Theory of Classical Metastability in Open Quantum Systems

Theory of classical metastability in open quantum systems

Katarzyna Macieszczak,1 Dominic C. Rose,2, 3 Igor Lesanovsky,4, 2, 3 and Juan P. Garrahan2, 3 1TCM Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom 2School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 3Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 4Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany We present a general theory of classical metastability in open quantum systems. Metastability is a consequence of a large separation in timescales in the dynamics, leading to the existence of a regime when states of the system appear stationary, before eventual relaxation toward a true stationary state at much larger times. In this work, we focus on the emergence of classical metastability, i.e., when metastable states of an open quantum system with separation of timescales can be approximated as probabilistic mixtures of a finite number of states. We find that a number of classical features follow from this approximation, for the manifold of metastable states, long-time dynamics between them, and symmetries of the dynamics. Namely, those states are approximately disjoint and thus play the role of metastable phases, the relaxation toward the stationary state is approximated by a classical stochastic dynamics between them, and weak symmetries correspond to their permutations. Importantly, the classical dynamics is observed not only on average, but also at the level of individual quantum trajectories: We show that time coarse-grained continuous measurement records can be viewed as noisy classical trajectories, while their statistics can be approximated by that of the classical dynamics. Among others, this explains how first-order dynamical phase transitions arise from metastability. Finally, to verify the presence of classical metastability in a given open quantum system, we develop an efficient numerical approach that delivers the set of metastable phases together with the effective classical dynamics. Since the proximity to a first-order dissipative phase transition manifests as metastability, the theory and tools introduced in this work can be used to investigate such transitions—which occur in the large size limit—through the metastable behavior of many-body systems of moderate sizes accessible to numerics.

CONTENTS 1. Long-time dynamics 12 2. Classical generator 12 I. Introduction3 3. Classical system dynamics 13 4. Hierarchy of classical long-time II. Metastability in open quantum systems5 dynamics 13 A. Dynamics of open quantum systems5 5. Classical observable dynamics 14 B. Spectral theory of metastability5 B. Classical characteristics of quantum C. Quantitative approach6 trajectories 14 D. Dissipative phase transitions7 1. Statistics of quantum trajectories 14 2. Classical tilted generator 15 III. Classical metastability in open quantum 3. Classical cumulants 15 systems7 4. Classical dynamics of quantum A. Deﬁnition of classical metastability7 trajectories 17 B. Test of classicality8 5. Classical metastability and dynamical C. Figures of merit9 phase transitions 18

IV. Classical metastable phases9 VI. Classical weak symmetries 19 A. Physical representation of metastable A. Symmetry and general metastability 20 arXiv:2006.01227v2 [cond-mat.stat-mech] 19 Jul 2021 manifold 10 1. Symmetry of metastable manifolds 20 B. Classical degrees of freedom 10 2. Symmetry of long-time dynamics 20 C. Approximate disjointness of metastable B. Symmetry and classical metastability 20 phases 11 1. Approximate symmetry of metastable D. Classical hierarchy of metastable phases 11 phases 20 2. No continuous symmetries 20 V. Classical long-time dynamics 11 3. Symmetric set of metastable phases 21 A. Classical average dynamics of system and 4. Symmetry of classical long-time observables 12 dynamics 21 2

5. Symmetric test of classicality 23 2. Trace-norm vs. L1-norm in classical metastable manifolds 42 VII. Unfolding classical metastability numerically 23 a. Distance between metastable states 42 A. Metastable phases from master operator b. Distance between metastable phases 42 spectrum 23 c. Norms of long-time generator 43 1. Metastable phases construction 23 d. Relaxation time 43 2. Construction for hierarchy of metastable 3. Orthogonality and disjointness of phases in manifolds 25 classical metastable manifolds 44 3. Construction for metastable manifolds a. Bounds on scalar products of metastable with symmetries 25 phases 44 4. Construction utilizing order parameters 25 b. Proof of Eqs. (31)-(33) in the main text 45 B. Metastable phases from biased quantum 4. Nonuniqueness of phases in classical trajectories 26 metastable manifolds 47

VIII. Conclusions and outlook 26 E. Classical long-time dynamics 47 1. Classical stochastic dynamics 47 Acknowledgments 27 a. Positivity and probability conservation 47 b. Stochastic trajectories 48 References 27 c. Weak symmetries 48 Supplemental Material 31 2. Classical dynamics of average system state 49 a. Best classical stochastic approximation of A. Classical metastability in proximity to long-time dynamics generator 49 dissipative phase transition at finite size 31 b. Derivation of Eq. (37) in the main text 49 1. Example in Figs.1–5 of the main text 31 c. Derivation of Eq. (39) in the main text 50 a. Model 31 d. Derivation of Eq. (41) in the main text 50 b. Dissipative phase transition 31 e. Approximation of dynamics resolvent 51 c. Plot parameters and numerical results 31 f. Classical discrete approximation of 2. General case 32 long-time dynamics 52 a. Dissipative phase transition and its 3. Classical statistics of quantum trajectories 53 proximity 32 a. Activity in quantum trajectories 53 b. Perturbation theory 33 b. Homodyne current in quantum trajectories 54 B. Metastability in open quantum systems 35 c. Time-integrals of system observables in 1. Projection on low-lying modes 35 quantum trajectories 56 ˜ a. Derivation of Eq. (12) in the main text 35 d. Corrections in approximations of Ws, ˜ ˜ b. Bound on metastability of states closest Wh, and Wr 57 to Eq. (4) in the main text 35 e. Rates of average and fluctuations in 2. Metastable regime 35 quantum trajectories after initial 3. Relaxation times 36 relaxation 59 a. Definitions 36 f. Rates of average and fluctuations in b. Relation to master operator spectrum 37 quantum trajectories during metastable c. Relation to metastable regime 37 regime 65 g. Asymptotic rates of fluctuations in C. Classical metastability in open quantum quantum trajectories 69 systems 38 h. Multimodal distribution of quantum 1. Definition of classical metastability 38 trajectories 70 2. Test of classicality 38 a. Distance of barycentric coordinates to F. Classical hierarchy of metastabilities 75 probability distributions 38 1. Hierarchy of metastabilities 75 b. Derivation of Eq. (22) in the main text 39 2. Hierarchy of classical metastable manifolds 75 c. Optimality of test of classicality 39 3. Hierarchy of classical metastable phases 77 a. Supports and basins of attraction 77 D. Classical metastable phases 40 b. Decay subspace 78 1. Properties of dual basis in classical 4. Hierarchy of classical long-time dynamics 78 metastable manifolds 40 a. Hierarchy of continuous approximations a. Properties of dual basis 40 of classical long-time dynamics 78 b. Distance to POVM 41 b. Hierarchy of discrete approximations of c. Cross-correlations of dual basis 41 classical long-time dynamics 80 3

