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Decoherence-Free Subspaces and Subsystems Daniel A. Lidar1 and K. Birgitta Whaley2 1 Chemical Physics Theory Group, Chemistry Department,University of Toronto, 80 St.George St., Toronto, Ontario M5S 3H6, Canada 2 Department of Chemistry and KennethPitzer Centerfor Theoretical Chemistry, University of California, Berkeley, California 94720, USA 3 0 0 Abstract. Decoherence is the phenomenon of non-unitary dynamics that arises as 2 a consequence of coupling between a system and its environment. It has important n harmfulimplicationsforquantuminformation processing, andvarioussolutionstothe a problem have been proposed. Here we provide a detailed a review of the theory of J decoherence-free subspaces and subsystems, focusing on their usefulness for preserva- 9 tion of quantum information. 1 v 1 Introduction 2 3 Recent results indicating that quantum information processing (QIP) is inher- 0 1 ently more powerful than its classical counterpart[1–3] have motivated a resur- 0 genceofinterestintheproblemofdecoherence[4].Decoherenceisaconsequence 3 of the inevitable coupling of any quantum system to its environment (or bath), 0 causinginformationlossfromthe systemto this environment.Itwasrecognized / h earlyonthatdecoherenceposesaseriousobstacletophysicalrealizationofquan- p tum information processors [5,6]. Here we define decoherence as non-unitary - t dynamics that is a consequence of system-environment coupling. This includes n (but is not limited to) both dissipative and dephasing contributions, tradition- a u allyknownasT1andT2 processes,respectively.Dissipationreferstoprocessesin q which the populations of the quantum states are modified by interactions with : theenvironment,whiledephasingreferstoprocessesthatrandomizetherelative v i phases of the quantum states. Both are caused by entanglement of the system X with environmental degrees of freedom, leading to non-unitary system dynam- r ics [7]. For QIP and other forms of quantum control, any such interaction that a degrades the unitary nature of the quantum evolution is undesirable, since it causes loss of coherence of the quantum states and hence an inevitable decay of their interference and entanglement. Interference is crucial for coherent control schemes [8,9], while entanglement is believed to be as important an ingredient of the quantum computational speed-up [10]. In fact, a sufficiently decohered quantum computer can be efficiently simulated on a classical computer [11]. In this review we provide a detailed introduction to the theory of decoherence-free subspaces and subsystems (DFSs), which have been conceived as one of the possible solutions to decoherence in QIP. Space limitations prevent us from dis- cussingindetailthe interestingproblemofperformingcomputation usingDFSs. Insteadwe focus hereonthe use ofDFSs as awayto preserve delicate quantum information. 2 Lidar & Whaley 2 Historical Background Environment-induced decoherence is a very extensively studied phenomenon in the quantum theory of measurement, and of the transition from quantum to classical behavior. Zurek has shown that environmental superselection results in the establishment of certain special states that show redundancies in their correlationswiththeenvironment,andthatmayconsequentlybeessentiallyun- perturbed by this [12].These states,referredto as “pointer states”, are defined by a “predictability sieve” [13]. While analysis of measurements are compli- cated by the presence of a measurement apparatus in addition to the system and environment, the notion of a special set of states that are defined by some underlying symmetry in the global physical description is a common element. Another noteworthy early study employing symmetries in order to reduce the coupling of subsets of system states to the environment is Alicki’s work on lim- itedthermalization[14].Identifyingthepowerofsuchsymmetriesinthephysical Hamiltonian, and systematizing the way to find such