization of DQLS states, which leads to simplelinear-algebraic algorithm to test whether a state is DQLS,has been provided and it is essentially based on theproperties of the reduced states on the neighborhoods. Letus define:

⇢Nk = traceNk(⇢d), (4)

where traceNkindicates the partial trace over the tensor

complement of the neighborhood Nk, namely HNk=N

a/2NkHa. We thus have the following result.

Theorem 1. A pure state ⇢d = | ih | is DQLS if and onlyif

supp(⇢d) =\

k

supp(⇢Nk ⌦ INk) ⌘

\

k

HNk . (5)

The proof of the Theorem leads to explicit connectionswith the concept of parent Hamiltonians. In fact, considera QL Hamiltonian H =

Pk Hk, Hk = HNk ⌦ INk

. Apure state ⇢d = | ih | is called a frustration-free groundstate if

h |Hk| i = min �(Hk), 8k,where �(·) denotes the spectrum of a matrix. A QLHamiltonian is called a parent Hamiltonian if it admits aunique frustration-free ground state (Perez-Garcia et al.2007). Suppose that a pure state admits a QL parentHamiltonian H. Then the QL structure of H may benaturally used to derive a stabilizing semigroup, and itis easy to show that this condition is also necessary:

Corollary 2. A state ⇢d = | ih | is DQLS if and only if itis the ground state of a QL parent Hamiltonian.

We refer the reader to Ticozzi and Viola (2012) for a moredetailed discussion of the relation of these results withthose derived relying on the Matrix-Product State (MPS)representation, see e.g. Perez-Garcia et al. (2007, 2008).

3. CAN A QL HAMILTONIAN STABILIZENON-DQLS STATES?

The first question that we will address is the following:are all QLS states also DQLS states? In other words, canthe addition of a QL Hamiltonian to some engineereddissipation make a non-DQLS state GAS for the newdynamics? In this section we provide a set of conditionsthat H and {Dk} have to satisfy for this to happen.

The first result regards the {Dk}: this variation of Lemma4 in Ticozzi and Viola (2012) shows that a QLS statesupport must be, as in the DQLS, contained in the kernelof the noise operators in standard form.

Lemma 2. Assume that the operators H, {Dk} make ⇢d =| ih | QLS. Then, for each k, we have supp(⇢Nk) ✓ker(DNk).

Proof. In order to be QLS, the state must be invariant forL. Hence, by Proposition 1, | i must be in the kernelof each Dk. Thus, with respect to the decompositionHQ = H � H?

, with H = span{| i}, every Dk mustbe of block form:

Dk =

0 DP,k

0 DR,k

�,

which immediately implies Dk⇢dD†k = 0. It then follows

that traceNk(Dk⇢dD

†k) = 0 = traceNk

(DNk ⌦ INk⇢dD

†Nk

⌦INk

). Therefore, it also follows thatDNk⇢NkD†Nk

= 0. If weconsider the spectral decomposition ⇢Nk ⌘Pj qj |�jih�j |,with qj > 0, the latter condition implies that, for each

j, DNk |�jih�j |D†Nk

= 0. Thus, it must be supp(⇢Nk) ✓ker(DNk), as stated. 2

Of course, given Theorem 1 above, if the support of ⇢d isexactly the intersection of these kernels, it would be DQLS.Let us then assume that

H0 =\

k

HNk , dim (H0) = n0 � 2,

so that ⇢d is not DQLS. With this notation, we have thefollowing:

Proposition 2. ⇢d is QLS but not DQLS only if H is aninvariant subspace for the e↵ective Hamiltonian H, and noother invariant subspace is contained in (or equal to) H0.

In particular, one can choose it so H| i = 0.

Proof. If there were another invariant subspace for theHamiltonian with support in H0, it would be, by definitionof H0, also in the kernel of each Dk, and hence wouldbe invariant. The dynamics in this invariant subspacewould be Hamiltonian, and hence it would admit as manyinvariant states as the the dimension of the subspace. Then⇢d could not be GAS. If H| i = �| i, � 6= 0 we can alwayschoose H 0 � �I instead, which is also QL if H was so. 2

When such a H exists, one must seek for noise operators{Dk} such that H is the only invariant subspace forthe whole generator, and hence make ⇢d GAS. Lemma 2suggests that the most e↵ective choice of noise operatorscan stabilize H0, but cannot go further. In order to specifywhat the action of an stabilizing Hamiltonian would be, itis convenient to pick an orthonormal basis of generator forH0, including the target:

H0 = span{| i, |�1i, . . . , |�ri}, r = n0 � 1.