G. Classical weak symmetries 81 however, more sophisticated (albeit still approximate) 1. Symmetries of low-lying eigenmodes 81 techniques such as variational approaches [26–28], per- 2. Symmetries of classical metastable turbative expansions in lattice connectivity [25, 29], infi- manifolds 81 nite tensor network simulations [13] or a field-theoretical a. Discrete symmetries of classical analysis [16] can still indicate a unique stationary state. metastable manifolds 82 While it is unusual to see phase transitions at finite b. No nontrivial continuous symmetries of system sizes [30–33], first-order phase transitions in sta- classical metastable manifolds 82 tionary states manifest at large enough finite system c. Symmetric set of metastable phases 83 sizes [34] through the occurrence of metastability, i.e., dis- 3. Symmetries of classical long-time dynamics 84 tinct timescales in the evolution of the system statistics: a. Derivation of Eq. (77) in the main text 84 classically, in the probability distribution over configura- b. Classical dynamics of symmetric degrees tion space [7, 22, 35, 37, 39]; quantum mechanically, in of freedom 85 the density matrix [5, 11]. The statistics of such systems 4. Example of classicality test with weak at long times can be understood in terms of metastable symmetry 85 phases which generally correspond to the phases on either side of the transition being distinct from the unique H. Quantitative analysis of algorithm in Sec.VIIA stationary state for a given set of parameters. Therefore, of the main text 86 already at a finite system size the structure of a possi- 1. Extreme eigenstates of dynamics ble first-order dissipative phase transition can be fully eigenmodes for metastable phases 86 determined by investigating metastable states of the sys- 2. Rotations of eigenmodes to expose tem [11, 42], which is of particular importance for many- metastable phases 86 body open quantum systems, where exact methods are 3. Maximal simplex in coefficient space as often limited to numerical simulations of finite systems simplex of metastable phases 87 of modest size. 4. Hierarchy of metastable manifolds 87 Metastability can also emerge in complex relaxation 5. Weak symmetries 88 toward a unique stationary state, even without a phase a. Symmetric eigenstates of dynamics transition present in the thermodynamic limit. This is eigenmodes 88 the case in classical kinetically constrained models [43– b. Degeneracy of coefficients 88 48] and spin glasses [49] and recent open quantum gen- c. Refined metastable phase construction 88 eralizations of these models [50–52] and [53]. Here, the study of metastability can unfold the long-time dynamics References 89 responsible for the complex relaxation to the stationary state [54], with metastable phases corresponding to dynamical rather than static phases. I. INTRODUCTION For classical systems with Markovian dynamics [7, 22, 35, 37, 39] and open quantum systems [5, 55] described With continuing advances in the control of experimen- by the master equation formalism [56, 57], metastability tal platforms used as quantum simulators, such as ultra- necessarily requires a large separation in the spectrum of cold atomic gases, Rydberg atoms and circuit quantum- the master operator governing the system evolution. This electrodynamics [1–7], a broad range of nonequilibrium separation leads to metastable states residing in a space phenomena of open many-body quantum systems has of a reduced dimension given by the slow eigenmodes been observed recently. Theoretical studies have proof the master operator, and long-time dynamics taking gressed via the combination of methods from atomic place within that space. Since the slow modes themselves physics, quantum optics and condensed matter, giving do not represent system states, a general structure of the rise to a range of techniques including quantum jump metastable manifold (MM) is not known, but conjectured Monte Carlo (QJMC) [8–12] simulations via tensor net- to feature disjoint phases, decoherence free subspaces and work [13], and field theoretical approaches [14–16]. noiseless subsystems, while the long-time dynamics is ex- Often the focus of studies on nonequilibrium open pected to be analogous to perturbative dynamics on such many-body quantum systems is a phase diagram of the states [5]. stationary state, and the related question of the struc- In this work, we comprehensively prove this conjec- ture of dissipative phase transitions occurring in the ther- ture for classical metastability in open quantum sys- modynamic limit of infinite system size. This includes tems. We define classical metastability as the case, where whether such systems can exhibit bistability (or multista- metastable states can be approximated as probability dis- bility) of the stationary state, a topic covered both the- tributions over a set of m states, where m−1 is the num- oretically [17–21] and experimentally [22, 23], and which ber of slow eigenmodes in the dynamics. We then show order parameters are relevant for distinguishing the co- that this definition is equivalent to a simple geometric cri- existing phases. Mean-field results often suggest multiple terion, which can be verified using the exact diagonaliza- stationary states in the thermodynamic limit [16, 24, 25], tion of the master operator. Crucially, the corresponding 4