symmetries has been one of the major innovative features of the recent development of DFS theory and its applications for quantum computation. Early discussions of the effects of decoherence on quantum computation fo- cusedonputtingconditionsonthestrengthofcouplingofthesystem-environment coupling and on the duration of quantum gates [5,6]. The searchfor systematic ways to bypass decoherence in the context of QIP, based on identification of states that might be immune to certain decohering interactions, started with observations of Palma, Suominen, and Ekert [15] in a study of the effects of pure dephasing, that two qubits possessing identical interactions with the en- vironment do not decohere. Palma et al. used the term “subdecoherence” to describe this phenomenon, suggested using the corresponding states to form a “noiseless” encoding into logical qubits, and noted that the set of states ro- bust against dephasing will depend on the specific form of qubit-environment coupling. This model was subsequently studied using a different method by Duan and Guo [16], with similar conclusions and a change of terminology to “coherence preserving states”. The idea of pairing qubits as a means for pre- serving coherence was further generalized by Duan and Guo in [17], where it was shown that both collective dephasing and dissipation could be prevented. However this assumed knowledge of the system-environment coupling strength. These early studies were subsequently cast into a general mathematical frame- workforDFSsofmoregeneralsystem/environmentinteractionsbyZanardiand Rasetti, first for the spin-boson model in [18], where the important “collective decoherence” model was introduced (where several qubits couple identically to the same environment, while undergoing both dephasing and dissipation), then for general Hamiltonians [19]. Their elegant algebraic analysis established the importance of identifying the dynamical symmetries in the system-environment interaction, and provided the first generalformal condition for decoherence-free (DF) states,thatdidnotrequireknowledgeofthe system-environmentcoupling strength. In the work of Zanardi and Rasetti these are referred to as “error avoidingcodes”. Severalpapers focusing on collective dissipationappearedsub- Decoherence-FreeSubspaces and Subsystems 3 sequently [20–22], as well as applications to encoding information in quantum dots [23,24]. Zanardi [25] and independently Lidar, Chuang, and Whaley [26] showed that DF states could also be derived from very general considerations of Markovian master equations. Lidar et al. introduced the term “decoherence- freesubspace”,analyzedtheirrobustnessagainstperturbations,andpointedout thatthe absenceofdecoherenceforDF statescanbe spoiledbyevolutionunder the system Hamiltonian, identifying a second major requirement for viable use of the DF states for either quantum memory or quantum computation [26]. A completely general condition for the existence of DF states was subsequently provided in terms of the Kraus operator sum representation (OSR) by Lidar, Bacon and Whaley [27]. All these studies share essentially the same canonical example of system-environment symmetry: A qubit-permutation symmetry in the system-environment coupling, that gives rise to collective dephasing, dissi- pation, or decoherence. The other main example of a symmetry giving rise to large DFSs was provided by Lidar et al. in [28]. This is a much weaker form of spatial symmetry than permutation-invariance, termed “multiple-qubit errors”, which we describe in detail in Section 6.4. Several papers reported various generalizations of DFSs, e.g., by extending DFSstoquantumgroups[29],usingtheriggedHilbertspaceformalism[30],and deriving DFSs from a scattering S-matrix approach[31].However,the next ma- jor stepforwardin generalizingthe DFS conceptwastakenbyKnill, Laflamme, and Viola [32], who introduced the notion of a “noiseless subsystem”. Whereas previous DFS work had characterizedDF states as