FFate of symmetry protected coherence in open quantum system
Tian-Shu Deng and Lei Pan ∗ Institute for Advanced Study, Tsinghua University, Beijing,100084, China
We investigate the fate of coherence in the dynamical evolution of a symmetry protected quan-tum system. Under the formalism of system-plus-bath for open quantum system, the anti-unitarysymmetry exhibits significant difference from the unitary one in protecting initial coherence. Specif-ically, taking advantage of Lindblad master equation, we find that a pure state in the symmetryprotected degenerate subspace will decohere even though both the system Hamiltonian and system-environment interaction respect the same anti-unitary symmetry. In contrast, the coherence willpersist when the protecting symmetry is unitary. We provide an elaborate classification table toillustrate what kinds of symmetry combinations are able to preserve the coherence of initial state,which is confirmed by several concrete models in spin-3 / I. INTRODUCTION
Symmetry is one of the greatest unifying themes inmodern physics and plays a fundamental role in classi-fying quantum phase of matter. In condensed matterphysics, the existence of periodical table in topologicalfree fermions is a great example in which the anomalousquantum Hall effect, topological insulators and topolog-ical superconductors, and many other interesting topo-logical phenomena are unified into a single frameworkof classification theory.
Despite high degree of univer-sality on the classification theory, the previous researchesabout symmetry analysis and topology in condensed mat-ters mainly focused on the isolated systems or thosesystems coupled to non-Hermitian potentials .However, realistic quantum systems inevitably coupleto external degrees of freedom, which, in general, shouldbe described by open quantum systems. Recently, effortshave been made to study the classification problem on thesteady states in Lindblad equations . The key pointfor experimentally searching stable symmetry-protectedtopological states is maintaining the coherence of quan-tum state in presence of surroundings. Thus, a naturalquestion is whether these symmetry protected topologi-cal states can survive from the decoherence induced byenvironment. In particular, one of the central issues foropen quantum system is the decoherence dynamics con-cerning how the quantum coherence evolves and vanisheswhich is particularly important for the quantum informa-tion and quantum computation . A crucial question ishow to avoid decoherence which is unavoidable roadblockto quantum information processing.Therefore, diagnosing the stability of coherence ofsymmetry-protected quantum state has fundamental im-portance both in theory and practice. Symmetry anal-ysis plays an important role in this issue. According toWigner’s theorem , a symmetry transformation is ei-ther unitary or anti-unitary. Recent theoretical researchrevealed that the coherence of states underlying manysymmetry protected features, could be fragile when thesymmetry is anti-unitary, even though both the envi-ronment and system-environment interaction respect the same symmetry as the Hamiltonian of system . Butthese features are always robust as long as the protectedsymmetry is unitary. Strikingly, when applying this totopological systems protected by time-reversal symmetry,one concludes that the topological phases could be unsta-ble to the perturbation of environment . Understandingthese nonequilibrium dynamics is of fundamental interestand has potential application in quantum information.In this