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81

Quantum Markovian Subsystems:Invariance, Attractivity, and Control

Francesco Ticozzi and Lorenza Viola

Abstract— We characterize the dynamical behavior ofcontinuous-time, Markovian quantum systems with respect to asubsystem of interest. Markovian dynamics describes a wideclassof open quantum systems of relevance to quantum informationprocessing, subsystem encodings offering a general pathway tofaithfully represent quantum information. We provide expl icitlinear-algebraic characterizations of the notion of invariant andnoiseless subsystem for Markovian master equations, underdif-ferent robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantumsubsystem is introduced, and sufficient existence conditions areidentified based on Lyapunov’s stability techniques. As a maincontrol application, we address the potential of output-feedbackMarkovian control strategies for quantum pure state-stabilizationand noiseless-subspace generation. In particular, explicit resultsfor the synthesis of stabilizing semigroups and noiseless subspacesin finite-dimensional Markovian systems are obtained.

I. I NTRODUCTION AND PRELIMINARIES

Quantum subsystems are the basic building block for de-scribing composite systems in quantum mechanics [1]. Fromboth a conceptual and practical standpoint, renewed interesttoward characterizing quantum subsystems in a variety ofcontrol-theoretic settings is motivated by Quantum Informa-tion Processing (QIP) applications [2]. In order for abstractlydefined QIP protocols to be useful, information needs to berepresented by states of a physical system, in ways whichminimize the impact of errors and decoherence due to theinteraction of the system with its surrounding environment.Subsystem encodingsprovide the most general mathematicalstructure for realizing quantum information in terms of physi-cal degrees of freedom, and a main tool for achieving a unifiedunderstanding of strategies for quantum error control inopenquantum systems [3], [4]. In particular, the idea that noise-protected subsystems may be identified in the overall statespace of a noisy physical system under appropriate symmetryconditions underlies the method of passive quantum stabi-lization via decoherence-free subspaces(DFSs) [5], [6] andnoiseless subsystems(NSs) [3], [7], [8]. In situations whereno such symmetry exists, open-loop dynamical decouplingtechniques can still ensure active protection through dynamicalNS synthesis [9] in thenon-Markovianregime.

While substantial progress has been made toward defin-ing and exploiting subsystems within general quantum errorcorrection theories (see also Section II-A), the subsystemnotion and its implications have not yet reached out to the

F. Ticozzi is with the Dipartimento di Ingegneria dell’Informazione,Universita di Padova, via Gradenigo 6/B, 35131 Padova, Italy(ticozzi@dei.unipd.it).

L. Viola is with the Department of Physics and Astronomy, Dart-mouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA(Lorenza.Viola@Dartmouth.edu).

quantum control community at large. It is one of the goalsof this paper to take a step in this direction, by offering alinear-algebraiccharacterization of the subsystem idea, whichallows easier contact with standard system-theoretic conceptssuch as invariance and stability. We focus oncontinuous-timeMarkovian quantum dynamics [10], which both accuratelydescribes a wide class of open quantum systems and presentsdistinctive quantum stabilization challenges compared toitsnon-Markovian counterpart. In the process, we elucidate thekey role played by differentmodel robustnessproperties inensuring that desired dynamical features may emerge, and ininfluencing the interplay between Hamiltonian and dissipativecomponents. Beside leading to a streamlined derivation ofMarkovian DFS- and NS- conditions which have only partiallyappeared in the literature [8], [11], our analysis motivates theconcept ofattractive subsystemas a strategy for “dissipation-assisted” asymptotic initialization in an intended subsystem.As a main application of our work, we begin exploring the po-tential of Markovian, output-feedback techniques for the robustsynthesis of pure states and NSs in finite-dimensional systems– complementing the above-mentioned open-loop dynamicalschemes [9] as well as closed-loop feedback approaches usingcontinuous-time state estimation [12].

A. Notations

Consider a separable Hilbert spaceH over the complex fieldC. B(H) represents the set of linear bounded operators onH,H(H) denoting the real subspace of Hermitian operators, withI, O being the identity and the zero operator, respectively.We indicate withA† the adjoint ofA ∈ B(H), with c∗ theconjugate ofc ∈ C. The commutatorand anti-commutatorof X,Y ∈ B(H) are defined as[X, Y ] := XY − Y X ,andX, Y := XY + Y X , respectively. In the special casewhereH is two-dimensional, a convenient operator basis forthe traceless sector ofB(H) is given by the Pauli operators,σα, α = x, y, z, satisfying[σα, σβ ] = 2iεαβγσγ , σα, σβ =2δαβI, ε, δ denoting the completely antisymmetric tensor andthe Kronecker delta, respectively. We choose the standard rep-resentation whereσx = ( 0 1

1 0 ) , σy =(

0 −ii 0

)

, σz =(

1 00 −1

)

.

Throughout the paper we shall use the Dirac notation [1].Given the inner product〈 , 〉 in H, a natural isomorphismexists between vectors inH (denoted|ψ〉, and called aket)and linear functionals in the dual spaceH∗ (denoted〈ψ|, andcalled abra), so that〈ψ|ϕ〉 = 〈ψ, ϕ〉. If A ∈ B(H), lettingA|ψ〉 := A(ψ) and 〈ψ|A := 〈A†(ψ), ·〉 yields 〈ψ|A|ϕ〉 =〈φ,A(ϕ)〉 = 〈A†(φ), ϕ〉. The “outer” product|ψ〉〈ϕ| standsfor 〈ϕ, ·〉ψ ∈ B(H). Moreover, if |〈ψ|ψ〉| = 1, |ψ〉〈ψ| =〈ψ, ·〉ψ is the orthogonal projector onto the one-dimensionalsubspace spanned by|ψ〉.

2

B. Basic notions of statistical quantum mechanics

In the standard formulation of quantum mechanics [1], aquantum systemQ is associated with a separable HilbertspaceH, whose dimension is determined by the physicsof the problem.Physical observablesare modeled as self-adjoint operators onH, the set of possible outcomes theycan assume in a von Neumann measurement process beingtheir spectrum. In what follows, we consider only observableswith finite spectrum, that can be represented as Hermitianmatrices acting onH ≃ C

d, d < ∞. Our (possibly uncertain)knowledge of the state ofQ is condensed in adensity operatorρ, with ρ ≥ 0 and trace(ρ) = 1. Density operators form aconvex setD(H) ⊂ H(H), with one-dimensional projectorscorresponding to extreme points (pure states, ρ|ψ〉 = |ψ〉〈ψ|).

If Q comprises two quantumsubsystemsQ1, Q2, thecorresponding mathematical description is carried out in thetensor product space,H12 = H1 ⊗ H2 [1], observables anddensity operators being associated with Hermitian, positive-semidefinite, normalized operators onH12, respectively. Inparticular, a joint pure stateρ12 = |ψ〉12〈ψ| which cannotbe factorized into the product of two pure states on theindividual factors is calledentangled. Let X be an elementof B(H12). The partial trace over H2 is the unique linearoperator trace2(·) : B(H12) → B(H1), ensuring that for everyX1 ∈ B(H1), X2 ∈ B(H2), trace2(X1⊗X2) = X1trace(X2).If ρ12 represents the joint density operator ofQ, the state of,say, subsystemQ1 alone is described by thereduced densityoperator ρ1 = trace2(ρ12) ∈ H(H1), so that trace(ρ1X1) =trace(ρ12X1 ⊗ I2), for all observablesA1 ∈ H(H1).

We consider quantum dynamics in the Schrodinger picture,with pure states and density operators evolving forward intime, and time-invariant observables. The evolution of anisolated(closed) quantum system is driven by the Hamiltonian,H , according to the Liouville-von Neumann equation:

d

dtρ(t) = −i[H, ρ(t)] ,

in units where~ = 1. Thus, ρ(t) = Utρ(0)U †t , where the

unitary evolution operator (orpropagator) Ut = e−iHt.In general, in the presence of internal couplings, quantum

measurements, or interaction with a surrounding environment,the evolution of a subsystem of interest is no longer unitaryand reversible, and the general formalism ofopen quantumsystemsis required [13], [10], [14]. The physically admissiblediscrete-time evolutions for a quantum system may be char-acterized axiomatically and are calledquantum operations, orcompletely positive (CP) maps [15]. LetI denote the physicalquantum system of interest, with associated Hilbert spaceHI ,dim(HI) = d. The class of trace-preserving (TP) quantumoperations is relevant to our purposes:

Definition 1 (TPCP map):A TPCP-mapT (·) on I is amap onD(HI), that satisfies:(i) T (·) is convex linear: given statesρi ∈ D(HI),

T(

∑

i

piρi

)

=∑

i

piT (ρi),∑

i

pi = 1, pi ≥ 0, ∀i;

(ii) T (·) is trace-preserving:

trace(T (ρI)) = trace(ρI) = 1;

(iii) T (·) is completely positive:

T (ρI) ≥ 0, (Im ⊗ T )(ρIE) ≥ 0,

for everym-dimensional ancillary spaceHE joint to HI ,and for everyρIE ∈ D(HI ⊗HE) 1.

A more concrete characterization of the dynamical maps ofinterest is provided by the following:

Theorem 1 (Hellwig-Kraus representation theorem):T [·]is a TPCP map onI iff for every ρI ∈ D(HI):

T (ρI) =∑

k

EkρIE†k,

where Ek is a family of operators inB(HI) such that∑

k E†kEk = I.

As a consequence of the above Theorem, every TPCP mapT (·) on D(HI) may be extended to a well-defined linear,positive, and TP map on the wholeB(HI).

C. Quantum dynamical semigroups

A relevant class of open quantum systems obeys Markoviandynamics [10], [16], [17]. Assume that we have no accessto the quantum environment surroundingI, and that thedynamics inD(HI) is continuous in time, the state changeat eacht > 0 being described by a TPCP mapTt(·). Adifferential equation for the density operator ofI may bederived provided that a forward composition law holds:

Definition 2 (QDS):A quantum dynamical semigroupis aone-parameter family of TPCP mapsTt(·), t ≥ 0 thatsatisfies:

(i) T0 = I,(ii) Tt Ts = Tt+s, ∀t, s > 0,(iii) trace(Tt(ρ)X) is a continuous function oft, ∀ρ ∈ D(HI),

∀X ∈ B(HI).