singlets (one dimensional ir- reducible representations)of the algebragenerating the dynamicalsymmetry in the system-bath interaction,the work by Knill et al. showedthat higher dimen- sional irreducible representations can support DF states as well. An important consequence was the reduction of the number of qubits needed to construct a DFS under collective decoherence from four to three. This was noted indepen- dentlybyDeFilippo[33]andbyYangandGea-Banacloche[34].Thegeneraliza- tion from subspaces to subsystems has provided a powerful and elegant tool on which the full theory of universal, fault tolerant quantum computation on DF states has now been established [35,36]. It has also provided a basis for unify- ing essentially all known decoherence-suppression/avoidance strategies [37,38]. In the remainder of this chapter we shall follow our usual convention to refer to both subsystems and subspaces interchangeably with the acronym DFS, the distinction being made explicit when necessary. FollowingtheinitialstudiesestablishingtheconditionsforDFSs,Bacon,Lidar, and Whaley made a thorough investigation of the robustness of DF states to symmetry breaking perturbations [39]. These authors also showedthat the pas- sive error correction (“error avoidance”) properties of a DFS can be combined with the active error correction protocols provided by quantum error correc- tionbyconcatenationofaDFS inside aquantumerrorcorrectingcode(QECC) [2,40],resultinginanencodingcapableofprotectingagainstbothcollectiveand independenterrors[27].AcombinedDFS-QECCmethodwasshowntobeneces- sary in order to be enable universal,fault tolerant quantum computation in the 4 Lidar & Whaley multiple-qubit errors model [41]. Interestingly, the DFS for collective decoher- ence offers a natural energy barrier against other decoherence processes, a phe- nomenon termed “supercoherence” by Bacon, Brown and Whaley [42]. Several more recentstudies, by Alber et al. and Khodjasteh and Lidar, have considered other hybrid DFS-QECC schemes, focusing in particular on protection against spontaneousemission[43–45].DFSs havealsobeencombinedwiththe quantum ZenoeffectbyBeigeet al.[46,47].Mostrecently,acombinationofDFSsandthe method of dynamical decoupling [48] was shown to offer a complete alternative to QECC [49,50]. These explicit theoretical demonstrations that, firstly, DF states exist and can provide stable quantum memory, and second, that fault tolerant universal computation canbe performedon states encoded into a DFS, while initially su- perficially surprising to many, have generated several experimental searches for verificationofDFSs.Thefirstexperimentalverificationcamewithademonstra- tionby Kwiatet al.ofa2-qubitDFSprotectingphotonstatesagainstcollective dephasing [51]. The same 2-qubit DFS was subsequently constructed and veri- fied to reduce decoherence in ion trap experiments by Kielpinski et al. [52]. In the latter experiment an atomic state of one ion was combined with that of a spectator ion to form a 2-ion DF state that was shown to be protected against dephasing deriving from long wavelength ambient magnetic field fluctuations. This experiment is significant in showing the potential of DFS states to protect fragile quantum state information against decoherence operative in current ex- perimental schemes. More recently, universal control on the same 2-qubit DFS forcollectivedephasinghasbeendemonstratedinthecontextofliquidstatenu- clear magnetic resonance by Fortunato et al. [53]. Nuclear magnetic resonance hasalsoledtothe firstexperimentaldemonstration,byViolaet al., ofa3-qubit DF subsystem, providing immunity against full collective decoherence deriving from a combination of collective spin flips and collective dephasing [54]. With these first experimental demonstrations, further experimental efforts towards implementation ofDFSs in active quantumcomputation, ormerely as quantum memoryencodingsto transportsinglequbits fromonepositiontoanotherwith- outincurringdephasing,seemsassured.Forexample,theuse