of attraction form the set of m order parameters to distinguish them. Furthermore, we find that the long-time dynamics of the system can be approximated as a classical stochastic dynamics between the metastable phases. This holds in the average system dynamics as well as in individual quantum trajectories [12, 58], as obtained via individual runs of an experimental system or from QJMC simulations, where classical trajectories arise via coarse graining of these in time. The classical dynamics between the metastable phases is then responsible for the occurrence of intermittence [11, 24, 51] or dynamical heterogeneity [50, 52] in quantum trajectories, leading to multimodal statistics of continuous measurements and system proximity to a first-order dynamical phase transition [59]. Therefore, classical metastability is a phenomenon occurring not only on average, but in dynamics of individual quantum trajectories. All these results are also discussed in the presence of further hierarchy of relaxation timescales. Finally, while our approach does not rely on the presence of any symmetries of the dynamics [1, 23, 24] (cf. Ref. [42]), we also show that the set of metastable phases is approximately invariant under any present symmetries. Thus, weak symmetries lead to approximate cycles of metastable phases and permutation symmetries of the classical long-time dynamics, and, as such, we find that any nontrivial continuous symmetries of slow eigenmodes of the dynamics preclude classical metastability. To verify the classicality of metastability present in Figure 1. Metastability in Markovian open quantum a general open quantum system and uncover the set systems: (a) Metastability corresponds to a separation in of metastable phases together with the effective struc- the real part of master operator spectrum, between m − 1 ture of long-time dynamics, we develop an efficient nu- slow (blue dashed) and fast modes (black solid), while the merical technique, which can be further simplified when stationary state corresponds to 0 eigenvalue (red solid); here a weak symmetry is present. Our approach relies on m = 4. (b) The manifold of metastable states is described the ability to diagonalize the system master opera- by coefficients ck [Eq. (4)], k = 2, ..., m, of decomposition be- tor, which is usually possible only for moderate sys- tween slow modes (dots for random initial pure states). The tem sizes, while metastability may become prominent long-time dynamics takes place within that manifold, with only for large system sizes. To mitigate this poten- the exponential decay of the coefficients toward the stationary state (red sphere) [Eqs. (3) and (5)]. Metastability can tial issue, we show that for classical metastability ac- be observed experimentally as a plateau in the dynamics of companied by intermittence or dynamical heterogeneity observable averages (c) or two-point correlations (d) appear- in quantum trajectories, metastable phases can be ex- ing during the metastable regime [Eqs. (8) and (9)]. Black tracted from quantum trajectories through the use of (solid) lines show observable dynamics, blue (dashed) lines large-deviation methods, such as the “thermodynamics the approximation by slow modes holding after the initial re- of trajectories” [24, 51, 59, 63, 64]. Therefore, there is laxation, and red (solid) lines the stationary value achieved potential to study classical metastability using QJMC after the final relaxation. (e) Long timescales can also be ob- simulations, which are generally feasible at quadratically served in continuous measurement records, e.g., as intermit- larger system sizes than exact diagonalization of the gen- tence in detection of quanta emitted due to jumps occurring in erator. the system (two types shown in blue and red; gray—without associated quanta), with regimes of jump activity having a This paper is organized as follows. In Sec.II, we re- length comparable to the long-time relaxation timescale. See view the results of Ref. [5]. In Sec.III, we introduce Sec.A1 in the SM for details on the model. the general approach to classical metastability in open quantum systems. We then discuss the resulting classical corrections play the role of a figure of merit in emergent structure of the MM in Sec.IV. The effectively classical classical properties of the manifold of metastable states system dynamics emerging at large times is discussed in and its long-time dynamics. Namely, we show that, for Sec.V. We refine these general results considering sym- classical metastability, m states can be considered as dis- metries of the system dynamics in Sec.VI. Finally, we tinct metastable phases, as they are approximately dis- introduce numerical approaches to unfold the structure joint and orthogonal to one another, while their basins of classical metastability in Sec.VII. Details and proofs 5 of our results are presented in the Supplemental Mate- R1 = ρss [30–33]. The system state at time t can be then rial (SM). Our results are illustrated in Figs.1–5 with decomposed as a system in proximity to an effectively classical dissipa- tL X tλk tive phase transition occurring at finite size. We discuss ρ(t) = e [ρ(0)] = ρss + cke Rk, (2) classical metastability arising for such systems in Sec.A k≥2 of the SM. The application of the general methods in- where the coefficients ck ≡ Tr[Lkρ(0)] are bounded by troduced in this paper to a many-body system beyond † this class is given in the accompanying paper [54] which the eigenvalues of Lk, with Lk being eigenmatrices of L studies in detail the metastability of the open quantum normalized such that Tr(LkRl) = δkl (there is a freedom East glass model [50]. of choice to normalize by scaling either Rk or Lk). The values of these coefficients for a given physical state are closely tied, such that the corresponding linear combination of Rk results in a positive matrix. We refer to II. METASTABILITY IN OPEN QUANTUM Lk and Rk as left and right eigenmatrices (eigenmodes), SYSTEMS respectively. Note that the trace preservation of the dynamics implies that L1 = 1, and thus beyond ρss other We begin by reviewing the spectral theory of metasta- right eigenmatrices do not correspond to quantum states, bility of Ref. [5]. We then introduce a quantitative de- Tr(Rk) = Tr(L1Rk) = 0 for k ≥ 2. The timescale τ of scription of those results by considering the corrections the final relaxation to ρss from Eq. (2) can be seen to to the stationarity during the metastable regime. In R depend on the gap in the spectrum, τ ≥ −1/λ2 . the next section, we build on this to understand when metastability in open quantum systems becomes classical, which is the main focus of this work. B. Spectral theory of metastability