Due to the trace and positivity preserving assumptions, a QDSis a semigroup of contractions2. It has been proved [16], [18]that the Hille-Yoshida generator for a QDS [19] exists and canbe cast in the following canonical form:

d

dtρ(t) = L(ρ(t)) = −i[H, ρ(t)] +

d2−1∑

k,l=1

Lkl(ρ(t)) (1)

= −i[H, ρ(t)] + 1

2

d2−1∑

k,l=1

akl

(

2Fkρ(t)F†l − F †

kFl, ρ(t))

,

where Fkd2−1k=0 is a basis ofB(HI), the space of linear

operators onHI , with F0 = I. The positive definite(d2 − 1)-dimensional matrixA = (akl), which physically specifiesthe relevant relaxation parameters, is also called the Gorini-Kossakowski-Sudarshan (GKS) matrix. Thanks to the Hermi-tian character ofA, Eq. (1) can be rewritten in a symmetrized

1The CP assumption is necessary to preserve positivity of arbitrary purifica-tions of states onHI , including entangled states, see e.g. [2] for a discussionof a well-known counter-example, based on the transpose operation.

2A map f from a metric spaceM with distanced(·, ·) to itself is acontraction if d(f(X), f(Y )) ≤ kd(X, Y ) for all X, Y in M, 0 < k ≤ 1.

3

form, moving to an operator basis that diagonalizesA:

L(ρ(t)) = −i[H, ρ(t)] +d2−1∑

k=1

γkD(Lk, ρ(t)) (2)

= −i[H, ρ(t)] +d2−1∑

k=1

γk

(

Lkρ(t)L†k −

1

2L†

kLk, ρ(t))

,

with γk denoting the spectrum ofA. Theeffective Hamilto-nian H and theLindblad operatorsLk specify the net effectof the Markovian environment on the dynamics. In general,His equal to the isolated system Hamiltonian,H0, plus a cor-rection,HL, induced by the coupling to the environment (so-called Lamb shift). The non-Hamiltonian termsD(Lk, ρ(t))in (2) account for non-unitary dynamics induced byLk.

D. Phenomenological Markovian models and robustness

In principle, the exact form of the generator of a QDS maybe rigorously derived from the underlying Hamiltonian modelfor the joint system-environment dynamics under appropriatelimiting conditions (the so-called “singular coupling limit” orthe “weak coupling limit,” respectively [10], [14]). In mostphysical situations, however, carrying out such a procedure isunfeasible, typically due to lack of complete knowledge ofthe full microscopic Hamiltonian. A Markovian generator ofthe form (1) is then assumed on a phenomenological basis3.In practice, it is often the case that direct knowledge of thenoise effect is available, allowing one to specify the Markoviangenerator by either giving a GKS matrix in (1) or a set ofnoise strengthsγk and Lindblad operatorsLk (not necessarilyorthogonal or complete) in (2). Each of the noise operatorsLkmay be thought of as corresponding to a distinctnoise channelD(Lk, ρ(t)), by which information irreversibly leaks from thesystem into the environment.

Example 1:Consider a two-level atom, with ground andexcited states|ψg〉, |ψe〉, respectively. Assume that there isan average rate of decay from the excited to the ground stateγ > 0, that is, the survival probability of the excited state|ψe〉decays ase−γt. The resulting dynamics is well described bya semigroup master equation of the form

d

dtρ(t) = −iω[σz, ρ(t)] +

γ

2

(

2σ−ρ(t)σ+ − σ+σ−, ρ(t))

,

where ω > 0 determines the energy splitting between theground and excited state, andσ− = |ψg〉〈ψe|, σ+ = σ†

− arepseudo-spin lowering and raising operators, respectively. Infact, computing the probabilityPe(t) = trace(ρ(0)ρ(t)), withinitial stateρ(0) = |ψe〉〈ψe|, yieldsPe(t) = e−γt.

In the above example, a single noise channel,D(σ−, ρ(t)),is relevant. In physical situations, it often happens that knownproperties of the error process naturally restrict the relevant

3The Markovian generator can be inferred from experimentally availabledata via quantum process tomography, see e.g. [2]. By measuring the effectof the environment on known states for fixed times, one may reconstructthe corresponding set of CP maps and the underlying infinitesimal generator.Special care must be paid to the fact that this procedure may lead to non-CP maps in the presence of measurement or numerical errors [20]. In somesituations, approximate models may be obtained upon formalquantization ofa classical master equation, e.g. the so-called Pauli master equation [10].

Lindblad operators or the admissible GKS descriptions to areduced formwhich incorporates the existing constraints. Aparadigmatic QIP-motivated example is the following:

Example 2: Consider aquantum registerQ, that is, aquantum system composed byq two-dimensional systems(qubits), with associated stateHQ = C2

(1)⊗· · ·⊗C2(q), d = 2q.

For an arbitrary Markovian error process, combined errors onany subset of qubits may occur, corresponding to an errorbasisFk which spans the full traceless sector ofB(HQ) or,equivalently, the (d2−1)-dimensional Lie algebrasu(d). Underthe assumption thatlinear decoherencetakes place, errors canindependently affect at most one qubit at the time, reducingthe relevant error set to operators of the form [6], [3]:

Fk,l = I(1) ⊗ · · · I(k−1) ⊗ σ

(k)l ⊗ I

(k+1) ⊗ · · · ⊗ I(q),

where k = 1, . . . , q, and l = x, y, x. Completing these3qorthonormal generators to the above basis for the tracelessoperators inB(HQ), Eq. (1) formally holds. Clearly, theresulting matrixA differs from zero only in a(3q × 3q)-dimensional block. If the noise process is additionally re-stricted to obey permutational symmetry (so-calledcollectivedecoherence), the relevant error set is further reduced tocompletely symmetric generators of the form

Fl =

q∑

k=1

I(1) ⊗ · · · ⊗ σ

(k)l ⊗ · · · ⊗ I

(q), l = x, y, z,

in which case spanFk ≃ su(2), andA may effectively betaken as a3 × 3 positive-definite matrix.

The above examples show how, in practice, a compactversion of (1) typically suffices, in terms of (orthonormal)errorgeneratorsFk spanning am-dimensionalerror subalgebra,m ≤ d2 − 1. The corresponding Markovian generator in (1)is then completely specified by areduced GKS matrixofdimensionm×m.

In the following sections, we are interested at characterizingdynamical properties of finite-dimensional QDSs in terms oftheir generator. As in most situations only limited or approxi-mate knowledge about the model is available, we are naturallyled to consider two kinds ofstructured robustness[21], [22]:

Definition 3 (Model robustness):Assume that a systemIundergoes QDS dynamics, under a nominal generator of theform (1) or (2), with uncertain knowledge of the parametersA = (ajk)

mj,k=1 or Γ = (γ1, . . . , γm). Let A and V denote

the uncertainty sets, that is, the sets of parameters identifyingthe admissible models in the form (1) and (2), respectively.ApropertyP is said to be

(i) A-robust if it holds for everyA = (aij) ∈ A;(ii) γ-robust if it holds for everyΓ = (γ1, . . . , γm) ∈ V .

The study ofA- or γ-robustness is important to establishwhether a desired feature of the model (e.g. invariance of asubsystem, existence of attractive states) may be ensured byavoidingfine-tuningon the noise parameters [6].

Remarks:A-robustness impliesγ-robustness. However theconverse is not true. In fact,γ-robustness corresponds torobustness only with respect to variation in the spectrum ofA.While studyingA-robustness, we shall always imply a reduced

4

description as explained above, withA = A = (ak,l)mk,l=1

denoting the set of relevant reduced GKS matrices. Clearly,only properties that are completely independent of the partic-ular noise model can beA-robust ifA is the wholeB(HI).

II. T HEORY

A. Quantum subsystems and their role in QIP

In order for the physical systemI to implement a QIP task,it is necessary that at every point in time well-defined “logical”degrees of freedom exist, which carry the desired quantuminformation and support a basic set of control capabilities.Within the standard quantum network model [2], such set mustinclude the ability to:

• Unitary control : Implement a set of control actions thatensure universal control.

• Initialization : Realize a quantum operation that preparesthe system in an intended pure state.

• Read-out: Perform measurements of appropriate sys-tem’s observables4.

According to thesubsystem principle[7], [4], the mostgeneral structure which can faithfully embody quantum in-formation is a subsystem ofI. Intuitively, a subsystem maybe thought of as a “portion” of the full physical system,whose states, in the simplest setting, obeyperfectlythe criteriaabove. Logical subsystems may or may not directly coincidewith physically natural degrees of freedom. IfI is noisy,in particular, it suffices that the action of noise be eithernegligible or correctable on subsystem where the informationresides. Thus, protecting information need not require thefull state of the physical system to be immune to noise,although this typically involves encodings which are entangledwith respect to the natural subsystem degrees of freedom. Aparadigmatic example is the protected qubit encoded in threespin-1/2 particles subject to collective decoherence [3], [7].

Formally, the following definition is suitable to QIP settings:

Definition 4 (Quantum subsystem):A quantum subsystemS of a systemI defined onHI is a quantum system whosestate space is a tensor factorHS of a subspaceHSF of HI ,

HI = HSF ⊕HR = (HS ⊗HF ) ⊕HR, (3)

for some factorHF and remainder spaceHR. The set of linearoperators onS, B(HS), is isomorphic to the (associative)algebra onHI of the formXI = XS ⊗ IF ⊕ OR.

Let n = dim(HS), f = dim(HF ), r = dim(HR), and let|φSj 〉nj=1, |φFk 〉

fk=1, |φRl 〉rl=1 denote orthonormal bases

for HS , HF , HR, respectively. Decomposition (3) is thennaturally associated with the following basis forHI :

|ϕm〉 = |φSj 〉 ⊗ |φFk 〉n,fj,k=1 ∪ |φRl 〉rl=1.

This basis induces a block structure for matrices acting onHI :

X =

(

XSF XP

XQ XR

)

, (4)

4In the simplest setting, the ability to effect strong, von Neumann projectivemeasurements is assumed, which together with unitary control implies theability to initialize the state.

where, in general,XSF 6= XS ⊗XF . Let ΠSF be the projec-tion operator ontoHS ⊗HF , that is,ΠSF =

(

ISF 0)

.For a noisy systemI, the goal ofpassivequantum error

control is to identify subsystems ofI where the dominant errorevents have minimum (ideally no) effect. Loosely speaking,each error operator belonging to the fixed error set for whichprotection is sought must have an “identity action” once ap-propriately restricted to the intended subsystem. Historically,the first kind of subsystems considered to this purpose havebeennoiseless subspacesof the system’s Hilbert space, oftencalled DFSs in the relevant literature [5], [6]:

HI = HDFS ⊕HR,

which corresponds to a special instance of decomposition (3)with one-dimensional “syndrome” co-subsystemF , HF ≃ C.The possibility for genuinenoiseless subsystem-encodings toexist and be useful was recognized in [3], in which case wespecialize the notation of (3) to

HI = (HNS ⊗HF ) ⊕HR, (5)

by explicitly identifying the noiseless factor withHNS5.