work, we investigate the fate of coherence indegenerate subspace protected by the unitary or anti-unitary symmetry. When the system is weakly cou-pled to the environment, the dissipative dynamics un-der the Born-Markovian approximation in this open sys-tem is governed by Lindblad master equation. Indeed,the anomalous decoherence or degeneracy breaking inanti-unitary symmetry has already arisen from the non-Hermitian linear response theory . At this level, dy-namical evolution of density matrix in degenerate sub-space is able to distinguish the difference between anti-unitary symmetry and unitary symmetry in maintainingcoherence . Besides, the decoherence is also related tothe number of coupling channels between system and en-vironment when the all coupling operators are Hermitianones . We will give a classification about the mainte-nance of coherence with different symmetry combinationsrespected by system and system-environment interaction.Furthermore, we take several representative examples inspin-3 / / a r X i v : . [ qu a n t - ph ] F e b II. GENERAL FORMALISM
In this section, we derive general formalism from mi-croscopical Hamiltonian to study the dynamical evolu-tion in the system protected by unitary or anti-unitarysymmetry. In order to investigate the robustness of co-herence in degenerate space, we consider a quantum sys-tem coupled to a Markov bath whose Hamiltonian readsˆ H T = ˆ H S + ˆ H B + ˆ H SB , (1)where ˆ H S , ˆ H B are Hamiltonians belonging to the systemand bath respectively, and ˆ H SB denotes the interactionbetween them. The coupling part ˆ H SB can be decom-posed as ˆ H SB = M (cid:88) j =1 ˆ A † j ⊗ ˆ B j + ˆ A j ⊗ ˆ B † j , (2)where ˆ A j , ˆ B j are operators belonging to system andbath respectively, and M denotes the number of couplingchannels.Let us focus on the situation that ˆ H S has related sym-metry in consideration, and then a natural question iswhether the symmetry protected feature such as degen-eracy is maintained or not. It is generally expected thatthe symmetry protected feature would be destroyed if thesystem-bath coupling ˆ H SB breaks the related symmetryand it survives as long as ˆ H SB respects the same sym-metry. This intuitive principle seems to be always true.However, recently, in their seminal work , McGinley andCooper found that it can fail when the respecting symme-try is anti-unitary even each part ( ˆ A j , ˆ B j and ˆ H B ) obeysthe same symmetry. This unexpected discovery is deeplyrooted in the Schur’s Lemma for anti-unitary group and immediately indicates the fragility of time-reversalsymmetry protected topological edge states . Here wewill investigate the fate of coherence in degenerate sub-space of the system. For concreteness, we consider thefollowing total Hamiltonianˆ H T = ˆ H S + M (cid:88) j =1 (cid:88) α g j,α ˆ O † j ˆ b α + g ∗ j,α ˆ O j ˆ b † α + (cid:88) α ω α ˆ b † α ˆ b α , (3)where the bath is considered as the reservoir of harmonicoscillators in thermal equilibrium and ˆ b α (ˆ b † α ) is annihi-lation (creation) operator of the bath for α mode withbosonic commutation relation (cid:104) ˆ b α , ˆ b † β (cid:105) = δ αβ and thebath is considered as the reservoir of harmonic oscilla-tors ˆ H B = (cid:80) α ω α ˆ b † α ˆ b α . Here we focus on the scenario inwhich the system Hamiltonian ˆ H S possesses the symme-try in question exhibiting degeneracy and the couplingˆ H SB and bath ˆ H B respect the same symmetry. That isto say the symmetry considered here is represented by agroup G and ˆ O , ˆ b α are all invariant under the symmetryoperations