Metastability corresponds to a large separation in the R R A. Dynamics of open quantum systems real part of the spectrum [5], λm/λm+1 1, which denotes the ratio of eigenvalues being of a lower order than We consider a finitely dimensional open quantum sys- 1; see Fig.1(a). Time t after the initial relaxation corre- tem with ts average state at time t described by a density spond to the terms beyond the m-th in the sum in Eq. (2) tλ matrix ρ(t) evolving according to a master equation as being negligible, e k ≈ 0 for k ≥ m + 1, and the reduced dρ(t)/dt = L[ρ(t)], where the master operator [56, 57] expansion m X tλk X † 1 n † o ρ(t) = ρss + cke Rk + ..., (3) L(ρ) = −i[H, ρ] + JjρJj − Jj Jj, ρ . (1) 2 k=2 j where ... stands for negligible corrections [cf. Eq. (2)]. Here, H is the system Hamiltonian, while the jump oper- When the separation in the spectrum is big enough, ators Jj provide coupling of the system to the surround- it is possible to further consider times when decay of ing environment (in this work we explicitly denote any tλR the remaining terms can be neglected, e k ≈ 1 for k ≤ time dependence; in particular, the master equation is m; cf. Ref. [67]. This is the metastable regime, during time-independent). If the interactions between the sys- which the system state is approximately stationary, i.e., tem and the environment are associated to emissions of metastable, as captured by an energy quanta, then the action of jump operators can m be detected through continuous measurements [58], e.g., X counting of photons emitted by atoms coupled to the ρ(t) = ρss + ckRk + ... ≡ P[ρ(0)] + ..., (4) vacuum electromagnetic field [24, 50–52]. Equation (1) k=2 is a general dynamics of a time-homogeneous Markovian where we defined P as the projection onto the low-lying open quantum system [56, 57], which arises for systems eigenmodes of the master operator, which is trace and interacting weakly with an effectively memoryless envi- Hermiticity preserving [18]. From Eq. (4) the manifold ronment [58]. of metastable states is fully characterized by the bounded Since the master operator L acts linearly on ρ(t), the coefficients (c2, ..., cm) and thus it is (m−1)-dimensional. evolution can be understood in terms of its eigenmatrices The MM is also convex, as a linear transformation of the R I Rk and their corresponding eigenvalues λk = λk +i λk [6]. convex set of initial states [see Fig.1(b)]. The real parts of these eigenvalues are not greater than At later times, only the slow modes contribute to the R 0, λk ≤ 0, as the dynamics in Eq. (1) is (completely) evolution [cf. Eq. (3)]. Therefore, the dynamics toward positive and trace preserving; we order the eigenvalues the stationary state takes place essentially inside the MM R by decreasing real part λk . In particular, zero eigenval- [see Fig.1(b)], ues correspond to stationary states [1,3]. In this work, we assume a generic case of a unique stationary state ρ(t) = etLMM P[ρ(0)] + ..., (5) 6 and is generated by [cf. Figs.1(c) and1(d)] main results: emerging classical features of metastable manifold, long-time dynamics, and weak symmetries in LMM ≡ PLP. (6) the case of classical metastability. Denoting by τ 00 the timescale of the initial relaxation, We consider errors of the approximation of the system 00 R 0 from Eq. (3) we have τ ≥ −1/λm+1. Similarly, for τ dynamics by the projection on the low-lying modes of the being the smallest timescale of the long-time dynamics we spectrum in Eq. (4) during a time regime t00 ≤ t ≤ t0, 0 R have τ ≤ −1/λm from Eq. (5). Then for times within 00 0 the metastable regime we have τ 00 ≤ t τ 0 from Eq. (4) CMM(t , t ) ≡ sup sup kρ(t) − P[ρ(0)]k (10) ρ(0) t00≤t≤t0 [cf. Fig.1(b)]. tL Metastability can be observed in the behavior of sta- = sup ke − Pk, 00 0 tistical quantities such as expectation values or autocor- t ≤t≤t relations of system observables [5, 11, 69]. For a system which we refer to√ as the corrections to the stationarity. observable, e.g., spin magnetization, we have Here, kXk = Tr( X†X) denotes the trace norm for an hO(t)i ≡ Tr [O ρ(t)] = Tr O etL[ρ(0)] operator X, while for a superoperator it denotes the norm induced by the trace norm [19]. For the time regime such X tλk = hOiss + bk ck e , (7) that k 00 0 where we introduced decomposition of the observable into CMM(t , t ) 1, (11) the left eigenmodes with the coefficients bk ≡ Tr(ORk) [cf. Eq. (2)], and b = hOi = Tr(O ρ ) is the static the corrections in Eq. (4) are negligible (note that den- 1 ss ss sity matrices are normalized in the trace norm), which re- average. After the initial relaxation, the contribution 00 R 0 R R 0 from fast modes can be neglected [cf. Eq. (3)], quires t > −1/λm+1 and t −1/λm (i.e., −λmt 1); see Sec.B2 in the SM. We refer to such a time regime as m X a metastable time regime. hO(t)i = hOi + b c etλk + ... (8) ss k k We now argue that the corrections to the stationarity k=2 can be considered as the central figure of merit in the tLMM = Tr O e P[ρ(0)] + ..., theory of metastability. Indeed, the corrections to the and the observable dynamics in Eq. (7) is accurately cap- positivity of metastable states projected on the low-lying tured by the effective long-time dynamics in Eq. (6). Im- modes are defined by the distance to the set of density portantly, during the metastable regime, the observable matrices, average is approximately stationary [cf. Eq. (3)], before C+ ≡ sup inf kP[ρ(0)] − ρk (12) the final relaxation to hOiss [see Fig.1(c)], allowing for a ρ(0) ρ direct observation of the metastability. This, however, re- = sup kP[ρ(0)]k − 1 ≡ kPk − 1, quires preparation of an initial system state different from ρ(0) the stationary state, ρ(0) 6= ρss, something often difficult to achieve in experimental settings. Nevertheless, for the with ρ and ρ(0) being density matrices [19] (see Sec.B1 system in the stationary state, metastability can be ob- in the SM), and can be bounded by the corrections to served as double-step decay in the time-autocorrelation the stationarity in Eq. (10), by considering the distance of a system observable. This is a consequence of the first to ρ ≡ ρ(t) within the metastable regime, measurement perturbing the stationary state, thus caus- tL 00 0 00 0 ing its subsequent evolution, which for times after the ini- C+ ≤ inf ke − Pk ≡ C˜+(t , t ) ≤ CMM(t , t ). (13) 00 0 tial relaxation follows the effective dynamics [cf. Eq. (8)], t ≤t≤t 2 tL 2 Furthermore, the corrections to the stationarity in hO(t)O(0)iss − hOiss = Tr[Oe O(ρss)] − hOiss (9) Eq. (10) establish a bound not only on Eq. (12), but = Tr[OetLMM PO(ρ )] − hOi2 + ..., ss ss also on the corrections in Eqs. (2), (5), (8), and (9). In where O denotes the superoperator representing the mea- fact, beyond the metastable regime, the corrections in surement of the observable O on a system state [5, 11]. Eq. (5) decay exponentially, as in the leading order they n 00 0 The autocorrelation initially decays from the observ- can be shown to be bounded by 2CMM(t , t ), where n 2 2 able variance in the stationary state, hO iss − hOiss, to is an integer such that t/n belongs to the metastable 2 the plateau at Tr[OPO(ρss)] − hOiss in the metastable regime [25]. Similarly, the corrections to observable av- regime, and afterwards to 0 during the final relaxation erages and correlations in Eqs. (8) and (9) are bounded n 00 0 n 00 0 2 [see Fig.1(d)]. by 2CMM(t , t )kOkmax and 2CMM(t , t )kOkmax, respectively, where kOkmax denotes the maximum singular value of O. We thus conclude that the corrections to C. Quantitative approach the stationarity in Eq. (10) are a figure of merit in the theory of metastability. This is further confirmed by the In this work, we introduce a quantitative description role played by them in the errors of classical approxima- of metastability. We later use this approach to prove our tions for the structure of the metastable states and the 7 long-time dynamics when classical metastability occurs, A. Definition of classical metastability which we discuss in later sections. We note that due to the way the condition in Eq. (11) We define classical metastability to take place when is formulated, a choice of the metastable regime is not any state of the system during the metastable regime unique. Indeed, the corrections in Eq. (10) grow when t00 ≤ t ≤ t0 can be approximated as a probabilistic mix- 0 00 t increases or t decreases. In particular, extending the ture of m states ρl ≥ 0 with Tr(ρl) = 1, l = 1, ..., m length t0 − t00 of the metastable regime n times leads to m the corresponding corrections the stationarity bounded in X 00 0 ρ(t) = plρl + ..., (14) the leading order by (n + 3)CMM(t , t ) [72]. In the rest of this work, we consider a given choice of the metastable l=1 regime for an open quantum system displaying metasta- where p ≥ 0 with Pm p = 1 represent the probabilities 00 0 ˜ 00 0 l l=1 l bility and denote CMM(t , t ) and C+(t , t ) by CMM and that depend only on an initial system state ρ(0), while ˜ C+, respectively. As a pronounced metastable regime is ρl, l = 1, ..., m, are independent from both time and the a hallmark of metastability phenomenon, however, some initial state. That is, the corresponding corrections in of our results rely on it being much longer than the ini- the trace norm tial relaxation time: the classical hierarchy of metastable 0 00 00 m phases discussed in Sec.IV on t − t ≥ t , and the cor- X C(ρ1, ..., ρm) ≡ sup inf sup ρ(t) − plρl , (15) respondence of coarse-grained quantum trajectories to ρ(0) p1,..,pm t00≤t≤t0 classical stochastic trajectories discussed in Secs. VB3 l=1 andVIIB on t0 − t00 ≥ nt00 with 1/n 1. fulfill Finally, an analogous approach to Eq. (10) can be in- C(ρ , ..., ρ ) 1. (16) troduced to formally define the timescales τ 00, τ 0 and 1 m τ; see Sec.B3 in the SM. In particular, the relations Here, the corrections depend on the choice of a R 00 00 0 0 R R −1/λm+1 ≤ τ < t , t τ ≤ −1/λm, and −1/λ2 ≤ τ metastable regime, but, for simplicity, we do not include follow (cf. Sec.IIB). it in the notation. We refer to ρl as metastable phases (although their metastability is not assumed, but it is proven to follow together with their approximate disjointness in Sec.IV, where we also discuss their nonunique- D. Dissipative phase transitions ness). The number of phases in Eq. (14) is motivated by uniqueness of the decomposition (see also Sec.C1 When metastability is a consequence of approach- of the SM) and the structure of first-order phase tran- ing a first-order dissipative phase transition, we have sitions in classical Markovian dynamics, where m dis- R R by definition λm/λm+1 → 0 and the ratios of the joint stationary probability distributions constitute sta- timescales for the final and the initial relaxation diverge ble phases of the system, and the system is asymptot- (τ 0/τ 00, τ/τ 00 → ∞). Therefore, the ratios t00/τ 00 and τ 0/t0 ically found in a probabilistic mixture of those phases, for the metastable regime can be chosen arbitrarily large with probabilities depending on the initial system con- leading to all corrections arbitrarily small, C+, CMM → 0; figuration. In later Secs.IV–VB we show that classical cf. Sec.B3 in the SM. properties of metastable phases and long-time dynamics akin to those in proximity to a first-order transition in a classical system follow as well. Remarkably, any metastable state in classical Marko- III. CLASSICAL METASTABILITY IN OPEN vian dynamics can be approximated by a probabilistic QUANTUM SYSTEMS mixture of approximately disjoint metastable phases [7, 22, 35, 37, 39], whether metastability results from prox- We now introduce the notion of classical metastability, imity to a first-order phase transition, or from con- by the virtue of approximation of metastable states by strained dynamics as in glassy systems. In open quantum probabilistic mixtures of a finite number of system states. dynamics, for the bimodal case m = 2, it is known that We show that this definition can be translated into a any metastable state is a probabilistic mixture of two ap- geometric criterion on the decomposition of metastable proximately disjoint metastable phases [5, 11]. For higher states in projections of those states. The correspond- dimensional MMs, however, the general structure is not ing corrections, together with the corrections to the known. Furthermore, it may be no longer classical [5], as stationarity and the positivity, play the central role in not only disjoint phases, but also decoherence free sub- emerging classical approximations to the structure of the spaces [73–75] and noiseless subsystems [76, 77] can be metastable manifold, long-time dynamics, and weak sym- metastable, e.g., when perturbed away from a dissipa- metries, discussed in Secs.IV–VI. Therefore, our crite- tive phase transition at a finite system size [1] (see also rion identifies the parent feature and the figure of merit Supplemental Material in Ref. [5] and cf. Refs. [78–81]). that govern the phenomenon of classical metastability in Therefore, it is important to be able to verify whether a open quantum systems. MM of an open quantum system is classical as defined 8