DFS and NS theory has received extensive attention to date.A relatively straightforward characterization is possible forerror sets which are effectively†-closed, in which case elegantresults from the representation theory of C*-algebras apply6.The operator-algebraic approach is suitable for investigatingNSs within both a Hamiltonian formulation of open-systemdynamics and a large class of TPCP maps, see e.g. [3],[23]. However, explicit characterizations for arbitrary quantumoperations and Markovian dynamics are more delicate and, tosome extent, less consolidated. While a number of definitionsand results are provided in [8], [11], [24], the increasinglyprominent role that quantum subsystems play within quantumerror correction theory [25], along with continuous experimen-tal advances in implementing DFSs [26], [27], [28] and NSs[29], heighten the need for a fully consistent system-theoreticapproach. It is our goal in the remaining of this Section toprovide such a framework for the case of Markovian dynamics,by paying special attention to the key role played by modelrobustness notions as stated in Definition 3.

B. Invariant subsystems

Definition 5 (State initialization):The systemI with stateρ ∈ D(HI) is initialized in HS with stateρS ∈ D(HS) if theblocks ofρ satisfy:

(i) ρSF = ρS ⊗ ρF for someρF ∈ D(HF );(ii) ρP = 0, ρR = 0.

5In principle, multiple NSs may exist for a given dynamical system. Whilewe do not explicitly address such a scenario, generalization is possible alongthe lines presented here.

6A C*-algebra is a complex normed algebraA with a conjugate linearinvolution (* or †, anadjoint operation), which is complete, satisfies‖AB‖ ≤‖A‖‖B‖, and ‖A†A‖ = ‖A‖2, for all A, B ∈ A. Any norm-closedsubspace of bounded operators onH is a C*-algebra if it closed under theusual adjoint operation. Up to unitary equivalence, every finite-dimensionaloperator *-algebra is isomorphic to a unique direct sum of ampliated fullmatrix algebras. Such a decomposition directly reveals thesupported NSs,wheneverA represents anerror algebra for the noisy systemI [3].

5

Condition (ii) in the above Definition guarantees thatρS =traceF (ΠSF ρΠ

†SF ) is a valid state ofS, while condition (i)

ensures that measurements or dynamics affecting the factorHF have no effect on the state inHS . We shall denote byIS(HI) the set of states initialized in this way. The larger setof states obeying condition (ii) alone will correspondingly bedenoted byISF (HI).

Definition 6 (Invariance):Let I evolve under TPCP maps.S is an invariant subsystemif the evolution ofρ ∈ IS(HI)obeys:

ρ(t) =

(

T St (ρS) ⊗ T F

t (ρF ) 00 0

)

, t ≥ 0, (6)

∀ ρS ∈ D(HS), ρF ∈ D(HF ), and with T St (·) and T F

t (·),t ≥ 0, being TPCP maps onHS andHF , respectively.

Thus, a subsystem is invariant if time evolution preservesthe initialization of the state, that is, the dynamics is confinedwithin IS(HI). For Markovian evolution ofI, Definition 2requires bothT S

t andT Ft to be QDSs on their respectivedomain. We begin with the following elementary Lemma:

Lemma 1:Let a linear operatorL : H1 ⊗ H2 → H3 bedifferent from the zero operator. Then there exist factorizedpure states inH1 ⊗H2 ⊖ ker(L).Proof.Assume thatL|ψ〉 = 0 for all factorized|ψ〉 ∈ H1⊗H2.Since such|ψ〉’s generate the wholeH1⊗H2, then by linearityit must beL = 0 and we conclude by contradiction.

Theorem 2 (Markovian invariance):HS supports an in-variant subsystem under Markovian evolution onHI iff forevery initial stateρ ∈ IS(HI), with ρS ∈ D(HS), ρF ∈D(HF ), the following conditions hold:

d

dtρ(t) =

(

LSF (ρSF (t)) 00 0

)

, ∀t ≥ 0, (7)

traceF [LSF (ρSF (t))] = LS(ρS(t)), ∀t ≥ 0, (8)

whereLSF andLS are QDS generators onHS⊗HF andHS ,respectively.Proof. Since Definition 6 is obeyed, computing the infinitesi-mal generator of (6) (att = 0) yields

d

dtρ(t)

∣

∣

∣

∣

t=0

=

(

(LS ⊗ IF + IS ⊗ LF )(ρS ⊗ ρF ) 00 0

)

.

(9)Then the time-invariant generator must have the form (7). Takethe partial trace overHF , and observe that trace(ρF ) = 1,trace(LF (ρF )) = 0. Then (8) holds.

To prove the opposite implication, assume that (7) and (8)hold, andρ ∈ IS(HI). Sinceρ evolves under a QDS generatorthat can be written in the form (2), with HamiltonianHand noise operatorsLk partitioned as in (4), computing thegenerator at a generic timet by blocks yields:

d

dtρ =

(

LSF (ρ) LP (ρ)LQ(ρ) LR(ρ)

)

,

where

LSF (ρ) = −i[HSF , ρSF ] +1

2

∑

k

(

2LSF,kρSFL†SF,k

−

L†SF,kLSF,k + L†

Q,kLQ,k, ρSF

)

,

LQ(ρ) = −iH†PρSF +

1

2

∑

k

(

2LSF,kρSFL†Q,k

− ρSF (L†SF,kLP,k + L†

Q,kLR,k))†,

LP (ρ) = iρSFHP +1

2

∑

k

(

2LSF,kρSFL†Q,k

− ρSF (L†SF,kLP,k + L†

Q,kLR,k

)

,

LR(ρ) =1

2

∑

k

2LQ,kρSFL†Q,k.

In order to satisfy (7), it must be12∑

k 2LQ,kρSFL†Q,k =

0, for every ρSF = ρS ⊗ ρF . Consider for example pureproduct statesρSF = |ψ〉〈ψ|⊗|φ〉〈φ|. Then, by observing thatLQ,kρSFL

†Q,k is positive for everyk and by using Lemma 1, it

must beLQ,k = 0 for everyk. Next, to ensure thatρP (t) = 0for everyt, the remaining contribution toLP (ρ) must vanish,that is, by using Lemma 1 again, it must be:

iHP − 1

2

∑

k

L†SF,kLP,k = 0. (10)

This leaves aSF -block of the form:

−i[HSF , ρSF ] +1

2

∑

k

(

2LSF,kρSFL†SF,k

−

L†SF,kLSF,k, ρSF

)

,

which indeed satisfies (7). To see this, notice that we mayalways writeLSF,k =

∑

iMk,i⊗Nk,i, with Mk,i (Nk,i) beingoperators onHS (HF ), respectively (this follows e.g. from theoperator Schmidt-decomposition [30]). Thus, we obtain

2LSF,kρSFL†SF,k −

L†SF,kLSF,k, ρSF

=

=∑

i,j

(

2Mk,iρSM†k,j ⊗Nk,iρFN

†k,j

−(M †k,jMk,iρS ⊗N †

k,jNk,iρF

+ρSM†k,jMk,i ⊗ ρFN

†k,jNk,i)

)

.

(11)

By tracing overHF , and using the cyclic property of the (full)trace, we obtain:

traceF[

2LSF,kρSFL†SF,k −

L†SF,kLSF,k, ρSF

]

=∑

i,j

bkij

(

2Mk,iρSM†k,j − (M †

k,jMk,iρS + ρSM†k,jMk,i)

)

,

where bkij = trace(ρFN†k,jNk,i). Since we require (8) to be

independentof ρF , LSF,k must take one of the following twoforms:

LSF,k =∑

j

Mk,j ⊗ IF = LS,k ⊗ IF ,

LSF,k =∑

j

IS ⊗Nk,j = IS ⊗ LF,k.

6

A similar reasoning shows thatH is constrained to

H = HS ⊗ IF + IS ⊗HF .

Thus, the generator has the form declared in (9).

As a byproduct, Theorem 2’s proof gives explicit necessaryand sufficient conditions for the blocks ofH andLk to ensureinvariance. We collect them in the following:

Corollary 1 (Markovian invariance):Assume thatHI =(HS ⊗HF ) ⊕HR, and letH, Lk be the Hamiltonian andthe error generators of a Markovian QDS as in (2). ThenHS

supports an invariant subsystem iff∀ k:

Lk =

(

LS,k ⊗ LF,k LP,k0 LR,k

)

,

iHP − 1

2

∑

k

(L†S,k ⊗ L†

F,k)LP,k = 0, (12)

HSF = HS ⊗ IF + IS ⊗HF ,

where for eachk eitherLS,k = IS or LF,k = IF (or both).

If we require parametric model robustness, as specified inDefinition 3, additional constraints emerge from imposing that(7) and (8) hold irrespective of parameter uncertainties. Theresults may be summarized as follows:

Theorem 3 (Robust Markovian invariance):AssumeHI =(HS⊗HF )⊕HR. (i) Let Fk be the error generators in (1).ThenHS supports anA-robust invariant subsystemiff ∀j, k,

Fk =

(

FS,k ⊗ FF,k FP,k0 FR,k

)

,

F †P,k(FS,j ⊗ FF,j) = 0, (13)

HSF = HS ⊗ IF + IS ⊗HF , HP = 0, (14)

where eitherFS,k = IS for everyk, or FF,k = IF for everyk,or both. (ii) If Lk are the noise operators in (2), thenHNS

supports aγ-robust invariant subsystemiff ∀ k,

Lk =

(

LS,k ⊗ LF,k LP,k0 LR,k

)

,

L†P,k(LS,k ⊗ LF,k) = 0, (15)

HSF = HS ⊗ IF + IS ⊗HF , HP = 0, (16)

and for eachk, eitherLS,k = IS or LF,k = IF (or both).

Proof. Consider case (i). Given Theorem 2, conditions (7)-(8) must hold irrespective ofA = (ajk) in (1). The lower-diagonal block is now

∑

jk ajkFQ,jρSFF†Q,k. Considering

only the diagonal termsj = k, we are again led to requireFQ,k = 0 for everyk. Thus, condition (10) must be replacedby iHP −

∑

jk ajkF†SF,kFP,j = 0, which is true for every

A = (ajk) iff HP andF †P,k(FS,j⊗FF,j) vanish independently,

as stated in (13), (14).To complete the proof, as before we writeFSF,k =

∑

k,lMk,l ⊗Nk,l, with Mk,l (Nk,l) operators onHNS (HF ),respectively. Then we have:

2FSF,jρSFF†SF,k −

F †SF,kFSF,j , ρSF

=

=∑

l,m

(

2Mj,lρNSM†k,m ⊗Nj,lρFN

†k,m

− (M †k,mMj,lρNS ⊗N †

k,mNj,lρF

+ ρNSM†k,mMj,l ⊗ ρFN

†k,mNj,l)

)

.