in G . Under this circumstance, a significant practical ques-tion is that whether symmetry protected properties suchas degeneracy in system are fundamentally stable againstto perturbations of the environment. To explore this, weconsider the one-channel case M = 1 for simplicity. Fol-lowing the standard procedure to integrate out the bathunder Markovian approximation (low temperature bathand constant noise spectrum) and Born approximation(up to g order), one can derive the Lindblad masterequation as d ˆ ρ ( t ) dt ≡ L ˆ ρ ( t )= − i (cid:104) ˆ H S , ˆ ρ ( t ) (cid:105) − γ (cid:110) ˆ ρ ( t ) , ˆ O † ˆ O (cid:111) + 2 γ ˆ O ˆ ρ ( t ) ˆ O † , (4)where L is Liouvillian superoperator and { ˆ A, ˆ B } ≡ ˆ A ˆ B + ˆ B ˆ A denotes the anti-commutator. This equationdominates the dynamics of density matrix. In this way,the symmetry protected properties in degenerate spaceare connected with the dynamics governed by Lindbladmaster equation. Here we focus on the von Neumannentropy in degenerate subspace to characterize decoher-ent process and the corresponding response of entropy isgiven by δS ( t ) = S ( t ) − S ( t ) (5)where S ( t ) is the unperturbed entropy with coherentevolution determined by ˆ H S . Specifically, we will studyvon Neumann entropy S v ( t ) = − Tr[ˆ ρ ( t ) log ˆ ρ ( t )] whichcan be obtained once the master equation is solved. Thedecoherence dynamics in degenerate subspace and en-tropy growth can be derived by non-Hermitian linear re-sponse theory. Moreover, the break of degeneracy can bereflected in matrix representation of Liouvillian L . Us-ing the Choi-Jamio(cid:32)lkowski isomorphism , Eq.(4) canbe mapped to the following equation ddt | ρ (cid:105) = ˆ L | ρ (cid:105) , (6)with vectorized density matrix | ρ (cid:105)| ρ (cid:105) = (cid:88) i,j ρ i,j | i (cid:105) ⊗ | j (cid:105) (7)where ρ i,j = (cid:104) i | ˆ ρ | j (cid:105) is matrix element of ˆ ρ . And ˆ L denotes the matrix representation of Liouvillian which iswritten asˆ L = − i (cid:16) ˆ H S ⊗ ˆ I − ˆ I ⊗ ˆ H T S (cid:17) + γ (cid:20) O ⊗ ˆ O ∗ − ˆ O † ˆ O ⊗ ˆ I − ˆ I ⊗ (cid:16) ˆ O † ˆ O (cid:17) T (cid:21) . (8)The Liouvillian superoperator shares the symmetry asso-ciated with ˆ O which determines the symmetry of Eq.(4).The fate of coherence in degenerate subspace depends onwhether ˆ L is proportional to identity matrix in degen-erate subspace. III. COHERENCE ANALYSIS FOR DIFFERENTSYMMETRY COMBINATIONS
In this section, we classify the coherent dynamics indegenerate subspace regarding different combinations ofsymmetries respected by ˆ H S and the ˆ O . Supposing thesystem respects unitary or anti-unitary symmetry char-acterized by group G . This means the matrix represen-tation of ˆ H S satisfies [ ˆ H S , ˆ U G ] = 0 for any group ele-ment ˆ U G ∈ G . If | ψ (cid:105) is one of eigenstates of ˆ H S , i.e.,ˆ H S | ψ (cid:105) = E | ψ (cid:105) , then ˆ U G | ψ (cid:105) would also be the eigenstateof ˆ H S . Thus { ˆ U G | ψ (cid:105)} spans a degenerate subspace whenˆ U G ∈ G satisfying ˆ U G | ψ (cid:105) (cid:54) = | ψ (cid:105) exists, which is also irre-ducible representation subspace of G . We call this sym-metry protected degeneracy. A notable example in half-odd integer spin system is the Kramers’ degeneracy when G represents time reversal symmetry group.In order to describe the coherent dynamics in degen-erate space, we assume that the { U G | ψ (cid:105)} contains twoorthogonal basis which are denoted by | φ + (cid:105) and | φ − (cid:105) cor-responding to two-fold degenerate subspace. The initialstate is prepared to be a