Figure 2. Classical metastability: (left) In the space of coefficients (c2, c3, c4) [Eq. (4)], the MM features the stationary state at (0, 0, 0) (red sphere) and is approximated by the simplex (blue lines) of m = 4 metastable phases (green spheres at the vertices). The dots represent metastable states from randomly generated pure states found inside (blue) and outside (black) P3 the simplex. (right) Barycentric coordinates (˜p1, p˜2, p˜3) (with p˜4 = 1 − l=1 p˜l) obtained by the transformation C [Eq. (18)] to the physical basis of metastable phases [Eq. (17)] yield probability distributions for states inside the simplex (green), while for states outside the simplex (black) the maximal distance Ccl becomes the figure of merit for classical metastability [Eq. (21)]. in Eq. (14). In this section, we introduce such a system- and checking that they are found approximately within atic approach based on a geometric criterion equivalent the chosen simplex (cf. Fig.2). to the definition in Eq. (14), and refer to it as the test of Motivated by the structure of classical MMs in the classicality. coefficient space, we now introduce the test of classicality—a geometric way of checking whether degrees of freedom describing metastable states during the metastable B. Test of classicality regime correspond, approximately, to probability distributions. Degrees of freedom in the MM are described by For a given set of m candidate system states, the test the coefficients of decomposition into the eigenmodes Rk, of classicality enables one to verify the approximation of k = 1, ..., m, so that, with c1 = 1, their number is m − 1. Eq. (14) and thus the classical metastability. Further- Motivated by Eq. (14), here we instead consider the de- more, it facilitates a check of whether a given set of m composition in the new basis given by the projections of initial states evolve into such metastable phases. Based metastable phases in Eq. (17), which is encoded by the on this, in Sec.VIIA we introduce an efficient numeri- transformation cal technique delivering candidate states which, with the (l) help of the test of classicality, can be postselected into (C)kl ≡ ck , k, l = 1, ..., m, (18) metastable phases forming classical MMs. Pm We first note that the definition of classicality in so that ρ˜l = k=1(C)kl Rk. In particular, the volume Eq. (14) leads to the MM in the space of coefficients being of the corresponding simplex in the coefficient space is approximated by a simplex (see Fig.2). When the MM is |det C|/(m−1)! [83]. The decomposition of a metastable state in this new basis [cf. Eq. (4)] classical, the coefficients ck of a general initial state ρ(0) Pm (l) can be approximated from Eq. (4) as plc , up to m l=1 k X C+C˜+, where C is the correction in trace norm in Eq. (14) P[ρ(0)] = p˜lρ˜l, (19) ˜ (l) l=1 and C+ is given in Eq. (13)[82]. Here, ck = Tr(Lkρl) represent the metastable phases ρl in the coefficient space is given by the barycentric coordinates p˜l = [cf. Eq. (4)], Pm −1 k=1(C )lkck of the simplex in the coefficient space, m so that p˜ = Tr[P˜ ρ(0)] with the new dual basis X (l) l l ρ˜l ≡ P(ρl) = ρss + ck Rk, l = 1, ..., m. (17) m k=2 X −1 P˜l ≡ (C )lk Lk, l = 1, ..., m. (20) Thus, the MM is approximated by a simplex in the coeffi- k=1 cient space with vertices given by the metastable phases. For low-dimensional MMs (m ≤ 4), this can be verified When ρ˜l are linearly independent for l = 1, ..., m, visually by projecting a randomly generated set of initial |det C| > 0 and C is invertible so that Eq. (20) is well ˜ Pm −1 conditions on their metastable states (to sample the MM) defined. In this case, Tr(Pkρ˜l) = n=1 (C )kn(C)nl = 9