By tracing overHF and using cyclicity yields∑

l,m

bkjlm

(

2Mj,lρNSM†k,m− (M †

k,mMj,lρNS + ρNSM†k,mMj,l)

)

,

where bkjlm = trace(ρFN†k,mNj,l). Since we wish (8) to be

independent ofρF , it follows that for everyk, FSF,k takesone of two possible forms:

FSF,k =∑

m

INS ⊗Nm,k = INS ⊗ FF,k,

FSF,k =∑

l

Mk,l ⊗ IF = FNS,k ⊗ IF ,

which establishes the conditions forA-robustness.For γ-robustness, it suffices to specialize the above proof to

diagonalA, that is, to consider onlyj = k. This let us identifytheLk ’s with theFk ’s, and the result follows.

C. Noiseless subsystems

As remarked after Definition 5, givenρ ∈ D(HI), thereduced projected state:

ρS = traceF (ΠSF ρΠ†SF ), (17)

need not be a valid reduced state ofS if ρR 6= 0, sinceits trace might be less than one. Still, to the purposes ofdefining noiseless behavior, there is no reason for requiringthat the evolution has to be confined toIS(HI), as long as theinformation encoded in the intended subsystem is preserved,i.e. it undergoes unitary evolution. This motivates aweakerdefinition of initialization:

Definition 7 (Reduced state initialization):The systemIwith stateρ is initialized in HSF with reduced stateρS ∈D(HS) if the blocks ofρ satisfy:

(i′) traceF (ρSF ) = ρS ;(ii) ρP = 0, ρR = 0.

Remark:The above definition is equivalent to state initial-ization as in given in Definition 5 if we restrict topure statesin HS , but it allows for entangled states otherwise.

Definition 8 (Noiselessness):Let I evolve under TPCPmaps. A subsystemS is a NS for the evolution if for everyinitial stateρ(0) of I initialized inS with reduced stateρS(0):

ρNS(t) = U(t)ρS(0)U(t)†, t ≥ 0, (18)

whereU(t) is a unitary operator onHS , independent of theinitial state on HI . If HF ≃ C, the NSS reduces to a DFS.

Following [11], we shall say thatperfect NS initializationoccurs when (i′) and (ii) are obeyed, and call subsystemS imperfectly initialized whenever (i′) or (ii) is violated.Physically, Definition 8 requires the state componentρNS

7

carried byHS to evolve unitarily, independently of the rest.The following proposition establishes how, in fact, a perfeclyinitialized NS is a special case of an invariant subsystem:

Proposition 1: Let HI = (HNS ⊗HF ) ⊕HR. ThenHNS

supports a NS for some given TPCP dynamics iff for allinitial condition ρ ∈ INS(HI), with ρNS ∈ D(HNS), ρF ∈D(HF ), the evolved state ofI obeys

ρ(t) =

(

U(t)ρNSU†(t) ⊗ T F

t (ρF ) 00 0

)

, t ≥ 0, (19)

whereU(t) is a unitary operator onHNS andT Ft are TPCP

maps onHF alone.Proof. AssumeHNS to support an NS and to be initializedwith reduced stateρNS , where it could beρNSF (0) 6=ρNS(0) ⊗ ρF (0). Let ρNSF (0) =

∑

k Sk ⊗ Fk, with Sk andFk operators onHS andHF , respectively. Thus,ρNS(0) =∑

k trace(Fk)Sk. Notice that, by linearity:

traceF (ΠSF ρ(t)Π†SF ) = traceF

[

∑

k

U(t)SkU†(t) ⊗ T F

t (Fk)]

=∑

k

U(t)SkU†(t) trace

(

T Ft (Fk)

)

= U(t)ρNS(0)U †(t).

Thus, the condition is sufficient.To prove the other implication, notice that if the evolution

of ρNS is unitary, it preserves the trace. By the properties ofpartial trace and by (17), it then follows that trace(ρNS(t)) =trace(ρNSF (t)). Therefore,ρR = 0 must vanish at all times inorder to ensure TP-evolution in theSF -block and, similarly,ρP = 0 in order to guarantee positivity of the whole state7.Then for everyt ≥ 0 the evolution must take the form:

ρ(t) =

(

T NSFt (ρNSF ) 0

0 0

)

,

where T NSFt is a TPCP map onHNS ⊗ HF . Now use

the Kraus representation theorem and the operator-Schmidtdecomposition, by employing a basis forB(HS), sayMj,such that,∀ ρNS ∈ D(HNS), ρF ∈ D(HF ),

T NSFt (ρNS ⊗ ρF ) =

∑

klm

MlρNSM†m ⊗Nk,lρFN

†k,m.

Thus,

traceF [ΠNSF ρ(t)Π†NSF ] = T S

t (ρNS) =∑

lm

αlmMlρNSM†m,

whereαlm = trace(

∑

kN†k,mNk,lρF

)

is a positive matrix.

By exploiting the fact thatMj is a basis, and decomposing∑

kN†k,mNk,l in Hermitian and skew-Hermitian parts, one

can see that a necessary and sufficient condition in order forT St to be independent ofρF is that

∑

kN†k,mNk,l = αmlIF

for every j, k. By imposing that traceF [ΠSFρ(t)Π†SF ] =

U(t)ρS(0)U †(t), (αjk) must have rank one, thusαjk = αjαkfor someαj. If we additionally choose the operator basisMj so thatM1 = U(t), then α11 is the only non-zeroentry, in particular,

∑

kN†k,lNk,l = 0 for every l 6= 1. This

7This follows from the fact that ifρ = (ρij ) ≥ 0, then |ρij | ≤ √ρiiρjj .

impliesNk,l = 0 for every l 6= 1, thus yielding to the desiredconclusion:

T SFt (ρS ⊗ ρF ) = U(t)ρSU(t)† ⊗

∑

k

NkρFN†k .

On one hand, as a consequence of the above Proposition,if reduced state initialization is assumed (as in Definition7),the factorHNS supports an NS only if it is invariant. On theother hand, under the stronger condition of initializationofDefinition 5, if HNS is invariant and unitarily evolving, thenit supports a NS. Accordingly, most of the results concerningNSs may be derived as a specialization of conditions forinvariance. Remarkably, this also implies that in the particularcase of a NS, the invariance property is robust with respectto the initialization in theNSF -block, that is, condition(i) may be effectively relaxed to (i′). This is not true forgeneral invariant subsystems. Explicit characterizations of theMarkovian noiseless property may then be established assummarized in the rest of this Section.

Corollary 2 (Markovian NS):Let HI = (HNS ⊗ HF ) ⊕HR. Then HNS supports a NS under Markovian evolutionon HI iff for every initial stateρ ∈ INS(HI), with ρNS ∈D(HNS), ρF ∈ D(HF ), and∀t ≥ 0:

d

dtρ(t) =

(

LNSF (ρNSF (t)) 00 0

)

, (20)

traceF [LNSF (ρNSF (t))] = −i[HNS, ρNS(t)], (21)

whereLNSF andLNS are QDS generators onHNS ⊗ HF

andHNS , respectively.Proof. Given Proposition 1 and Theorem 2, we need to ensurethat the evolution inHNS is unitary. That is, theNSF -blockmust be driven by an generator of the form−i[HNS, ρNS ]⊗ρF + ρNS ⊗LF (ρF ), which replaces (9) and ensures unitaryevolution on the NS-factor, while allowing for general non-unitary Markovian dynamics onHF . The proof of Theorem 2applies, with the noise operators constrained to have the form

LNSF,k =∑

j

IS ⊗Nk,j = IS ⊗ LF,k.

Accordingly, the necessary and sufficient conditions on thematrix blocks ofH andLk for NS-behavior are modified to:

Corollary 3: AssumeHI = (HNS ⊗ HF ) ⊕ HR, and letH , Lk be the Hamiltonian and the error generators of aMarkovian QDS as in (2). ThenHNS supports a NS iff∀k:

Lk =

(

INS ⊗ LF,k LP,k0 LR,k

)

,

iHP − 1

2

∑

k

(INS ⊗ L†F,k)LP,k = 0, (22)

HNSF = HNS ⊗ IF + INS ⊗HF .

While derivations differ, Corollary 3 provides the same NS-characterization of Theorem 5 in [11]. Beside more directlytying to the CP context, our approach shows how the NSnotion may emerge as a specialization of the conditions forinvariance. Following the same lines as in Section II-B, wenext proceed to a general result forA- andγ-robust NSs, whichcompletes the partial conditions proposed in [8]:

8

Corollary 4 (Robust Markovian NS):Assume thatHI =(HNS ⊗HF ) ⊕ HR. (i) Let Fk be the error generators in(1). ThenHNS supports anA-robust NS iff∀j, k:

Fk =

(

INS ⊗ FF,k FP,k0 FR,k

)

,

F †P,k(INS ⊗ FF,j) = 0, (23)

HNSF = HNS ⊗ IF + INS ⊗HF , HP = 0. (24)

(ii) If Lk are the error generators in (2), thenHNS supportsa γ-robust NS iff∀k:

Lk =

(

INS ⊗ LF,k LP,k0 LR,k

)

, ∀k, (25)

L†P,k(INS ⊗ LF,k) = 0, (26)

HNSF = HNS ⊗ IF + INS ⊗HF , HP = 0. (27)

Proof. Given Theorem 3, it suffices to ensure that evolution inHNS be unitary (as in Corollary 2), for everyA. This is true iffHP = 0 andF †

P,k(INS⊗FF,j) = 0 independently. From (17),FSF,k = IS ⊗ FF,k must hold for everyk. The specializationto a γ-robust NS follows from similar observations.

Clearly, anA-robust NS may exist only if theFk do notgenerate the wholeB(HI), that is, we are restricting to a setof possible noise generators as remarked in Section I-D. Forapplications, it may be useful to further specialize the resultto the case of aγ-robust DFS, for whichHF is trivial:

Corollary 5 (γ-robust DFS): AssumeHI = HDFS ⊕HR.Let Lk be the error generators in (2). ThenHDFS is aγ-robust DFS iff∀k:

HP = 0, Lk =

(

ckIDFS LP,k0 LR,k

)

, (28)

with LP,k = 0 if ck 6= 0.Proof. In Corollary 4 above, setHF = C.