One may hope that, for example, H|�ji /2 H0 for eachj. This set of conditions is clearly not necessary, and ingeneral it may not be possible given the QL constraints anddetermining if a QL H exists for any non-DQLS state is inan open problem. However, in the simple case of n0 = 2,this idea leads to specialized formulation of Proposition 2.

Corollary 3. ⇢d is QLS but not DQLS, with H0 =span{| i, |�1i}, only if there exists a QL Hamiltonian H

such that H| i = 0 and

H|�1i /2 H0. (6)

The necessary conditions summarized in the results aboveclearly show that an e↵ective Hamiltonian cannot be afrustration-free Hamiltonian, i.e. it must destabilize H0.In the following section we will employ Corollary 3 toshow that such an Hamiltonian does exist for two notableclasses of states: maximally entangled GHZ states andtranslationally invariant W states for multi-qubit systems.Both these classes are never DQLS, i.e. for any number ofqubits and non-trivial locality constraint the conditions ofTheorem 1 cannot be satisfied. However, with respect toa suitable locality notion, we can show that there exists aH proving they are QLS.

Remark: In the case we find a H satisfying Proposition2, we still do not have a constructive procedure for deter-mining the stabilizing Dk. What can be shown, as we willillustrate with an example in Section 4.1, is that not allchoices of {Dk} satisfying Lemma 2 are e↵ective. In fact,one has to be careful to ensure that the interplay betweenH and the noise operators introduces enough “mixing” anddoes not allow for the existence of other invariant sets.We claim that a generic choice of noise operators thatsatisfy Lemma 2 and stabilize H0 will su�ce: the proofwill provided elsewhere, and would follow the argument ofSection III.B of Ticozzi et al. (2011).

4. DQLS 6= QLS: EXAMPLES

4.1 GHZ states

Let us first consider a n-qubit system, with total dimensionN = 2n, and the possibility of rendering a Greenberger-Horne-Zeilinger (GHZ) state ⇢GHZ = | ih |, with

| i ⌘ | GHZi = (|000 . . . 0i+ |111 . . . 1i)/p2,

globally asymptotically stable for a QL generator.

As it has been already shown in Ticozzi and Viola (2012),GHZ states are not DQLS. In fact, any reduced state onany (nontrivial) neighborhood is an equiprobable mixtureof |000 . . . 0i and |111 . . . 1i. It is then immediate to seethat

H0 = span{|000 . . . 0i, |111 . . . 1i} =\

k

supp(⇢Nk ⌦ INk),

and hence ⇢GHZ is not DQLS. However, the introductionof a QL Hamiltonian can indeed render ⇢GHZ GAS: givenCorollary 3, we need H such that:

H(|000 . . . 0i+ |111 . . . 1i)/p2 = 0,

H(|000 . . . 0i � |111 . . . 1i)/p2 /2 H0.

In order for this to happen, we need to find H such that

H|000 . . . 0i = �H|111 . . . 1i.We now take into account the QL structure of H =

Pk Hk,

and the fact that each QL component acts on at most afinite number nk = #(Nk) of “symbols”, i.e. 0 or 1 in thefactorized components of the states |0, 1i⌦n, respectively.

If we assume nk < n/2, we have that

H|000 . . . 0i 2 span{|x1, . . . , xni, xj 2 {0, 1},X

j

xj < n/2},

H|111 . . . 1i 2 span{|x1, . . . , xni, xj 2 {0, 1},X

j

xj > n/2}.

Hence, the two vectors must be orthogonal, since theybelong to subspaces spanned by two orthogonal sets ofvectors. Hence an Hamiltonian satisfying the requirementsof Corollary 3 does not exists. In other words, we need Hto be able to “flip” at least N/2 bits in the completelyfactorized basis states |000 . . . 0i, |111 . . . 1i.In fact, if we take the neighborhood to include nk = n/2qubits, we can always construct a QL Hamiltonian suchthat

H(|000 . . . 0i = (|1 . . . 10 . . . 0i � |0 . . . 01 . . . 1i)/p2,

H(|111 . . . 1i = (�|1 . . . 10 . . . 0i+ |0 . . . 01 . . . 1i)/p2,

with the vectors in the r.h.s. containing exactly n/2 zeroesand n/2 ones, which clearly satisfies the requirements. Thiscan be obtained by considering two disjoint sets S`=1,2

each including one half of the qubits (or (n + 1)/2, (n �1)/2 in the odd-n case), and for ` = 1, 2, choose H` =