−1 (C C)kl = δkl and the normalization of the dual basis the metastability is classical. Moreover, it can be ˜ in Eq. (20) is fixed by the traces of metastable states in shown that Ccl(˜ρ1, ..., ρ˜m) . C(ρ1, ..., ρm) + C+, pro- Eq. (17) being 1. vided that m[C(ρ1, ..., ρm) + C˜+] 1, which also im- Although for the barycentric coordinates we have plies C˜ (˜ρ , ..., ρ˜ ) 1 (cf. Eqs. (15) and (23), and see Pm Pm ˜ 1 cl 1 m l=1 p˜l = 1, and thus l=1 Pl = , they do not in Sec.C2 in the SM). Since for the classical metastability general correspond to probability distributions. Indeed, we have Eq. (16), assuming these corrections decrease they are not all positive whenever a metastable state lies when changing a dynamical parameter or when increas- outside the simplex in the coefficient space corresponding system size, but m remains constant, Eq. (25) follows. ing to ρ˜l, l = 1, ..., m (see Fig.2), as the distance in L1 We thus conclude that it is a necessary and sufficient con- norm of barycentric coordinates to the simplex is given dition for classical metastability. by k˜p−pk1 = k˜pk1 −1, where p is the closest probability Interestingly, the bimodal case of m = 2 is always distribution to barycentric coordinates ˜p [here, (p) = p l l classical as the metastable phases ρ˜1 and ρ˜2 leading to and (˜p) =p ˜ , l = 1, ..., m]. Nevertheless, when the max- l l Ccl(˜ρ1, ρ˜2) = C˜cl(˜ρ1, ρ˜2) = 0 can be constructed explic- imum distance to the simplex (cf. Fig.2), itly [5, 11]. For higher dimensional MMs, the presence of classical metastability can be uncovered using Ccl(˜ρ1, ..., ρ˜m) ≡ max k˜pk1 − 1, (21) ρ(0) the bound in Eq. (23) for candidate states generated by the numerical approaches of Sec.VII, see, e.g., Ref. [54]. is small, it follows that the metastability is classical, with In particular, approaching an effectively classical first- the corrections in Eq. (14) bounded as order dissipative phase transition with a finite m requires the possibility of choosing m candidate states such that m X Ccl(˜ρ1, ..., ρ˜m) → 0 (cf. Sec.IID). Importantly, the con- ρ(t) − plρl . Ccl(˜ρ1, ..., ρ˜m) + C+ + CMM, (22) dition in Eq. (25) is independent from the presence of l=1 weak symmetries (which, nevertheless, can be efficiently 00 0 where t ≤ t ≤ t and . stands for ≤ in the leading incorporated; cf. Sec. VI B 5). order of the corrections (see Sec.C2 in the SM), while In Sec.IV, we show that the metastable phases in pl is chosen as the closest probability distribution to the Eq. (17) are approximately disjoint, while the operators barycentric coordinates, ρl as the closest state to ρ˜l, so in Eq. (20) take the role of basins of attractions. More- that kρ˜l − ρlk ≤ C+ [cf. Eq. (12)], and CMM bounds the over, in Sec.V, we explain how the long-time dynamics approximation of ρ(t) by the projection on the low-lying toward the stationary state corresponds approximately to modes [cf. Eqs. (10) and (19)]. Similarly, the average classical stochastic dynamics between metastable phases. distance to the simplex can be considered (cf. Refs. [8,9] Since corrections in those results depend only on the cor- and see Sec.C2 in the SM). rections to the stationarity, the positivity, and the classi- Finally, the corrections in Eq. (21), which we refer to cality defined in Eqs. (10), (12), and (21), these quantities as the corrections to the classicality, can be efficiently can be viewed as a complete set of figures of merit charac- estimated using the dual basis, terizing classical metastability in open quantum systems. As in the most of this work, we consider a given choice of m X min ˜ m states, we denote C(ρ1, ..., ρm) by C and Ccl(˜ρ1, ..., ρ˜m) Ccl(˜ρ1, ..., ρ˜m) ≤ 2 (−p˜l ) ≡ Ccl(˜ρ1, ..., ρ˜m), (23) by Ccl for simplicity. l=1

min ˜ where p˜l ≤ 0 is the minimum eigenvalue of Pl in Eq. (20), so that IV. CLASSICAL METASTABLE PHASES

C˜cl(˜ρ1, ..., ρ˜m) ≤ m Ccl(˜ρ1, ..., ρ˜m). (24) In Sec.III, we introduced the definition of classical metastability of when MMs of open quantum systems Apart from being easy to compute, C˜cl(˜ρ1, ..., ρ˜m) also can be approximated as probabilistic mixtures of a set of carries the operational meaning of being an upper bound states. We now show that in this case, those states are on the distance of the operators P˜l to the set of POVMs; necessarily metastable and constitute a physical basis of cf. Sec.IVB and see Sec.D1 in the SM. the MM as distinct phases of the system. To this aim, we demonstrate that the probabilities that represent the degrees of freedom in the MM can be accessed with neg- C. Figures of merit ligible disturbance, so that the metastable phases can be distinguished with a negligible error. We further argue From Eq. (21), we obtain a criterion for verification of that their supports and basins of attraction are approxi- whether for a given set of states, the MM can be approx- mately disjoint, in analogy to first-order phase transitions imated as a probabilistic mixture of the corresponding and metastability in classical Markovian systems [7]. Fi- metastable states [Eq. (17)]. In particular, whenever nally, we also discuss how, in the case of any further separation in the low-lying spectrum, later MMs are nec- Ccl(˜ρ1, ..., ρ˜m) 1, (25) essarily classical as well. 10