An alternative formulation of Corollary 5 also holds:Proposition 2 (Alternativeγ-robust DFS condition):

HDFS = span|φDFSj 〉 is a γ-robust Markovian DFSsubspace ofHI iff ∀j, k the following conditions hold:

HP = 0, Lk|φDFSj 〉 = ck|φDFSj 〉,

L†kLk|φDFSj 〉 = |ck|2|φDFSj 〉.

Proof. Assume that∀k, Lk|φDFSj 〉 = ck|φDFSj 〉. Then it mustbe

Lk =

(

ckIDFS LP,k0 LR,k

)

.

Since

L†kLk =

(

ckc∗kIDFS c∗kLP,k

ckL†P,k L†

P,kLP,k + L†R,kLR,k

)

,

thenL†kLk|φDFSj 〉 = |ck|2|φDFSj 〉 is true iff the conditions of

the proof of Corollary 5 are obeyed.

Remark:According to the above Corollary,γ-robust DFS-states are joint (right) eigenvectors ofboth each LindbladoperatorLk and each “jump” operatorL†

kLk. Such charac-terizations ofA- and γ-robust DFSs link our analysis to the

definition presented in [31]. In fact, the definition of DFSproperty invoked there imposes more constraints than ourDefinition 7: it requires the Hamiltonian to preserve the DFSindependentlyfrom the dissipative component of the generator.This may be regarded as a yet different kind of robustness,weaker than bothγ- andA-robustness investigated here.

D. Imperfect initialization

Thus far, we have addressed model robustness of theinvariance and the noiselessness properties. As the relevantsubsystem dynamics also depends on the initial state, it isnatural to ask how critical initialization is to the purposes ofensuring a desired behavior. This motivates the introductionof a different robustness notion:

Definition 9: Assume thatHI = (HS⊗HF )⊕HR, and letI undergo QDS dynamics with a generator of the form (1) or(2). Let a given propertyPS hold for the dynamical modelinitialized in HS , according to Definition 5. IfPS holds forρS = traceF (ΠSF ρΠSF ) for every ρ ∈ D(HI), then PS issaid to beρ-robust.

This approach leads to the same conditions for imperfect-initialization Markovian NSs (initialization-free NS) obtainedin [11]. By Definition 9, consideringHI = (HNS⊗HF )⊕HR,a subsystem supported onHNS is a ρ-robust NS if for everyt ≥ 0, ∀ ρ(0) with reduced stateρNS(0) ∈ D(HNS):

ρNS(t) = traceF (ΠNSFρ(t)Π†NSF ) = U(t)ρNS(0)U †(t),

whereU(t) is unitary onHNS . This means that the dynamicsof theNSF -block cannot be influenced byρP , ρR. Notice thatProposition 1 has already clarified how imperfect initializationin the NSF block does not affect the unitary character ofthe evolution of the reduced state of the NS. Computing thegeneratorL(ρ) by blocks, by inspection one sees that for allk it must beLP,k = 0, LQ,k = 0, henceHP = 0. By theproof of Theorem 4, it also follows thatLNSF,k = I ⊗Mk.Thus, the main difference with respect to the perfect NS-initialization case is the constraintLP,k = 0, which decouplesthe evolution of theNSF -block from the rest. Notice that thisalso automatically ensuresγ-robustness in our framework.

E. Attractive subsystems

The analysis developed so far indicates how initializationrequirements may be relaxed by requiringρ-robustness. How-ever, this implies in general tighter conditions on the noiseoperators, which may be demanding to ensure and leave lessroom for Hamiltonian compensation of the noise action (seeSection III-B). In order to both address situations where suchextra constraints are not met, as well as a question which isinteresting on its own, we explore conditions for a NS to benot only invariant, but also attractive:

Definition 10 (Attractive Subsystem):Assume thatHI =(HS ⊗HF )⊕HR. ThenHS supports an attractive subsystemwith respect to a familyTtt≥0 of TPCP maps if∀ρ ∈ D(HI)the following condition is asymptotically obeyed:

limt→∞

(

Tt(ρ) −(

ρS(t) ⊗ ρF (t) 00 0

))

= 0, (29)

9

where ρS(t) = traceF [ΠSFTt(ρ)Π†SF ], ρF (t) =

traceS [ΠSF Tt(ρ)Π†SF ].

An attractive subsystem may be thought of as a subsystemthat “self-initializes” in the long-time limit, by somehowreabsorbing initialization errors. Although such a desirablebehavior only emerges asymptotically, for QDSs one can seethat convergence is exponential, as long as some eigenvaluesof L have strictly negative real part.

We begin with a negative result which, in particular, showshow the initialization-free and attractive characterizations aremutually exclusive.

Proposition 3: AssumeHI = (HNS⊗HF )⊕HR,HR 6= 0,and letH, Lk be the Hamiltonian and the error generatorsas in (2), respectively. LetHNS support a NS. IfLP,k =

L†Q,k = 0 for everyk, thenHNS is not attractive.

Proof. Consider a block-diagonal state of the form:

ρB =

(

ρSF 00 ρR

)

.

It is straightforward to see that the generator has the form

d

dtρB =

(

LSF (ρSF ) 00 LR(ρR)

)

,

which preserves the trace ofρR. Thus, if ρR 6= 0, ρB doesnot satisfy Eq. (29).

Remark: The conditions of the above Proposition areobeyed, in particular, for NSs in the presence of purelyHermi-tian noise operators, that is,Lk = L†

k, ∀k. As a consequence,attractivity is never possible for this kind ofunital Markoviannoise, as defined by the requirement of preserving the fullymixed state. Still, even if the conditionLP,k = L†

Q,k = 0condition holds, attractive subsystems may exist in the pure-factor case, whereHR = 0. Sufficient conditions are providedby the following:

Proposition 4: AssumeHI = HS ⊗HF (HR = 0), and letHS be invariant under a QDS of the form

L = LS ⊗ IF + IS ⊗ LF .If LF (·) has a unique attractive stateρF , thenHS is attractive.Proof. Let ρ be a generic state onHI = HS ⊗ HF . ρ mayalways be expressed (recall the proof of Theorem 2) in theform ρ =

∑

i Pi ⊗ Qi. Without loss of generality, we maytake theQi to be Hermitian. If this is not the case, decomposeQi = QHi + iQAi into Hermitian and skew-Hermitian parts, sothat Pi ⊗ Qi = Pi ⊗ QHi + (iPi) ⊗ QAi . Each of theQH,Ai

may be further decomposed in a positive and a negative part,which one may normalize to unit trace. To do so, considerthe spectral representation of eachQi, separate the positiveand negative eigenvalues, and partition the matrix in a sumof two Qi = Q+

i + Q−i . NormalizeQ+

i , Q−i to trace1, −1,

respectively, and reabsorb the normalization coefficientsandthe minus sign, inPi. Thus, we can writeρ =

∑

i Pi ⊗ ρF,i,and ρNS =

∑

i Pi. By applying the above generator to sucha state and using the linearity of the evolution,

limt→∞

ρt =∑

i

limt→∞

(

T tS (Pi) ⊗ T t

F (ρF,i))

=(

∑

i

limt→∞

T tS (Pi)

)

⊗ ρF = limt→∞

T tS (ρNS) ⊗ ρF ,

thus the desired conclusion follows.In the mathematical-physics literature, QDSs with a unique

attractive stationary state are calledrelaxing, and have beenmostly studied in the ’70 in the context of rigorous approachesto quantum thermodynamics. Useful linear-algebraic condi-tions for determining whether a generatorLF (·) is relaxingare presented in [32], [33]. The uniqueness of the stationarystate turns out to benecessarywhen considering NSs:

Proposition 5: AssumeHI = HS ⊗ HF , (HR = 0), andlet HS support a NS under a QDS of the form

L = LS ⊗ IF + IS ⊗ LF .

If LF (·) admits at least two invariant states, thenHS is notattractive.Proof. It suffices to construct a state of the form:

ρ = pρ(1)S ⊗ ρ

(1)F + (1 − p)ρ

(2)S ⊗ ρ

(2)F ,

where ρ(1)S , ρ

(2)S are orthogonal pure states onHS , and

ρ(1)F , ρ

(2)F are the two invariant states forLF , and0 < p < 1.

Again, by using the linearity of the evolution,

ρ(t)= [pTS(ρ(1)S ) ⊗ TF (ρ

(1)F )+(1 − p)TS(ρ

(2)S ) ⊗ TF (ρ

(2)F )]

=pUS(t)ρ(1)S U †

S(t) ⊗ ρ(1)F +(1 − p)US(t)ρ

(2)S U †

S(t) ⊗ ρ(2)F ,

so it follows that the state does not factorize for anyt ≥ 0.

Note that proper initialization plays a more critical rolein the DFS- than in the NS-context, as a consequence ofProposition 1. IfHF 6= C and the initial state is not factorizedon theNSF -block, the reduced state onHNS still evolvesunitarily, provided that the generator satisfies the conditionsgiven in Corollary 3. Practically, this means that there is noactual need to require a (factorized) subsystem-initialized state,as long as a bounded error on the NS-component can betolerated. For an imperfectly-initialized DFS, unitary evolutionon the intended block can only occur ifLP,k ≡ 0, makingattractivity a compelling option if the latter does not hold.Accordingly, our main emphasis is on attractivity in the DFS-case, which may be guaranteed by invoking a specializationof the Krasowskii-LaSalle invariance principle (see e.g. [34]).

Theorem 4 (Attractive Subspace):Let HI = HS ⊕ HR

(HF = C), and letHS support an invariant subspace underL.Assume that there exists a continuously differentiable functionV (ρ) ≥ 0 on D(HI), such thatV (ρ) ≤ 0 on imperfectlyinitialized states inD(HI) \ IS(HI). Let

W = ρ ∈ D(HI)| V (ρ) = 0,Z = ρ ∈ D(HI)| trace[ΠRL(ρ)] = 0,

whereΠR is the orthogonal projector onHR. If W ∩ Z ⊆IS(HI), thenHS is attractive.Proof. ConsiderV1(ρ) = trace(ΠRρ) + trace(ΠRρ)V (ρ). It iszero iff ρR = 0, i.e. for perfectly initialized states. Computing

10

L(ρ), we get for theR block:

L(ρ)R = − i[HR, ρR] + iρ†PHP − iH†PρP

− 1

2

∑

k

(

2LR,kρRL†R,k − L†

R,kLR,k, ρR

+ LQ,k, ρSFL†Q,k − 2L†

P,kLP,k, ρR+ 2LR,kρ

†PL

†Q,k + 2LQ,kρPL

†R,k

− ρ†P (L†SF,kLP,k − L†

Q,kLR,k)

−(L†P,kLSF,k + L†

R,kLQ,k)ρP

)

.