ofaDFShasbeen identified as a major component in the construction of a scalable trapped ion quantum computer [50,55,56]. A common criticism of the theory of DFSs has been that the conditions re- quiredfora DFSto existareverystringentandthe assumptionsunderlying the theorymaybetoounrealistic.ItisimportanttoemphasizethattheDFSconcept was never meant to provide a full and independent solution to all decoherence problems. Instead, the central idea has been to make use of the dynamical sym- metries in the system-environment interaction first (if they exist), and then to consider the next level of protection against decoherence. The robustness prop- erties of DFSs ensure that this is a reasonable approach. In addition, while the experimental evidence to date [51–54,57,58], is a reason for cautious optimism, there have also been a number of theoretical studies showing that the condi- tions for DFSs may be created via the use of the dynamicaldecoupling method, Decoherence-FreeSubspaces and Subsystems 5 by symmetrizingthe system-bathinteraction[59–62].This holds true fora wide rangeofsystem-bathinteractionHamiltonians.Itseemsquiteplausiblethatsuch active“environmentengineering”methodswillbenecessaryforDFSstobecome truly comprehensive tools in the quest to protect fragile quantum information. In the remainder of this review we will now leave the historical perspective and provide instead a summary of the theory of DF subspaces and their gener- alizations, DF subsystems. We shall start, in Section 3 with a simple example of DFSs in physical systems that is then used as a basis for a rigorous analysis of decoherence and the conditions for lack of this in an open quantum system (Section 4). We then provide, in Section 5, a series of complementary charac- terizations of what a DFS is, using both exact and approximate formulations of open systems dynamics. A number of examples of DFSs in different physical systems follow in Section 6, some of them new. The later sections of the review deal with the generalization to DF Subsystems (Section 7), and the robustness of DFSs (Section 8). We conclude in Section 9. 3 A Simple Example of Decoherence-Free Subspaces: Collective Dephasing Let us begin by analyzing in detail the operation of the simplest DFS. This example, first analyzed by Palma et al. in [15] and generalized by Duan & Guo [16], will serve to illustrate what is meant by a DFS. Suppose that a system of K qubits (two-level systems) is coupled to a bath in a symmetric way, and undergoes a dephasing process. Namely, qubit j undergoes the transformation 0 0 1 eiφ 1 , (1) j j j j | i →| i | i → | i which puts a random phase φ between the basis states 0 and 1 (eigenstates | i | i of σ with respective eigenvalues +1 and 1). This can also be described by z − the matrix R (φ) = diag 1,eiφ acting on the 0 , 1 basis. We assume that z {| i | i} the phase has no space (j) dependence, i.e., the dephasing process is invariant (cid:0) (cid:1) under qubit permutations. This symmetry is an example of the more general situation known as “collective decoherence” . Since the errors can be expressed in terms of the single Pauli spin matrix σ of the two-level system, we refer z to this example as “weak collective decoherence” . The more general situation when errors involving all three Pauli matrices are present, i.e., dissipation and dephasing, is referred to as “strong collective decoherence” . Without encoding a qubit initially in an arbitrary pure state ψ = a0 +b1 will decohere. j j j | i | i | i Thiscanbeseenbycalculatingitsdensitymatrixasanaverageoverallpossible values of φ, ∞ ρ = R (φ)ψ ψ R (φ)p(φ)dφ, j z | ijh | z† Z−∞ wherep(φ)isaprobabilitydensity,andweassumetheinitialstateofallqubitsto beaproductstate.ForaGaussiandistribution,p(φ)=(4πα)−1/2exp( φ2/4α), − 6 Lidar & Whaley it is simple to check that a2 ab e α ρj = a|be| α ∗b2− . ∗ − (cid:18) | | (cid:19) The decayof the off-diagonalelements in the computationalbasisis a signature of decoherence. Let us now consider what happens in the two-qubit Hilbert space. The four basis states undergo the transformation 0 0 0 0 1 2 1 2 | i ⊗| i →| i ⊗| i 0 1 eiφ 0 1 1 2 1 2 | i ⊗| i → | i ⊗| i 1 0 eiφ 1 0 1 2 1 2 | i ⊗| i → | i ⊗| i 1 1 e2iφ 1 1 . 