pure state ˆ ρ (0) = | ψ (0) (cid:105)(cid:104) ψ (0) | with | ψ (0) (cid:105) = α | φ + (cid:105) + β | φ − (cid:105) and then the dynamics ofdensity matrix will be dominated by Lindblad masterequation (4). We focus on the density matrix in the sub-space ˆ ρ G ( t ) = ˆΠ G ˆ ρ ( t ) ˆΠ G , where the ground state projec-tive operator is defined by ˆΠ G = | φ + (cid:105)(cid:104) φ + | + | φ − (cid:105)(cid:104) φ − | .For a general total Hamiltonian, the bath and system-bath coupling don’t respect the related symmetry of ˆ H S ,in which case ˆ ρ G ( t ) evolves into a mixed state wherethe decoherence happens. Nevertheless, what we concernhere is whether the complete information of initial state,or at least the coherence, can be maintained when boththe operators { ˆ a α } and { ˆ O j } respect the same symmetryas the system Hamiltonian ˆ H S . As stated above, the fateof coherence is determine by unitarity or anti-unitarity ofthe symmetry. This conclusion had been pointed out inRef. and here we provide a brief explanation. For theunitary symmetry, ˆΠ G ˆ O † ˆ O ˆΠ G ∝ ˆΠ G , ˆΠ G ˆ O † ˆΠ G ∝ ˆΠ G ,and ˆΠ G ˆ O ˆΠ G ∝ ˆΠ G are guaranteed by Schur’s lemma as long as ˆ O † , ˆ O respect the same symmetry as ˆ H S . Inthis case, Lindblad superoperator L only acts on thesubspace density matrix as ˆΠ G L [ˆ ρ ( t )] ˆΠ G ∝ ρ G (0). Ac- cordingly, if we define ρ G ( t ) = (cid:18) ρ ++ ρ + − ρ − + ρ −− (cid:19) = (cid:18) (cid:104) φ + | ˆ ρ | φ + (cid:105) (cid:104) φ + | ˆ ρ | φ − (cid:105)(cid:104) φ − | ˆ ρ | φ + (cid:105) (cid:104) φ − | ˆ ρ | φ − (cid:105) (cid:19) , (9)then all of the matrix elements synchronously decay withthe same rate. That is to say, renormalized density ma-trix in subspace ˜ ρ G = ρ G ( t ) / tr( ρ G ( t )) is always equalto ρ G (0) under the time-evolution of Lindblad equationwhich means the initial coherence is maintained. In ad-dition, the matrix representation of ˆ L in subspace isproportional to identity, namely,ˆ L G ∝ , (10)which acts trivially on the state of sub-space ˆ L G (cid:2) ρ + , + ( t ) , ρ + , − ( t ) , ρ − , + ( t ) , ρ − , − ( t ) (cid:3) T ∝ (cid:2) ρ + , + (0) , ρ + , − (0) , ρ − , + (0) , ρ − , − (0) (cid:3) T . For the caseof anti-unitary symmetry, according to the Schur’sLemma , if ˆ O is Hermitian operator obeying thissymmetry, it will be proportional to identity matrix indegenerate subspace, i.e., ˆΠ G ˆ O † ˆΠ G = ˆΠ G ˆ O ˆΠ G ∝ ˆ I G andthe condition (10) is also true which means the densitymatrix will maintain its initial coherence. But if ˆ O isnon-Hermitian operator, then its matrix presentation isproportional to identity, i.e., ˆΠ G ˆ O ˆΠ G (cid:54)∝ ˆ I G . In this case,one can find immediately ˜ ρ G = ρ G ( t ) / tr( ρ G ( t )) (cid:54) = ρ G (0)from master equation (4) and decoherence occurs.Meanwhile ˆ L G is no longer proportional to identityˆ L G (cid:54)∝ . (11)The above statements for unitary or anti-unitary sym-metric systems can be naturally generalized to those withvarious combinations of symmetries. We summarize dif-ferent kinds of symmetry combinations and correspond-ing fate of coherence in Table I, we take time-reversalsymmetric group and quaternion group as candidates ofantiuntary and unitary group, respectively. Some repre-sentative examples will be illustrated in next section. IV. NUMERICAL RESULTS WITH SPIN- / MODELS
In this section, we take spin-3 / . The matrix representation of spin-3/2 angular momentum is chosen as S x = 12 √ √ √
30 0 √ ,S y = i −√ √ − −√
30 0 √ ,S z = −