Therefore,

trace[ΠRL(ρ)] = −1

2trace

(

∑

k

L†P,kLP,k, ρR

)

= −trace(

∑

k

L†P,kLP,kρR

)

,(30)

is always negative or zero. Hence

V1(ρ) = trace(ΠRL(ρ))(1 + V (ρ)) + trace(ΠRρ)V (ρ) ≤ 0,

for everyD(HI), and it is zero only inW ∩Z ∪ IS(HI). IfW∩Z ⊆ IS(HI), by applying Krasowskii-LaSalle invariancetheorem, we conclude.

The following result immediately follows:Corollary 6: Assume thatHI = HS ⊕HR (HF = C), and

let HS support an invariant subspace underL. Assume that∑

k

L†P,kLP,k > 0, (31)

where> means strictly positive. ThenHS is attractive.Proof. It suffices to note that (31) guarantees that (30) in theproof of the Theorem above is zero iffρR = 0. The conclusionfollows by taking aV (ρ) constant and positive onD(HI).

Remark:From considerations on the rank of the l.h.s. of (31)and then× r dimension ofLP,k, the condition of Corollary6 may be obeyed only ifn ≥ r, i.e. dim(HS) ≥ dim(HR).An application of this result will be given in Section III-B.

Proposition 4 and Theorem 4 (or Corollary 6), may becombined in order to obtainsufficient conditionsfor attractivityin the general NS case.

Proposition 6: Let HI = (HS⊗HF )⊕HR, and letHS bean invariant subsystem underL, with

LF (·) = traceS(ΠSFL(·)ΠSF ).

Assume that there exist a continuously differentiable func-tionalV (ρ) ≥ 0 onD(HI), such thatV (ρ) ≤ 0 on imperfectlyinitialized states inD(HI) \ ISF (HI). Let W ,Z be definedas in Theorem 4. IfLF (·) is relaxing andW∩Z ⊆ ISF (HI),thenHS is attractive.Proof. From Corollary 1,LF (·) is a Markovian generator onHF , thus it makes sense to require that it is relaxing. LetρFbe its unique attractive state. Observe that from Theorem 4,the state will asymptotically have support only onHS ⊗HF .Let ρSF (t) = ΠSF ρ(t)Π

†SF : By defining τ = t/2, s = t/2,

and by invoking the Markovian property, along with the aboveobservation, invariance, and Proposition 4 we may write:

limt→∞

ρSF (t) = limτ,s→∞

ΠSF T τ (ρ(s))Π†SF

= limτ→∞

T τS ⊗ T τ

F ( lims→∞

ρSF (s))

= limτ→∞

T τS ( lim

s→∞ρNS(s)) ⊗ ρF .

The last equality comes from the fact thatlims→∞ ρSF (s)is certainly bounded, and can be always written in the formlims→∞

∑

k PS,k(s) ⊗ ρF,k by choosing a basis ofB(HF )of density operatorsρF,k and by following ideas similar tothose in the proof of Proposition 4.

III. C ONTROL APPLICATIONS

A. Quantum trajectories and Markovian output feedback

Building on pioneering work by Belavkin [35], it has beenlong acknowledged for a diverse class of controlled quantumsystem that intercepting and feeding back the informationleaking out of the system allow to better accomplish a numberof desired control tasks (see [36], [37], [12], [38], [39], [40]for representative contributions). This requires the ability toboth effectively monitor the environment and control the targetevolution through time-dependent Hamiltonian perturbationswhich depend upon the measurement record. We begin by re-calling some well-established continuous-measurement modelsfor the conditioned dynamics, as originally developed in thequantum-optics setting by Wiseman and Milburn [36], [37].

The basic setting is a measurement scheme which mimicksoptical homo-dyne detection for field-quadrature measure-ments, whereby the target system (e.g. an atomic cloud trappedin an optical cavity) is indirectly monitored via measurementsof the outgoing laser field quadrature [36], [41]. Let themeasurement record be denoted byYt (e.g. a photo-current inthe above setting), and let(Ω, E , P ) a (classical) probabilityspace with an associatedWt, t ∈ R+ standardR-valuedWiener process. The homo-dyne detection measurement recordmay then be written as the output of a stochastic dynamicalsystem of the form:

dYt = η tr(Mρt + ρtM†)dt+

√ηdWt, (32)

whereρt ≡ ρ(t) is the system state at timet, M is the mea-surement operator determining the system-probe interaction,and0 ≤ η ≤ 1 quantifies the efficiency of the measurement.

The real-time knowledge of the photo-current providesadditional information on the dynamics, leading to astochasticmaster equation(SME) for the conditional evolution:

dρt = (F(H, ρt) + D(M,ρt))dt+ G(M,ρt)dWt (33)

=(

− i[H, ρt] + η(MρtM† − 1

2M †M, ρt)

)

dt

+√η[

Mρt + ρtM† − tr(Mρt + ρtM

†)ρt]

dWt.

Here,F is the Hamiltonian generator, whereasD(M,ρt) andG(M,ρt) are the Lindblad noise channel and the “diffusion”contribution due to the weak measurement ofM . Given aninitial condition ρ0, the solutionρt exists, is confined toD(HI), and is adapted to the filtration induced byWt, t ∈

11

R+(see e.g. [42], [43]). As the SME (33) is an Ito stochasticdifferential equation, to obtain the average evolution generatorit suffices to drop the martingale partG(M,ρt)dWt. Thus, the“unconditional” evolution obeys a deterministic QDS genera-tor of the form (2). Notice that the diffusion term plays therole of the innovation part of a nonlinear Kalman-Bucy filter,and that the conditional state follows continuous trajectories,whereby the name ofquantum trajectoriesapproach in thequantum-optics literature [44].

In what follows, we assume perfect detection, that is,η = 1,unless otherwise specified (see III-E and IV). In [37], it hasbeen argued that the photo-current can be instantaneouslyfed back to further modify the dynamics, still maintainingthe Markovian character of the evolution. This motivatesconsidering aHamiltonian feedback superoperatorof the form

d

dtρft =

d

dtYtF(F, ρt), F = F †, (34)

which is, however, ill-defined given the stochastic nature ofYt. In order to obtain a feedback Markovian evolution, (34)has been interpreted as an “implicit” Stratonovich stochasticdifferential equation [37]. Its Ito equivalent form is:

dρft = F(F, ρt)dYt +1

2F2(F, ρt)dt. (35)

Thus, one can consider the infinitesimal evolution resultingfrom the feedback followed by the measurement actionρt +dρt = T F

dt T Mdt(ρt), where:

T Fdt(ρt) = ρt + F(F, ρt)dYt +

1

2F2(F, ρt)dt,

T Mdt(ρt) = ρt + (F(H, ρt) + D(M,ρt))dt+ G(M,ρt)dWt.

Substituting the definitions and using Ito’s rule, it yields:

dρt =(

F(H, ρt) + D(M,ρt) + F(F,Mρt + ρtM†)

+1

2F2(F, ρt)

)

dt+(

G(M,ρt) + F(F, ρt))

dWt.(36)

Dropping again the martingale part and rearranging the re-maining terms leads to the Wiseman-MilburnMarkovian Feed-back Master equation(FME) [36], [37]:

d

dtρt = F

(

H+1

2(FM +M †F ), ρt

)

+D(M− iF, ρt). (37)

In the following sections, we will tackle state-stabilization andNS-synthesis problems for controlled Markovian dynamicsdescribed by FMEs.

B. Control assumptions

The feedback state-stabilization problem for Markoviandynamics has been extensively studied for the single-qubitcase [45], [46]. In particular, conditions for achieving a puresteady state have been identified in [47]. In the existingliterature, however, the standard approach to design a Marko-vian feedback strategy is to specify both the measurementand feedback operatorsM,F, and to treat the measurementstrength and the feedback gain as the relevant control parame-ters accordingly. Here we will pretend to have more freedom,considering, for a fixed measurement operatorM , bothF andH as tunable control Hamiltonians.

Definition 11 (CHC): A controlled FME of the form (37)supportscomplete Hamiltonian control(CHC) if (i) arbitraryfeedback HamiltoniansF ∈ H(HI) may be enacted; (ii)arbitraryconstantcontrol perturbationsHc ∈ H(HI) may beadded to the free HamiltonianH .

As we shall see, this leads to both new insights andconstructive control protocols for systems where the noiseoperator is a generalized angular momentum-type observable,for generic finite-dimensional systems. While assuming thatthe implementation of arbitrary coherent Hamiltonians posesno problem is in line with standard universality construc-tions for open quantum systems [48], [49], from a physicalstandpoint the CHC assumption is certainly demanding andshould, as such, be carefully scrutinized on a case by casebasis. In particular, constraints on the allowed Hamiltoniancontributions relative to the Lindblad dissipator may emerge,notably in so-called weak-coupling limit derivations of Marko-vian models [10]. A first, interesting consequence of assumingCHC emerges directly from the following observation:

Lemma 2:The Markovian generator

d

dtρt = −i[H, ρt] +

∑

k

D(Lk, ρt) (38)

is equivalent to

d

dtρt = −i[H +Hc, ρt] +

∑

k

D(Lk, ρt), (39)

where for allk, andck ∈ C:

Lk = Lk + ckI, Hc = −i∑

k

(c∗kLk − ckL†k). (40)

Proof. Considerk = 1:

D(L, ρ) = LρL† − 1

2L†L, ρ

= LρL† − 1

2L†L, ρ + [c∗L− cL†, ρ]

= −i[(ic∗L− icL†), ρ] + D(L, ρ). (41)

Notice thati(c∗L− cL†) is Hermitian. Fork > 1, it sufficesto add up the correction parts in (41) for differentk’s, and usethe linearity of the commutator.

Note that for HermitianL and realc, Hc = 0. In general,by exploiting CHC, we may vary the trace of the Lindbladoperators through transformations of the form (40)8, and, ifneeded or useful, appropriately counteract the HamiltoniancorrectionHc with a constantcontrol Hamiltonian. This mayallow to stabilize subsystems that are not invariant for theuncontrolled equation,without directly modifying the non-unitary part. In addition to this, restricting to such open-loop,constant control Hamiltonians avoids additional difficultieswhich are related to reconcile the Markovian limit with generictime-varying perturbations [10], [50].