1 2 1 2 | i ⊗| i → | i ⊗| i i Observe that the basis states 0 1 and 1 0 acquire the same phase. 1 2 1 2 | i ⊗| i | i ⊗| i This suggests that a simple encoding trick can solve the problem. Let us define encoded states by 0 = 0 1 01 and 1 = 10 . Then the state L 1 2 L | i | i ⊗| i ≡ | i | i | i ψ =a0 +b1 evolves under the dephasing process as L L L | i | i | i ψ a0 eiφ 1 +beiφ 1 0 =eiφ ψ , L 1 2 1 2 L | i→ | i ⊗ | i | i ⊗| i | i and the overall phase thus acquired is clearly unimportant. This means that the 2-dimensional subspace DFS (0) = Span 01 , 10 of the 4-dimensional 2 {| i | i} Hilbertspaceoftwoqubitsisdecoherence-free.ThesubspacesDFS (2)=Span 00 2 {| i} and DFS ( 2) = Span 11 are also (trivially) DF, since they each acquire a 2 − {| i} global phase as well, 1 and e2iφ respectively. Since the phases acquired by the different subspaces differ, there is decoherence between the subspaces.1 ForK =3qubitsasimilarcalculationrevealsthatthesubspacesDFS (1)= 3 Span 001 , 010 , 100 and DFS ( 1) = Span 011 , 101 , 110 are DF, as 3 {| i | i | i} − {| i | i | i} wellthe(trivial)subspacesDFS (3)=Span 000 andDFS (3)=Span 111 . 3 3 {| i} {| i} More generally, let λ =number of 0s minus the number of 1s K ′ ′ inacomputationalbasisstate(i.e.,abitstring)overK qubits.Thenitiseasyto check that any subspace spanned by states with constantλ is DF, and can be K denoted DFS (λ ) in accordance with the notation above. The dimensions of K K these subspaces are given by the binomial coefficients: d dim[DFS (λ )] = K K K and they each encode log d qubits. ≡ λK 2 The encoding for the “collective phase damping” model discussed here has (cid:0) (cid:1) beentestedexperimentally.Thefirst-everexperimentalimplementationofDFSs used the DFS (0) subspace to protect against artifially induced decoherence in 2 1 This conclusion is actually somewhat too strict: for a non-equilibriumenvironment, superpositions of different DFSs can be coherent in the collective dephasing model [63]. Decoherence-FreeSubspaces and Subsystems 7 a linear optics setting [51]. The same encoding was subsequently used to allevi- ate the problem of external fluctuating magnetic fields in an ion trap quantum computing experiment[52],andfiguresprominentlyin theoreticalconstructions of encoded, universal quantum computation [49,50,64]. 4 Formal Treatment of Decoherence Let us now present a more formal treatment of decoherence. Consider a closed quantum system, composed of a system S of interest defined on a Hilbert space (e.g., a quantum computer) and a bath B. The full Hamiltonian is H H=H I +I H +H , (2) S B S B I ⊗ ⊗ where H , H and H are, respectively, the system, bath and system-bath S B I interaction Hamiltonians, and I is the identity operator. The evolution of the closed system is given by ρ (t) = Uρ (0)U , where ρ is the combined SB SB † SB system-bath density matrix and the unitary evolution operator is U=exp( iHt) (3) − (we set ¯h =1). Assuming initial decoupling between system and bath, the evo- lution of the closed system is given by: ρ (t) = U(t)[ρ (0) ρ (0)]U (t). SB S B † ⊗ Without loss of generality the interaction Hamiltonian can be written as H = S B , (4) I α α ⊗ α X where S and B are, respectively, system and bath operators. It is this cou- α α pling between system and bath that causes decoherence in the system, through entanglement with the bath. To see this more clearly it is useful to arrive at a descriptionofthesystemalonebyaveragingouttheuncontrollablebathdegrees offreedom,aprocedureformallyimplementedbyperformingapartialtraceover the bath: ρ(t)=Tr [U(t)ρ(0) ρ (0)U (t)]. B B † ⊗ The reduced density matrix ρ(t) now describes the system alone. By diagonal- izing the initial bath density matrix, ρ (0) = λ ν ν , and evaluating