00 0 0 − . (12) TABLE I. Classification table for the fate of initial coherence protected by there kinds of symmetries in ˆ H s and coupledto operator O with different types of symmetry combinations. Here [ ˆ H S , Q ] = 0 ([ ˆ H S , T ] = 0) means the system respects( Q -symmetry) time-reversal symmetry. ˆ I G represents the identity matrix in ground-state subspace.Symmetry of ˆ H S Hermiticity of ˆ O Symmetry of ˆ O Coherence/Decoherence ˆ L G ∝ ˆ I G ⊗ ˆ I G (Yes/No)[ ˆ H S , Q ] = 0 Hermitian [ ˆ O , Q ] = 0 Coherence Yes[ ˆ O , Q ] (cid:54) = 0 Decoherence NoNon-Hermitian [ ˆ O , Q ] = 0 Coherence Yes[ ˆ O , Q ] (cid:54) = 0 Decoherence No[ ˆ H S , T ] = 0 Hermitian [ ˆ O , T ] = 0 Coherence Yes[ ˆ O , T ] (cid:54) = 0 Decoherence NoNon-Hermitian [ ˆ O , T ] = 0 Decoherence No[ ˆ O , T ] (cid:54) = 0 Decoherence No[ ˆ H S , T ] = 0[ ˆ H S , Q ] = 0 Hermitian [ ˆ O , T ] = 0 , [ ˆ O , Q ] = 0 Coherence Yes[ ˆ O , T ] = 0 , [ ˆ O , Q ] (cid:54) = 0 Coherence Yes[ ˆ O , T ] (cid:54) = 0 , [ ˆ O , Q ] = 0 Coherence Yes[ ˆ O , T ] (cid:54) = 0 , [ ˆ O , Q ] (cid:54) = 0 Decoherence NoNon-Hermitian [ ˆ O , T ] = 0 , [ ˆ O , Q ] = 0 Coherence Yes[ ˆ O , T ] = 0 , [ ˆ O , Q ] (cid:54) = 0 Decoherence No[ ˆ O , T ] (cid:54) = 0 , [ ˆ O , Q ] = 0 Coherence Yes[ ˆ O , T ] (cid:54) = 0 , [ ˆ O , Q ] (cid:54) = 0 Decoherence No The unitary and anti-unitary symmetries will be dis-cussed respectively.
A. Unitary symmetry
We first study the system with unitary symmetry. Inorder illustrate the symmetry protected coherence, forsimplicity, we choose the quaternion group Q as candi-date of the unitary group which is a non-Abelian unitarygroup whose irreducible representation can be two di-mensions, spanning a two-fold degenerate subspace. Thematrix representation of Q is displayed explicitly in Ap-pendix B. We then construct a Q -symmetric Hamilto-nian ˆ H S = E g ( ˆ S x ˆ S y ˆ S z + ˆ S z ˆ S y ˆ S x ) with twofold degener-ate ground states which forms the two-dimensional irre-ducible representation space of group Q . For Hermitianoperator ˆ O coupled to system respecting Q -symmetrysuch as ˆ O = ˆ S y , the coherence in ground-state subspacemaintains all the time but decoherence occurs in the caseof breaking this symmetry, say, ˆ O = ˆ S x ˆ S y + ˆ S y ˆ S x asshown in Fig.1(a). The same situation happens to thenon-Hermitian coupling operators that the symmetricand asymmetric ˆ O are chosen as ˆ O = ˆ S x ˆ S y ˆ S z and ˆ O =ˆ S y ˆ S z respectively (see Fig.1(b)). The increasing entropyreflects the fact subspace density matrix evolves into amixed state. The unchanged von Neumann entropy indi-cates that the density matrix in subspace is always a purestate, which means the coherence is maintained. Thisis reasonable consequence that the decoherence appearsonly when the system is perturbed by symmetry-brokencoupling. As mentioned above, the Schur’s lemma allowsus to infer that ˆΠ G ˆ O ˆΠ G ∝ ˆ I G if ˆ O respects Q -symmetry which will lead to ˆ L G ∝ ˆ I G ⊗ ˆ I G according to expres-sion (8). In contrast, ˆ L G (cid:54)∝ ˆ I G ⊗ ˆ I G if ˆΠ G ˆ O ˆΠ G (cid:54)∝ ˆ I G since ˆ O breaks Q -symmetry which destroys the coher-ence and the corresponding entropy of decoherent dy-namics reaches steady-state value S v ( ∞ ) = ln 2 ≈ . B. Anti-unitary symmetry