Example 3.Consider a generator of the form:

d

dtρ(t) = −i[σz, ρ(t)] +

(

Lρ(t)L† − 1

2L†L, ρ(t)

)

,

8Interestingly, this corresponds, in physical terms, to a variation of the localoscillator in the optical homo-dyne detection setting [37].

12

where L = σz + σ+. Suppose that the task is to makeρd = diag(1, 0) invariant. SinceHP = 0, LS = 1, Lp = 1,invariance is not ensured by the uncontrolled dynamics. Usingthe above result, it suffices to apply a constant HamiltonianHc = −i(L−L†) = σy . The desired state turns out to be alsoattractive, see Proposition 7 below.

C. Pure-state preparation with Markovian feedback: Two-level systems

As mentioned, the problem of stabilizing an arbitrary purestate for a two-level atom is discussed in detail in [45] forH = ασy , M =

√γσ−, andF = λσy , with the Hamiltonian,

measurement, and feedback strength parameters (α , γ, λ,respectively), treated as the control design parameters. In termsof a standard Bloch sphere parametrization of the state set,ρ = 1/2I2 + 1/2(xσx + yσy + zσz), with 0 ≤ |(x, y, z)| ≤ 1,it is proved there that any pure state in thexz plane can bemade invariant and attractive, with the only exception of thestates on the equator of the sphere. The possibility of relaxingthe perfect detection assumption is also addressed.

Our perspective differs not only because we mainly focuson continuous measurement ofHermitian spin observables,but more importantly because we start from identifying whatconstraints must be imposed to a Lindblad equation for a two-dimensional system as in (2) for ensuring that one of thesystem’s pure states is an attractive equilibrium. Withoutlossof generality, let such a state be written asρd = diag(1, 0),and write, accordingly,

Lk =

(

lk,S lk,Plk,Q lk,R

)

, H =

(

hS hPh∗P hR

)

.

Proposition 7: The pure stateρd = diag(1, 0) is a glob-ally attractive, invariant state for a two-dimensional quantumsystem evolving according to (2) iff:

ihP − 1

2

∑

k

l∗k,Slk,P = 0, (42)

lk,Q = 0, ∀k, (43)

and there exists ak such thatlk,P 6= 0.Proof. Eqs. (42)-(43) imply the invariance conditions ofCorollary 1, henceρd is stable. For the choiceLDk =diag(lS,k, lR,k), every diagonal state would clearly be station-ary (directly from the form ofL(·), or by Proposition 3).Hence it must belP,k 6= 0 for somek. To prove thatρd is theonly attractive point for (2), it suffices to note thatlk,P 6= 0is the two-dimensional version of the sufficient condition forattraction given in Corollary 6.

Remark: Observe that, even ifρd is stable for the un-conditional,averageddynamics over the trajectories of (33),because it is pure it cannot be obtained as a convex combi-nation of other states. Thus,ρd must be theasymptotic limitof each trajectory with probability one. In fact, any invariantset different fromρd alone could not have it as average. Weprovide next a characterization of the stabilizable manifold.

Proposition 8: Assume CHC. For any measurement oper-atorM , there exist a feedback HamiltonianF and a Hamil-tonian compensationHc able to stabilize an arbitrary desired

pure stateρd for the FME (37) iff

[ρd, (M +M †)] 6= 0. (44)

Proof. Consider as before a basis whereρd = diag(1, 0), andletMH andMA denote the Hermitian and anti-Hermitian partof M , respectively. By (44),MH cannot be diagonal in thechosen basis. In fact, assumeMH to be diagonal, then, byProposition 7,MS − F must be brought to diagonal form toensure invariance ofρd. Hence, by the same result, it followsthat ρd cannot be made attractive. However, ifMH is notdiagonal, we can always find an appropriateF in order toget an upper diagonalL = MH + i(MS − F ), andH ′ =H + (FM + M †F )/2. To conclude, it suffices to devise acompensation HamiltonianHc such that the conditioni(H ′ +Hc)P − 1

2 l∗SlP = 0 is satisfied.

The above proof naturally suggests a constructive algorithmfor designing the feedback and correction Hamiltonian re-quired to stabilize the desired state. From our analysis, wealsorecover the results of [45] recalled before. For example, thestates that are never stabilizable within the control assumptionsof [45] are the ones commuting with the Hermitian part ofM = σ+, that is,MH = σx. On thexz plane in the Bloch’srepresentation, the latter correspond precisely to the equatorialpoints. The following example serves to illustrate the basicideas we shall extend to thed-level case.

Example 4:The simplest choice to obtain an attractive gen-erator is to engineer a dissipative part determined byL =σ+ = ( 0 1

0 0 ) . Let H = n0I2 + nxσx + nyσy + nzσz , withn0, nx, ny, nz ∈ R. Consider e.g.M = 1

2σx andF = − 12σy.

Notice that in this case12 (FM +M †F ) = 0, thusH ′ = H .Substituting in the FME (37), one clearly obtain the desiredresult, provided thatHc = −nxσx − nyσy.

The spin measurement models considered above have beenalready exploited for stabilization problems (see e.g. [43]),although in the context of strategies necessitating a real-time estimate of the state [12] – so-calledBayesian feedbacktechniques in the physics literature [46]. Assume that it ispossible to continuously monitor a single observable, e.g.σx inthe above example. Since the choice of the reference frame forthe spin axis is conventional, by suitably adjusting the relativeorientation of the measurement apparatus and the sample, itisthen in principle possible to prepare and stabilize any desiredpure state with the same control strategy.

D. Extension to multi-level systems

The previous two-dimensional results naturally extend togenericd-level systems. This will also provide an exampleof an attractive state, which does not satisfy the sufficientcondition of Proposition 6. Let the pure state to be FME-stabilized be written asρd = diag(1, 0, . . . , 0). Under CHC,we may without loss of generality assumeH to be diagonalin this basis.

Proposition 9: The pure stateρd is a globally attractive, in-variant state for the FME (37) conditioned over the continuous

13

measurement of the operator:

M =1

2

0 m1 0

m1 0. . .

. . .. . . md−1

0 md−1 0

,

and a Markovian feedback Hamiltonian:

F =i

2

0 m1 0

−m1 0. . .

. . .. . . md−1

0 −md−1 0

,

with mi 6= 0, for i = 1, . . . , (d− 1).Proof. First, observe thatL = (li,j) = M − iF, the onlyelements different from zero areli,i+1 = mi. By writing ρ =(ρij)i,j=1,...,d, one gets:

D(L, ρ) = 4

m∗1m1ρ22 · · · m∗

d−1m1ρ2d 0...

. . ....

m∗1md−1ρd2 · · · m∗

d−1md−1ρdd 00 · · · 0 0

−

2

0 |m1|2ρ12 · · · |md−1|2ρ1d

|m1|2ρ21 2|m1|2ρ22 · · · (|m1|2+|md−1|2)ρ2d

......

. . ....

|md−1|2ρd1 (|m1|2+|md−1|2)ρd2 · · · 2|md−1|2ρdd

.

(45)DefineH = diag[0, 1, . . . , (d−1)]. Thus, the functionVd(ρ) =tr(Hρ) is a valid global Lyapunov function for the target stateρd in D(HI) [34]. Indeed,V (ρ) ≥ 0 andV (ρ) = 0 iff ρ = ρd.ComputingVd(ρ) = tr(HD(L, ρ)) using (45), one obtains:

Vd(ρ) =4

[ d−1∑

i=2

(i− 1)|mii|2ρi+1,i+1 −d−1∑

j=1

j|mjj |2ρj+1,j+1

]

− 4d−1∑

i=1

|mii|2ρi+1,i+1.

We conclude by applying Lyapunov stability theorem. Since|mii|2 > 0 andρi,i ≥ 0, the derivative is always non-positiveand can be zero iffρi,i = 0, i = 2, . . . , d, i.e. ρ = ρd.

The matrixH in the above proof is essentially a Hamilto-nian with energy gaps renormalized to one, whereasF andM play a role analogous to theσy andσx observables of thed = 2 case. Notice that their form is not different from thatof standard, higher-dimensional spin observables.

E. On Markovian-feedback state preparation

The feedback strategies we consider preserve the Markoviancharacter of the open-system evolution. Thus, in a sense,the corresponding control problem may then be seen as a“Markovian environment design” problem [45] – implying thatwe may write the QDS generatorindependentlyof the systemstate. This, along with the remark in Section III-C, ensures

the desired convergence feature foreachinitial state (in otherwords, robustness with respect to errors in the initial stateestimation is guaranteed).

The main advantage with respect to other feedback-designstrategies is represented by the potential ease in practicalimplementations, since virtually no signal-processing stageis required in the realization of the feedback loop. Thisshould be contrasted with Bayesian feedback strategies [43],[46], whereby an updated state estimate has to be obtainedthrough real-time integration of (33), and used to tailor a state-dependent feedback action on the underlying evolution. Sucha task becomes rapidly prohibitive as the dimensionality ofthetarget system grows.

As a potential disadvantage, howewer, the Markovianoutput-feedback we use requires strong control capabilities andperfect detection. On one hand, an infinite bandwidth is neededto feed back the measurement output in real time. On the otherhand, both the feedback and measurement parameters have tobe accurately tuned, along with both the system Hamiltonianand its control compensation, if needed. Nevertheless, forstatestabilization problems, one may assess the role of the perfect-detection hypothesis and the possibility to relax it. Ifη < 1,the FME is modified as follows [41]:

d

dtρt = F

(

H + 1/2(FM +M †F ), ρt

)

+ D(M − iF, ρt) + εD(F, ρt), (46)

where we definedε = (1 − η)/η.In [10], generators of the form (1)-(2) are rewritten in a

convenient way by choosing a suitable Hermitian basis inB(Hi) ≈ Cd×d. In fact, endowingCd×d with the innerproduct 〈X,Y 〉 := trace(X†Y ) (Hilbert-Schmidt), we mayuse a basis where the first element is1√

dId, and complete it

with a orthonormal set of Hermitian, traceless operators. Thiscan always be done for finited, for example by employingthe naturald-dimensional extension of the Pauli matrices [10],[49]. In such a basis, all density operators are representedbyd2-dimensional vectorsρ = (ρ0, ρ1, . . . , ρd2−1)

T , where thefirst componentρ0, relative to 1√

dId, is invariant and equal to

1√d

for TP-dynamics. Letρv = (ρ1, . . . , ρd2−1)T . Hence, any

QDS generatorL(ρ) must take the form:

d

dtρ =

(

0 0C D

)(

1/√d

ρv

)

. (47)

Assume that the dynamics has a unique attractive stateρ(0).ThusD must be invertible and we obtain:

ρ(0) =1√d

(

1−D−1C

)

.