(�1)`�1⇣N

a2S`�(a)x

⌘⌦ IS`

. If in our locality constraints

there exists two neighborhoods Nk1,k2 such that

S`=1,2 ⇢ Nk`=1,2

with a proper choice of the noise operators, a GHZ statecan then be rendered GAS. For example, in the 3-qubitcase, one can choose S1 = {1},S2 = {2, 3}, so that under2-body locality constraints N1 = {1, 2},N2 = {2, 3}, wecan implement:

H = �(1)x � �(2)

x ⌦ �(3)x ,

D1 = I2 ⌦ (|00ih01|+ |11ih10|),D2 = I2 ⌦ (|00ih01|+ i|11ih10|).

Remark: Notice the phase factor in D2: in fact, the “moresymmetric” choice D0

2 = I2 ⌦ (|00ih01| + |11ih10|) would

leave the �1 eigenspace ofN

a=1,2,3 �(a)x invariant for

the generator associated to H,D1, D02, while ⇢GHZ is the

unique invariant state for H,D1, D2.

This simple example is enough to prove that DQLS(QLS,namely that there exist pure entangled states that are notstabilizable by dissipation alone but they can be made GASby the addition of a QL Hamiltonian.

Furthermore, this way of making GHZ states GAS isgeneral, and, in the light of the observation above itrequires neighborhoods of the minimum possible size.Notice that the neighborhood size scales linearly (⇡ n/2)with the overall size of the system, and hence for n ! +1the needed interaction size goes to infinity.

4.2 W states

Consider the an n-qubit system with the same notation asin the previous example. The W states ⇢W = | ih |, with| i ⌘ | Wi = (|100 . . . 0i+|010 . . . 0i+. . .+|000 . . . 1i)/pnrepresent permutation-invariant superposition of the stan-dard basis states with a single 1, and has reduced states

on any neighborhood size that are statistical mixtures of|000 . . . 0i and a smaller W state | W0i, of the dimensionof the neighborhood. Thus,

H0 = span{|000 . . . 0i, | Wi} =\

k

supp(⇢Nk ⌦ INk),

and thus ⇢W is not DQLS. We are still in a case whereCorollary 3 can be applied.

It is easy to show that the following 2-body Hamiltoniansatisfies the requirements of Corollary 3:

H = �(1)x ⌦ P0 � P0 ⌦ �(n)

x (7)

= �(1)x ⌦

n�1X

a=2

(�(a)z � (n� 3)I(a))

!

+

n�1X

a=2

(�(a)z � (n� 3)I(a))

!⌦ �(n)

x , (8)

where P0 defined as above is an operator onNn�1

a=2 Ha suchthat:

P0| Wn�2i = 0, P0|0i⌦(n�2) = |0i⌦(n�2).

While we derived this particular Hamiltonian relying onsymmetry considerations, to our scope it is su�cient toverify by direct computation that, as we needed:

H| W i = 0, H|0i⌦n = |10 . . . 0i � |0 . . . 01i.This shows howW-states on an arbitrary number of qubitscould in principle be QLS by allowing for arbitrary 2-bodyinteractions.

5. INCLUDING DRIFT DYNAMICS

We will now establish some modified results that take intoaccount the possibility of pre-existing QL dynamics. Itwill be convenient to discuss the case of having a givenHamiltonian component first, and to introduce the moregeneral case next.

5.1 Drift Hamiltonian

Assume a drift, free QL Hamiltonian H0 is given for thesystem of interest, and the target state is ⇢d = | ih |.If we want to establish whether ⇢d can be made GAS bypurely dissipative dynamics added to the action of H0, wecan first, check whetherH0| i = �| i; If it is not, considera QL decomposition of H0 =

Pk Hk. Decompose Hk in

matrix blocks according to H = H �HR :

Hk =

HS HP

H†P HR

�.

Define for each k:

D0k =

1 D0

P,k

0 0

�, D0

P,k = 2iHP,k.