the B ν ν| ih | partial trace in the same basis one finds: P ρ(t) = µU(t) ρ(0) λν ν ν U†(t)µ h | ⊗ | ih |! | i µ ν X X = A ρ(0)A , (5) a †a a X where the “Kraus operators” are given by: A = λ µUν ; a=(µ,ν). (6) a ν h | | i p 8 Lidar & Whaley The expression (5) is known as the Operator Sum Representation (OSR) and can be derived from an axiomatic approach to quantum mechanics, without reference to Hamiltonians [65]. Since Trρ(t)=1 the Kraus operators satisfy the normalization constraint A†aAa =IS. (7) a X Because of this constraint it follows that when the sum in Eq. (5) includes only onetermthedynamicsisunitary.Thus a simple criterion for decoherence in the OSR is the presence of multiple independent terms in the sum in Eq. (5). While theOSRisaformallyexactdescriptionofthe dynamicsofthe system densitymatrix,itsutilityissomewhatlimitedbecausetheexplicitcalculationof theKrausoperatorsisequivalenttoafulldiagonalizationofthehigh-dimensional Hamiltonian H. Furthermore, the OSR is in a sense too strict. This is because as a closed-system formulation it incorporates the possibility that information whichis putinto the bathwillback-reactonthe systemandcausea recurrence. Suchinteractionswillalwaysoccurinthe closed-systemformulation(due to the the Hamiltonian being Hermitian). However, in many practical situations the likelihood of such an event is extremely small. Thus, for example, an excited atom which is in a “cold” bath will radiate a photon and decohere, but the bath will not in turn return the atom back to its excited state, except via the (extremely long) recurrence time of the emission process. In these situations a more appropriate way to describe the evolution of the system is via a quantum dynamical semigroup master equation [66,67]. By assuming that (i) the evolu- tion of the system density matrix is governed by a one-parameter semigroup (Markovian dynamics), (ii) the evolution is futher “completely positive” [67], and (iii) the system and bath density matrices are initially decoupled, Lindblad [66] has shown that the most general evolution of the system density matrix ρ (t) is governedby the master equation S dρ(t) = i[H˜ ,ρ(t)]+L [ρ(t)] S D dt − M 1 LD[ρ(t)]= 2 aαβ [Fα,ρ(t)F†β]+[Fαρ(t),F†β] (8) αX,β=1 (cid:16) (cid:17) whereH˜ =H +∆isthesystemHamiltonianH includingapossibleunitary S S S contributionfromthebath∆(“Lambshift”),theoperatorsF constituteaba- α sis for the M-dimensional space of all bounded operators acting on , and a αβ H are the elements of a positive semi-definite Hermitian matrix. The commutator involving H˜ is the ordinary,unitary, Heisenberg term. All the non-unitary,de- S cohering dynamics is accounted for by L , and this is one of the advantages of D the Lindblad equation: unlike the OSR, it clearly separates unitary from deco- heringdynamics.ForaderivationoftheLindbladequationfromtheOSR,using a coarse graining procedure, including an explicit calculation of the coefficients a and the Lamb shift ∆, see [68]. Note that the F can often be identified αβ α with the S of the interaction Hamiltonian in Eq. (4) [68]. α Decoherence-FreeSubspaces and Subsystems 9 5 The DFS Conditions With the above statement of the conditions for decoherence let us now show how to formally eliminate decoherence. It is convenient to do so by reference to the Hamiltonian, OSR, and semigroup formulations. This leads to a number of essentially equivalent formulations of the conditions for DF dynamics, whose utilityisdeterminedbytheapproachonewouldliketoemploytostudyaspecific problem. In addition, it is useful to give formulations which make contact with thetheoryofquantumerrorcorrectingcodes(QECC).First,letusgiveaformal definition of a DFS: Definition 1. AsystemwithHilbertspace issaidtohaveadecoherence-free subspace ˜ if the evolution inside ˜ isHpurely unitary. H H⊂H Note that because of the possibility of a bath-induced Lamb shift [∆ in Eq. (8)] this definition of a DFS does not entirely rule out adverse effects a bath may have on a system. Also, we are not excluding unitary errors that may be the result of inaccurateimplementation of quantum logic gates.Both of these problems, which in practice are inseparable, must be dealt with by other methods, such as concatenated codes [27]. 