Consider now a small perturbation of the generator depend-ing on the continuous parameterε, with 1 − δ < η < 1, andδ sufficiently small so that(D+ εD′) remains invertible. Thegenerator becomes:

dρ

dt=

[(

0 0C D

)

+ ε

(

0 0C′ D′

)](

1/√d

ρv

)

, (48)

and the new attractive, unique equilibrium state is:

ρ(ε) =1√d

(

1−(D + εD′)−1(C + εC′)

)

.

14

Becauseρ(ε) is a continuous function ofε, we are guaranteedthat for a sufficiently high detection efficiency the perturbedattractive state will be arbitrarily close to the desired one intrace norm. Therefore, if we relax our control task to a statepreparation problem with sufficiently high fidelity, this maybe accomplished with a sufficiently high detection efficiency,yet strictly less than 1.

Insofar as noise suppression is the intended task, we see howmonitoring a perfectly dissipative environment may be usefulfor control process. However, the ability to suppress the noisesource via feedback is necessarily limited by the form of (37).It is apparent that the feedback action is only able tomodifythe skew-Hermitian part of the noise operatorL, and evenin cases where this may suffice, (nearly) perfect detection isneeded. Nonetheless, Markovian feedback may prove to beextremely interesting when only partial noise suppressionisconsidered, for instance in order to achieve longer coherencetimes. In this spirit, we turn to analyze how our techniquesmay be employed to synthesize DFSs or NSs in the Markovianlimit where open-loop control is not an option.

F. DFS synthesis with Markovian feedback

As remarked, the feedback loop can only modify the skew-Hermitian part of the measurement operatorM , which im-poses strict constraints on the non-unitary generators that areable to be synthesized. A natural question is to what extent wemight be able to generate DFSs or NSs by closed-loop control.In the single-observable feedback setting under examination,the DFS notion turns out to be appropriate.

Theorem 5:Let p = d/2, if d is even,p = (d + 1)/2,if d is odd, and assume CHC for (37). Then a DFS of (atleast) dimensionp can be generated by Markovian feedbackfor every measurement operatorM .Proof.A DFS for (37) can be generated, under CHC hypotesis,iff there exist a choice of basis such that:

L = M − iF =

(

cIDFS P0 R

)

=

(

Re(c)IDFS P/2

P †/2 RH

)

+ i

(

Im(c)IDFS −iP/2iP †/2 RA

)

,

where we have decomposedL into Hermitian (H) and skew-Hermitian (A) parts as before. The skew-Hermitian part canbe arbitrarily modified under CHC hypothesis, by choosingthe appropriateF . Thus, it remains to prove that there existsa basis whereMH has the form ofLH above, and the blockproportional to the identity is (at least)p-dimensional.MH

is indeed Hermitian, and can be diagonalized. LetDH be thediagonal matrix of the (real) eigenvalues ofMH . Then we arelooking for aU and a Hilbert space decomposition such that:(

Real(c)IDFS P/2P †/2 RH

)

= UDHU †

=

(

UDFS UPUQ UR

)(

DHDFS 00 DH

R

)

(

U †DFS U †

Q

U †P U †

R

)

Hence, we want to findp orthonormal vectors to stack in(UDFS UP )† such that:(

UDFS UP)

(

DHDFS 00 DH

R

)

(

U †DFS

U †P

)

= c′IDFS ,

with c′ ∈ R. This is equivalent to ask that the compressionof MH to some subspace is equivalent to a scalar matrix.Let u1, u2 be normalized eigenvectors ofMH of eigenvaluesd1, d2, respectively. Then by takingu3 = αu1 + βu2, suchthat |α|2 + |β|2 = 1, we can construct a vector that is aneigenvector ofMH restricted to the one-dimensional subspacegenerated byu3 itself. u3 has then eigenvalue|α|2d1+ |β|2d2,a convex combination of the former eigenvaluesd1, d2. Takethe diagonal elements ofDH in descending order, and pairthe first with the last, the second to last but one, and so on. Ifd is odd, the eigenvalue in the middle will remain unpaired.Thus, from these pairs and the respective eigenvectors we canthen obtainp new vectors as illustrated above, which are alleigenvectors ofMH restricted to their linear span, with thesame eigenvalue (in general,c′ will be a convex combinationof the two middle eigenvalues. For oddd, it will be the middleeigenvalue). From the resulting linear span, we can then obtainthe desired DFS, by choosing a feedback HamiltonianF suchthat theQ block ofM − iF is zero.

Remarks:Note that the above proof provides a constructivealgorithm for generating the DFS. The result is potentiallyuseful in light of ongoing efforts for efficiently finding quan-tum information-preserving structures [4], [23], [51]. Both theCHC assumption and the ability to perfectly monitor the noisechannel are demanding for present experimental capabilities,however the promise of a new technique to generate DFSsmay prompt further developments in this direction. In par-allel, further study is needed in order to weaken the aboverequirements, as it is likely to be possible in specific contexts.

From another perspective, it is intriguing to compare The-orem 5 to the analysis of continuous-time quantum errorcorrection presented in [52]. In that case, the target system isassumed to be a quantum register, and under the assumptionthat independent errorsare occurring on different qubits, aMarkovian feedback strategy is identified such that the closed-loop behavior implements continuous-time quantum error cor-rection for a so-called “stabilizer code” [2]. This may be seenas equivalent to the generation of a DFS able, in particular,to encode(n − 1) logical qubits in a2n dimensional space.Even if our analysis follows different lines, the setting for ourDFS-generation problem is similar, and our result consistentlyleads to the same encoding efficiency ford = 2n – providedwe can compound the noise effect in a single measurementoperator. Interestingly, no assumption is made at this stage onthe structure of the Hilbert space, neither do we impose anyconstraint on the form of the “error”, that is, the measurementor noise operator in our case.

Before concluding, we present a simple example of gen-eration of attractive DFS for coupled qubits via Markovianfeedback, which further illustrates some of our results.

Example 4.ConsiderH = Hq ⊗Hq, Hq = span|0〉, |1〉,and a controlled closed-loop evolution driven by (37), withH diagonal andM = σx ⊗ σx. Assume that we are able tomonitorM and actuate the feedback HamiltonianF = −σy⊗σx. ThenL = M − iF = 2σ− ⊗σx. If we considerHDFS =span|0〉⊗|0〉, |0〉⊗|1〉, we obtain block-decomposition of theform (4), whereL is such thatLDFS = 0, LQ = 0, LP = σx.Hence, by Corollary 3HDFS is a DFS and by Corollary 6,

15

we can prove it is attractive. Notice that the noise operatorM = σx ⊗ σx does admit noiseless subspaces, e.g.H′

DFS =span(|+〉 ⊗ |+〉, |+〉 ⊗ |−〉), with |±〉 = 1/

√2(|0〉 ± |1〉), but

by Proposition 3, none of them can be attractive. This showshow the feedback-generated noiseless structure may offer anadvantage with respect to existing ones.

IV. D ISCUSSION ANDCONCLUSION

We have revisited some fundamental concepts about Marko-vian dynamics for quantum systems and restated the notionof a general quantum subsystem inlinear-algebraic terms.A system-theoretic characterization of invariant and noiselesssubsystems for Markovian quantum dynamical systems hasbeen provided, with special attention on key model-robustnessissues relevant for practical applications. In particular, wehave showed that, in order to avoid situations where onlyfine-tuning of the Hamiltonian and dissipative terms wouldensure invariance of a given subsystem, condition (12) mustbe replaced by the independent conditions (13) and (14). Thisinduces similar modifications on NS invariance conditions.This part of our work both puts on more rigorous mathematicalgrounds and completes the existing literature on the subject.

When imperfect subsystem initialization is considered, theconditions to be imposed on the Markovian generator becomemore demanding, which motivates the new notion of asymptot-ically stableattractivesubsystem. Interestingly, the possibilitythat the noise action may be useful or necessary for remainingin the intended subsystem has been independently consideredbefore for specific QIP settings [53]. However, a formalgeneralization of the idea and an analytical study were stillmissing. Our linear-algebraic approach, along with Lyapunov’stechniques, provides explicit stabilization results which havebeen illustrated in simple yet paradigmatic examples.

In the second part of the work, the conditions identifiedfor subsystem invariance and attractivity serve as the startingpoint for designing output-feedback Markovian strategiesableto actively achieve the intended quantum stabilization. Wehave completely characterized the state-stabilization problemfor two-level systems described by FMEs of the form (37).While the analysis assumed perfect detection efficiency, aperturbative argument indicated how unique attractive statesdepend in a continuous fashion on the model parameters. Oursuggested DFS generation strategy is also crucially dependenton the perfect detection condition, as otherwise the feedback-corrected FME would take the form (46), implying an ad-ditional error component due to finite efficiency. Nonetheless,the norm of this portion of the noise generator is bounded, andtends to zero forη → 1. Therefore, even if the non-unitarydynamics cannot be counteracted exactly, the time-scale oftheresidual noise action may still be significantly reduced in thedesired subspaces for sufficiently high detection efficiency.

The Markovian, output-feedback techniques we employhave also been compared, in terms of robustness features,with the Bayesian-feedback approach. A key advantage ofthe Markovian approach lies in its intrinsic design simplicity,which makes it possible to avoid a costy real-time integrationof the feedback-controlled master equation and thereby pavesthe way to implementation in higher-dimensional systems.

Further work is needed in order to establish completely gen-eral Markovian feedback stabilization results, includingfinitedetection efficiency andmulti-channel continuous monitoring.From an algorithmic standpoint, it also appears worthwhileto investigate the potential of the linear-algebraic approachin problems related to finding NSs for a given generator,either under perfect or imperfect knowledge. Among the mostinteresting perspectives, additional investigation is certainlyrequired to establish the full power of Hamiltonian controlandMarkovian feedback in generating NS structures. This maypoint to new venues for producing protected realizations ofquantum information for physical systems whose dynamics isdescribed by quantum Markovian semigroups.

V. ACKNOWLEDGMENTS

It is a pleasure to thank Angelo Carollo for early discussionson DFS conditions under Markovian dynamics, and ClaudioAltafini, Augusto Ferrante, Hideo Mabuchi, Michele Pavon,and Ramon van Handel for valuable feedback. F. T. acknowl-edges hospitality from the Physics and Astronomy Departmentat Dartmouth College, under the support of the Constance andWalter Burke Special Projects Fund in Quantum InformationScience.

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