A. Physical representation of metastable manifold consequences for measurements of system observables. The MM is determined by the coefficients ck of de- We begin by noting that phases given in Eq. (14) are composition into low-lying eigenmodes Lk, k = 2, ..., m uniquely defined up to the so far considered corrections [cf. Eq. (4)], or, equivalently, by the barycentric coor- when the condition in Eq. (16) is fulfilled. Indeed, for dinates p˜l of decomposition into the projections ρ˜l of metastable phases, l = 1, ..., m [cf. Eqs. (17) and (19)]. states ρl in Eq. (14), the distance to their projections ρ˜ in Eq. (17) is bounded by 2(C + C˜ ) when C˜ 1 Furthermore, system states can be probed by POVMs l + cl † P 1 (see Sec.C2 in the SM), so that they are metastable. It (Pl = Pl , Pl ≥ 0, l Pl = ), including von Neumann then follows that their distance to the states chosen in measurements (PkPl = δklPl) corresponding to measuring system observables. Considering the POVM Eq. (22) as closest states to the projections ρ˜l in Eq. (17) ˜ is bounded by . 2(C +C+)+C+. Finally, for two different ˜ min1 Pl − p˜l sets of m metastable phases corresponding to different Pl ≡ , l = 1, ..., m, (26) C˜cl projections in Eq. (17) and the corrections to the classi- 1 + 2 0 cality Ccl and Ccl, which fulfill Eq. (25), the distance in trace norm between the projection of a metastable phase the distance of the probability distribution (p)l ≡ in one set to the closest projection of a metastable phase Tr[Plρ(0)], l = 1, ..., m, to the barycentric coordinates is 0 0 ˜ in the other set is bounded by . Ccl + Ccl + min(Ccl, Ccl) bounded by k˜p − pk1 . Ccl (cf. Eq. (23) and see Sec.D1 (see Sec.D in the SM). in the SM). Therefore, by measuring a metastable state Furthermore, in contrast to the right eigenmodes of ρ(t), the barycentric coordinates can be accessed with ˜ the master operator with Tr(Rk) = 0 for k = 2, ..., m the error up to Ccl + CMM [cf. Eq. (10)], while the state [from L1 = 1 and Tr(LkRl) = δkl], the projections of can be reconstructed by preparing the closest state ρl to metastable phases in Eq. (17) feature normalized trace, ρ˜l upon obtaining lth outcome, with the resulting distur- Tr(˜ρl) = 1, l = 1, ..., m, are Hermitian, and approxi- bance [86] mately positive [see Sec.IIA and cf. Eq. (12)]. Moreover, m when the condition in Eq. (25) is fulfilled, any metastable X kρ(t) − Tr[P ρ(t)] ρ k C˜ + C + C , (27) state is approximated well by their probabilistic mixture l l . cl + MM [cf. Eq. (22) and Fig.2]. Thus, the projections in Eq. (17) l=1 can be considered as physical basis of the MM and ap- where t00 ≤ t ≤ t0. This result should be contrasted with proximate metastable phases. the case of measuring a general system state where no While the left low-lying eigenmodes Lk, k = 2, ..., m, information is available without disturbance (see, e.g., describe quantities conserved in the system during the Ref. [2]). initial relaxation and the metastable regime [cf. Eq. (8) The minimal average error of distinguishing equally 0 for t ≤ t , where bk = δkl for O = Ll] the dual basis probable two states according to Holevo-Helstrom theo- operators in Eq. (20) determine the decomposition of a rem is determined by the distance in the trace norm as metastable state into the basis in Eq. (17), and as such, 1/2 − kρ(1) − ρ(2)k/4 (see, e.g., Ref. [10]). For metastable when the condition in Eq. (25) is fulfilled, they represent states, this error is approximately determined by the dis- approximate basins of attraction for metastable phases tance between their barycentric coordinates as (see also Sec.D1 in the SM). Importantly, via barycentric coordinates in Eq. (35), they define order parameters ˜ (1) (2) (1) (2) Ccl ˜ kρ (t) − ρ (t)k k˜p − ˜p k1 1 − − 2CMM, that distinguish the metastable phases, Tr(Pkρ˜l) = δkl, & 2 with system observable averages being their linear com- (28a) binations [cf. Eq. (8)]. (1) (2) (1) (2) kρ (t) − ρ (t)k . k˜p − ˜p k1 (1 + C+) + 2CMM, Finally, the barycentric coordinates are a physical rep- (28b) resentation of m-1 degrees of freedom present in the metastable regime, as they approximate probability dis- where the first bound corresponds to the error when mea- tributions. As a consequence, we next show that they suring the POVM in Eq. (26). In particular, a pair of are classical from an operational perspective of measur- metastable phases can be distinguished with the error ing the system. ˜ . [min(Ccl/2, Ccl) + C+]/2, since

kρk − ρlk & 2 (1 − Ccl − C+) , k 6= l, (29) B. Classical degrees of freedom for ρl being the closest state to ρ˜l in Eq. (17), which corresponds to a measuring the POVM with two elements: ˜ min1 max min 1 We now argue that in the case of classical metasta- P ≡ (Pl − p˜l )/(˜pl − p˜l ) and − P . For ρl bility, the degrees of freedom determining the MM can that projects on ρ˜l, the bound in Eq. (29) reduces to be accessed with a negligible disturbance of metastable & 2(1 − Ccl). For derivations, see Sec.D2 in the SM. states. It then follows that metastable phases can be Finally, during the metastable regime, t00 ≤ t ≤ t0, the distinguished with a negligible error. We also discuss probability distribution for any measurement of the sys- 11