By Corollary 1 of Ticozzi and Viola (2008) it follows thatadding D0

k as a noise operator for each neighborhood makes⇢d invariant for the global dynamics. Hence, we can find analternative representation of the generator with the addednoise operators in standard form with H 0

0 and {D00k} such

that H 0| i = 0. Now we can proceed to:

(1) Determine if ⇢d would be DQLS in the absence of H 00

by Theorem 1;

(2) If it is DQLS, ageneric choice of stabilizing {Dk} willmake ⇢d GAS.

(3) If ⇢d is not DQLS, we can check if H 00 satisfies the

necessary conditions of Proposition 2; If it does not,⇢d cannot be made GAS. If it does, we know thatunder some suitable locality notions, there can existsstabilizing dissipators.

If we add QL Hamiltonians to our engineering capabilities,the situation is simpler, since if the QL H0 is known, acontrolled H1 = �H0 can be added to cancel the action ofthe drift. Then the problem becomes completely equivalentto the driftless one, for which we have Theorem 1 (DQLScase) or the set necessary conditions Lemma 2, Proposition2 and/or Corollary 3 (QLS case).

5.2 Drift Hamiltonian and noise

Assume that a QL drift dynamics is driving the system ofinterest, let us denote it by:

⇢ = �i[H0] + LD[⇢],

where LD[⇢] denotes some noise generator in Lindbladform, associated to QL noise operators {Lk}.The first thing to check is whether, for each neighborhoodNk, Lk| i = �k| i. In fact this is a necessary conditionfor the invariance of ⇢d that must be satisfied by allnoise operators irrespective of them being controlled or not(Corollary 1 in Ticozzi and Viola (2008)). Hence, if thereexists k such that Lk| i 6= �k| i, then ⇢d cannot be madeGAS.

If Lk| i = �k| i for all k, then following Lemma 1we can write an equivalent generator in standard formassociated to operators H0, Lk such that Lk| i = 0. Thesenoise operators thus must satisfy Lemma 2. In this way,we have now recast the problem into the one consideredin the previous section, where we had only Hamiltoniandynamics.

(1) In case we have only purely-dissipative control ca-pabilities, we need to determine if the Hamiltoniandestabilizes the desired state. A destabilizing Hamil-tonian can then be compensated following point (2.b)of the previous section, but now choosing

D0P,k = 2iHP,k � L†

S,kLP,k.

The free dynamics plus the QL {D0k} can now be

associated to operators in standard form H 00, D

0k.

(2) If the state is DQLS, a generic choice of {Dk} shouldsu�ce. If the state is not DQLS, we can check ifH 0

0 satisfies the conditions of Proposition 2 and/orCorollary 3. If this is the case, we are left looking fore↵ective QL noise operators, otherwise the state isnot QLS.

(3) In case we also have QL Hamiltonian engineeringcapabilities, then we can cancel H 0

0 as above, and theproblem is that of determining whether ⇢d is DQLSor QLS.

6. QL STABILITY AND STABILIZATION: WHEREDO WE STAND?

Building on the analysis and results of Ticozzi and Viola(2012), we have provided examples that show for certain

locality notions how the class of DQLS states is strictlycontained in the QLS class, as the latter contains also thesought-after GHZ or W states. This points to QL controlHamiltonians as key resources for dissipative engineeringentanglement in multipartite systems.

While a characterization of QLS states is still missing, wehave provided in Proposition 2 and/or Corollary 3 neces-sary conditions for an Hamiltonian to be an e↵ective aid instabilizing the desired pure state. If such an Hamiltonianexists, we have so far been able to easily find e↵ectiveQL noise operators that make the desired state GAS (andhence QLS) on a case by case basis. Our claim is that ageneric choice of the {Dk} is e↵ective, provided that theysatisfy Lemma 2 and that the only invariant subspace forthe dissipative part of the dynamics is:

H0 =\

k

supp(⇢Nk ⌦ INk) =

\

k

HNk .

A detailed proof of this claim will be worked out elsewhere,relying on the fact that the sub-manifold of states thatcan be made invariant by adding an Hamiltonian toa dissipative generator has a lower dimensionality withrespect to the whole set.

The key open problems are then to provide an explicitcharacterization of QLS states, possibly suggesting a wayto classify states at least for commonly-used locality con-straints, and to find a systematic way to construct, when-ever possible, a QL Hamiltonian satisfying Proposition 2.The possibility of considering switched dynamics, feedbackcontrol and time-dependent generators also o↵er promisingresearch directions, in the e↵ort of surpassing the stringentlimitations imposed by QL continuous-time quantum dy-namics.

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