From now on, we focus on the systems respecting time-reversal symmetry. Since time-reversal operation invertsthe angular momentum operators, i.e., ˆ T ˆ S x,y,z ˆ T − = − ˆ S x,y,z , one can construct time-reversal invariant Hamil-tonian as ˆ H S = E g { ˆ S x , ˆ S z } . The ground-state subspaceof ˆ H S is two-fold degenerate due to Kramers’ theoremand we denote two degenerate states as | φ ± (cid:105) . Simi-lar to the discussion of unitary symmetric system, herethe hermitian and non-hermitian coupling ˆ O will be in-vestigated respectively. In order to verify the case of[ ˆ H S , T ] = 0 in Table I, we prepare the initial state on | ψ (0) (cid:105) = α | φ + (cid:105) + β | φ − (cid:105) , and then calculate the time evo-lution of von Neumann entropy in ground-state subspaceby means of Lindblad equation shown in Fig. 2.For Hermitian case, as shown in Fig.2(a), the ini-tial coherence will maintain (vanish) if the operator ˆ O coupled by system respects (breaks) time-reversal sym-metry. However, there will be a dramatic differencewhen this time-reversal symmetric system couples to non-Hermitian operator ˆ O . Fig.2(c) plots the von Neumannentropy from which one can see clearly that the coher- (a)(b) FIG. 1. Time evolution of von Neumann entropy in degener-ate subspace protected by unitary-symmetry ( Q -symmetry).(a) Von Neumann entropy as function of time for hermitiancoupling operator: with Q -symmetry ˆ Q = ˆ S y (red solid line)and without Q -symmetry ˆ Q = ˆ S x ˆ S y + ˆ S y ˆ S x (blue dashedline). (b) Von Neumann entropy as function of time for non-hermitian coupling operator: with Q -symmetry ˆ Q = ˆ S x ˆ S y ˆ S z (red solid line) and without Q -symmetry ˆ Q = ˆ S x ˆ S y ˆ S z (bluedashed line). ence is always destroyed despite whether or not ˆ O re-spect time-reversal symmetry. Hence, we demonstratethe statement that coherence could survive only if thetime-reversal symmetric coupling operator is also Her-mitian. This is consistent with Schur’s Lemma for anti-unitary group that an anti-unitary symmetric and her-mitian operator is proportional to identity in degeneratesubspace, i.e., ˆΠ G ˆ O ˆΠ G ∝ ˆ I G which makes the relationˆ L G ∝ ˆ I G ⊗ ˆ I G true. When the situation that any ofsymmetry and hermiticity is not satisfied happens, it re-sults in ˆΠ G ˆ O ˆΠ G (cid:54)∝ ˆ I G which causes the decoherence andmeanwhile ˆ L G (cid:54)∝ ˆ I G ⊗ ˆ I G .With regard to density matrix ˆ ρ G ( t ) in ground-statespace, for coherent evolution, all matrix elements decaywith equal-rate during whole region of time as shown inFig.2(b). While for decoherent process, the nondiagonalelements could decay to zero but its diagonal elementskeep finite which means all coherence is lost and the sys-tem reaches maximum entanglement in groundstate sub-space (see Fig.2(d)). (a) (b) (d)(c) FIG. 2. Time evolution of von Neumann entropy and den-sity matrix in degenerate subspace protected by time-reversalsymmetry. (a) Von Neumann entropy as function of timefor hermitian coupling operator: with time-reversal symme-try ˆ Q = S x (red solid line) and without time-reversal sym-metry ˆ Q = S z (blue dashed line). (b) Matrix elements ofdensity matrix associated with red solid line in (a) evolves astime. (c) Von Neumann entropy as function of time for non-Hermitian coupling operator: without time-reversal symme-try ˆ Q = S x S y S z (red solid line) and with time-reversal sym-metry ˆ Q = iS z (blue dashed line). (d) Matrix elements ofdensity matrix associated with blue dashed line in (c) evolvesas time. The initial value of density matrix is chosen as | ψ (0) (cid:105) (cid:104) ψ (0) | with | ψ (0) (cid:105) = √ (cid:0) | φ + (cid:105) + | φ − (cid:105) (cid:1) . C. Both unitary and anti-unitary symmetry