5.1 Hamiltonian Formulation Asremarkedabove,intermsoftheHamiltonianHofEq.(2),decoherenceisthe result of the entanglement between system and bath caused by the interaction term H . In other words, if H = 0 then system and bath are decoupled and I I evolveindependently andunitarily under their respectiveHamiltonians H and S H . Clearly, then, a sufficient condition for decoherence free (DF) dynamics B is that H = 0. However, since one cannot simply switch off the system-bath I interaction, in order to satisfy this condition it is necessary to look for special subspaces of the full system Hilbert space . As shown first by Zanardi and Rasetti [19], such a subspace is found by assHuming that there exists a set k˜ {| i} of eigenvectors of the S ’s with the property that: α S k˜ =c k˜ α, k˜ . (9) α α | i | i ∀ | i Note that these eigenvectorsaredegenerate, i.e., the eigenvalue c depends only α ontheindexαofthesystemoperators,butnotonthestateindexk.IfH leaves S the Hilbert subspace ˜ =Span[ k˜ ] invariant, and if we start within ˜, then H {| i} H the evolutionof the system will be DF. To show this we follow the derivation in [27]:Firstexpandtheinitialdensitymatricesofthesystemandthebathintheir respectivebases:ρ (0)= s ˜ı ˜j andρ (0)= b µ ν .UsingEq.(9), S ij ij| ih | B µν µν| ih | one can write the combined operationof the bath and interactionHamiltonians over ˜ as: P P H IS HB+HI =IS Hc IS [HB+ aαBα]. ⊗ ⊗ ≡ ⊗ α X 10 Lidar & Whaley This clearly commutes with H over ˜. Thus since neither H (by our own S S H stipulation) nor the combined HamiltonianH takes states out of the subspace: c U[˜ı µ ]=U ˜ı U µ , (10) S c | i⊗| i | i⊗ | i where US = exp( iHSt) and Uc = exp( iHct). Hence it is clear, given the − − initially decoupled state of the density matrix, that the evolution of the closed system will be: ρSB(t) = ijsijUS|˜ıih˜j|U†S ⊗ µνbµνUc|µihν|U†c. It follows using simple algebra that after tracing over the bath: ρ (t) = Tr [ρ (t)] = S B SB P P USρS(0)U†S, i.e., that the system evolves in a completely unitary fashion on H˜: under the condition of Eq. (9) the subspace is DF. As shown in Ref. [19] by performing a short-time expansion, Eq. (10) is also a necessary condition for a DFS. Let us summarize this: Theorem 1. Let the interaction between a system and a bath be given by the Hamiltonian of Eq. (2). If H leaves the Hilbert subspace ˜ = Span[ k˜ ] in- S variant, and if we start within ˜, then ˜ is a DFS if anHd only if it{|sait}isfies H H Eq. (9). TheconditionofEq.(9)isveryusefulforcheckingwhetheragiveninteraction supports a DFS: The operators S often form a Lie algebra, and the condition α (9) then translates into the problem of finding the one-dimensional irreducible representations (irreps) of this Lie algebra, a problem with a textbook solution [69]. We will consider this in detail through examples below. 5.2 Operator-Sum Representation Formulation Let ˜ be an N-dimensional DFS. As first observed in [27], in this case it fol- H lows immediately from Eqs. (10) and (6) that the Kraus operators all have the following representation(in the basis where the first N states span ˜): H g U˜ 0 A = a ; g =√ν µU ν . (11) a 0 A¯a a h | c| i (cid:18) (cid:19) Here A¯ is an arbitrary matrix that acts on ˜ ( = ˜ ˜ ) and may cause a ⊥ ⊥ decoherencethere;U˜ isrestrictedto ˜.ThissHimpleHcondHiti⊕onHcanbesummarized H as follows: Theorem 2. A subspace ˜ is a DFS if and only if all Kraus operators have H an identical unitary representation upon restriction to it, up to a multiplicative constant. An explicit calculation will help to illustrate this condition for a DFS. Con- sider the set of system states k˜ satisfying: {| i} A k˜ =g U˜ k˜ a, (12) a a | i | i ∀

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