There is another case in which the Hamiltonian re-spects both time-reversal symmetry and Q -symmetry,such as ˆ H S = E g ˆ S z whose degeneracy of groundstatesis protected by both of symmetries. This highly sym-metric system allows for more types of symmetry combi-nations which contain four circumstances for both Hermi-tian and non-Hermitian couplings as shown in the thirdrow of Table I. The corresponding entropy dynamics as-sociated with different symmetries is shown in Fig. (3).For hermitian case, the initial coherence in ground-statesubspace is always maintained whichever symmetry thecoupling operator ˆ O has. This is consistent with the con-clusions in subsection IV A and IV B that ˆΠ G ˆ O ˆΠ G ∝ ˆ I G regardless of unitarity or antiunitarity which signifiesthe decoherence occurs only when the coupling opera-tor breaks both symmetries, as shown in Fig.3.(a). Fornon-Hermitian case, the coherence will be destroyed oncethe Q -symmetry is broken by coupling operator ˆ O asshown in Fig.3.(b). This reflects the fact that coherenceprotected only by antiunitary symmetry is fragile undernon-Hermitian perturbation. (a)(b) FIG. 3. The von Neumann entropy as a function of time withthe system Hamiltonian respecting both Q -symmetry andtime-reversal symmetry. The Hermitian and non-Hermitiancoupling ˆ O ’s are shown in (a) and (b) respectively. Forhermitian case, the four kinds of coupling ˆ O are chosenas ˆ S x (red solid line), ˆ S x ˆ S y + ˆ S y ˆ S x (blue dashed line),ˆ S x ˆ S y ˆ S z + ˆ S z ˆ S y ˆ S x (green dotted line), and ˆ S x (dot-dashedline). For non-Hermitian case, the corresponding choice is i ( ˆ S x ˆ S y ˆ S z − ˆ S z ˆ S y ˆ S x ), ˆ S x ˆ S y , ˆ S x ˆ S y ˆ S z , and ˆ S x ˆ S z . The ini-tial value of density matrix is chosen as | ψ (0) (cid:105) (cid:104) ψ (0) | with | ψ (0) (cid:105) = √ (cid:0) | φ + (cid:105) + | φ − (cid:105) (cid:1) . V. SUMMARY AND OUTLOOK
We study the fate of coherence in degenerate subspacewhich is protected by unitary symmetry or anti-unitarysymmetry. With symmetries lying in system Hamilto-nian, and the interaction between system and environ-ment, we have demonstrated that the coherence could befragile when the symmetry is anti-unitary. We elaborateon classification of various symmetry combinations andanalyze the stability of coherence detailedly which is con-firmed by several spin-3/2 models. The results of this in-vestigation are extensions of Ref. and could be appliedto the investigation of robustness in time-reversal invari-ant topological edge states. Recent experimental pro-gresses in controlling and manipulating dissipation in ul-tracold atoms provide unprecedented opportunity for un-derstanding the dynamics of open quantum system which is driven far from of equilibrium . We expect ourwork will guide the preparation for stable time-reversalsymmetry protected topological states in laboratory.There is another intriguing issue about the open quan-tum systems beyond the Born-Markov approximation. Inthis case, the coherence protected by anti-unitary sym-metry could also be fragile even the system couples her-mitian operators with the same symmetry . What’smore, the current study focuses on non-interacting sys-tems. An important issue in future study is exploring tothe stability of quantum degeneracy protected by anti-unitary in interacting systems, especially to interactingtopological phases in many-body system such as Haldanephase in AKLT model chain . ACKNOWLEDGEMENTS
We would like to thank Yu Chen for helpful discussions.This work is supported by Beijing Outstanding YoungScientist Program hold by Hui Zhai. L. P acknowledgessupport from the project funded by the China Postdoc-toral Science Foundation (Grant No. 2020M680496).
Appendix A: Response of density matrix and vonNeumann entropy growth
In this appendix, we derive the response of densitymatrix and von Neumann entropy growth by means ofNHLRT. As discussed in Ref. , the whole system cou-pling to a bath with white noise can be described by anon-Hermitian effective Hamiltonianˆ H eff = ˆ H S + ˆ H diss , (A1)where H diss = (cid:16) − iγ ˆ O † ˆ O + ˆ O † ˆ ξ + ˆ ξ † ˆ O (cid:17) and γ = π | g | ρ is dissipation strength where ρ is spectrum density ofbath. ˆ ξ ( t ), ˆ ξ † ( t ) present the Langevin noise operatorswhich obey the following relations (cid:104) ˆ ξ ( t ) ˆ ξ † ( t ) (cid:105) noise = 2 γδ ( t − t ) , (cid:104) ˆ ξ ( t ) ˆ ξ ( t ) (cid:105) noise = (cid:104) ˆ ξ † ( t ) ˆ ξ ( t ) (cid:105) noise = (cid:104) ˆ ξ † ( t ) ˆ ξ † ( t ) (cid:105) noise = 0 , (A2)where (cid:104)· · · (cid:105) noise denotes the noise average . This formal-ism is equivalent to the total Hamiltonian with a whitenoise bath. In the interaction picture, the time evolutionof density matrix can be expressed byˆ ρ ( t ) = ˆ U eff ( t )ˆ ρ ( t ) ˆ U † eff ( t ) (A3)where ˆ U eff ( t ) = (cid:101) T exp (cid:16) − i (cid:82) t ˆ H diss ( t (cid:48) ) dt (cid:48) (cid:17) with anti-time-ordered operator (cid:101) T and ˆ ρ ( t ) = e − i ˆ H S t ˆ ρ (0) e i ˆ H S t denotesthe time-evolution of density matrix determined by ˆ H S .Taking ˆ H diss as perturbation and then averaging the noise, one can obtain the density matrix with the first-order correction of γρ ( t ) ≡ (cid:104) ˆ ρ ( t ) (cid:105) noise = (cid:68) U (cid:48) eff ( t )ˆ ρ ( t ) U (cid:48)† eff ( t ) (cid:69) noise = (cid:42) (cid:32) ∞ (cid:88) n =1 ( − i ) n (cid:90) t < ··· This appendix provides matrix representation ofquaternion group Q discussed in maintext. The Q -group { Q j , j = 1 , · · · , } is a non-Abelian group which is iso-morphic to subset { , i, j, k, − , − i, − j, − k } whose multi-plication table is displayed in Table II. The Q -group con-tains a two-dimensional irreducible representation (seeTable.III) which can be described as a subgroup of thespecial linear group SL ( C ). We can construct the fol-lowing 4-dimensional reducible representation TABLE II. The multiplication table (Cayley table) of quater-nion group. Element e e i i j j k ke e e i i j j k k-1 -1 e i i j j k ki i i e e k k j j i i i e e k k j j j j j k k e e i ij j j k k e e i ik k k j j i i e ek k k j j i i e e Q = I ⊗ I, Q = − Q , Q = − I ⊗ iσ z , Q = − Q ,Q = − iσ x ⊗ σ y , Q = − Q , Q = − iσ x ⊗ σ x , Q = − Q , (B1)where σ x,y,z denote Pauli matrices and I is identity ma-trix. One can easily find that the matrix representation(B1) obeys the multiplication table (II) and contains two-dimensional irreducible representation. TABLE III. 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