1 tem is approximated as a probabilistic mixture of prob- that Tr[ Hl ρ(0)] & 1 − 2|1 − p˜l| − Ccl. Furthermore, 1 ability distribution for individual metastable phases, as we have Tr[ Hk ρ(0)] ≤ 2|1 − p˜l| + Ccl for k 6= l, and P 1 ˜ m m 1≤k≤m: k6=l Tr[ Hk ρ(0)] ≤ 2|1 − p˜l| + Ccl. Thus, we X X X Tr[P ρ(t)] − p Tr[P ρ ] ≤ ρ(t) − p ρ , conclude that the basins of attractions are approximately k l k l l l disjoint. We note, however, that in general subspaces H k l=1 l=1 l (30) themselves are not disjoint as they feature states that de- where {P } is a POVM and the right-hand side is cay into multiple metastable phases (it can be shown that k k ˜ bounded by corrections in Eqs. (14) or (22), depending on subspaces spanned by the Pl eigenstates with the eigen- the choice of metastable phases. This conditional struc- values separated from 0 and 1 by distance Ccl + C+, ture of the probability distribution, however, is not di- l = 1, ..., m, can be neglected in the support of metastable rectly related to the classicality, since it is present for phases; see Sec.D3 in the SM). any system state being a probabilistic mixture of not necessarily orthogonal states, while, as we argue next, metastable phases are approximately disjoint. D. Classical hierarchy of metastable phases

C. Approximate disjointness of metastable phases A second metastable regime corresponds to a further separation in the low-lying spectrum of the master operator in Eq. (1), λR /λR 1 with m < m m2 m2+1 2 Below we show that the metastable phases in Eqs. (14) (cf. Ref. [21]). In Sec.F of the SM, we show that and (22) are approximately disjoint, that is, they describe metastable states during the second metastable regime, states restricted to distinct regions of the system space. 00 0 t2 ≤ t ≤ t2, also form a classical MM, i.e., are mixtures of Furthermore, we also find that their basins of attraction 0 00 m2 metastable phases provided that t2 ≥ 2t2 . Those m2 are approximately disjoint. metastable phases are approximately disjoint mixtures of First, note that the distance in the trace norm equals m metastable phases of the first MM, and their supports 2 only for disjoint (mutually orthogonal, states). There- as well as their basins of attraction are approximately fore, the bound in Eq. (29) implies that the metastable disjoint [cf. Eqs. (29)-(33)]. Therefore, each metastable phases are approximately disjoint. This is further corrob- √ phase of the first MM evolves approximately into a single orated by similar bounds on scalar products of ρl or ρl; metastable phase in the second MM, unless the second see Sec.D3 of the SM. MM is not supported on that phase (the phase belongs Second, to capture approximately disjoint supports of to the decay subspace). metastable phases, we consider the subspaces H defined l These results are a direct consequence of long-time dy- as the space spanned by eigenstates of P˜ in Eq. (20) with l namics in a classical MM being well approximated by eigenvalues equal or above 1/2, l = 1, ..., m. We have (see classical stochastic dynamics, which we discuss in next, Sec.D3 in the SM) as metastable states of classical stochastic dynamics are 1 known to be mixtures of as many metastable phases as Tr ( Hk ρl) . Ccl + 2C+, k 6= l (31) 1 the number of low-lying modes [7, 22]. Tr ( Hl ρl) & 1 − Ccl − 2C+, (32) where ρl is the closest state to ρ˜l, k, l = 1, ..., m. Fur- thermore, we also have [cf. Eq. (23)] V. CLASSICAL LONG-TIME DYNAMICS X Tr (1 ρ ) C˜ + 2C . (33) Hk l . cl + The definition of classical metastability in Eq. (14) 1≤k≤m: k6=l determines not only the structure of metastable states. Remarkably, as a consequence of the long-time relax- The bounds in Eqs. (31)-(33) support the statement that ation toward the stationary state effectively taking place the metastable phases reside in approximately disjoint ar- inside the MM, the long-time dynamics is approxi- eas of the state space. The bounds in Eqs. (31)-(33) also mately classical as well. We prove that it corresponds 1 1 hold well for ρ˜l in Eq. (17) as |Tr( Hk ρl)−Tr( Hk ρ˜l)| ≤ to classical stochastic dynamics occurring between dis- C+, while for the states in Eq. (14) that project on ρ˜l, joint metastable phases and can be accessed by mea- ˜ they are further reduced to . Ccl, & 1 − Ccl and . Ccl, suring averages or time-correlations of system observ- respectively. For the bimodal case of open quantum dy- ables whenever metastable phases differ in the averages. namics, m = 2, approximate disjointness was already We also discuss the role played by a further separation argued in Refs. [5, 11]. of timescales in the long-time dynamics, i.e., another Finally, the subspace Hl in Eqs. (31)–(33) captures not metastable regime. Finally, we show that stochastic tran- only majority of ρl support, but by its definition also the sitions between metastable phases can be observed di- corresponding basin of attraction, i.e., the initial states rectly by means of continuous measurements of quanta which evolve into metastable states close to ρ˜l, i.e., with emitted during system interaction with the environment, 1 − p˜l 1. Indeed, in Sec.D3 of the SM, we show provided that metastable phases differ in the average 12 measurement rates. In that case, the statistics of integrated continuous measurement is generally multimodal for times within the metastable regime, which for times after the final relaxation can lead to a high fluctuations rate, reminiscent of the proximity to a dynamical phase transition [59].

A. Classical average dynamics of system and observables

1. Long-time dynamics

From Eq. (5) the evolution for times t ≥ t00 effectively takes place on the MM with the effective generator LMM defined in Eq. (6). This generator can be expressed in the basis of the metastable phases [Eqs. (17) and (20)] as

(W˜ )kl ≡ Tr[P˜kLMM(˜ρl)], (34)

−1 where k, l = 1, .., m, and thus W˜ = C ΛC with (Λ)kl ≡ λkδkl [cf. Eq. (18) and see Fig.3(a)]. The dynamics of the system state within the MM is then determined by the dynamics of the barycentric coordinates

˜p(t) = etW˜ ˜p, (35)

˜ Pm where (˜p)l ≡ Tr[Plρ(0)], so that P[ρ(t)] = l=1[˜p(t)]lρ˜l [cf. Eq. (19) and see Fig.3(b)]. By definition, the long-time evolution in Eq. (6) transforms the MM onto itself, see, e.g., Fig.1(b). This does Figure 3. Classical long-time dynamics: (a) The long- not guarantee, however, that the simplex of m metastable time dynamics [Eq. (5)] can be understood as dynamics be- phases is transformed onto itself, as the evolution may tween metastable phases [Eq. (35)], governed by the trace- ˜ cause states inside the simplex to evolve toward states preserving generator W [Eq. (34)], which can be approx- outside, and thus an initial probability distribution (posi- imated by a classical stochastic generator W [Eqs. (36) and (37)], which is both trace preserving and positive; here a tive barycentric coordinates) acquiring some negative val- negative transition rate from ρ˜2 to ρ˜3 (marked by red cross) ues at later times [see the inset in Fig.3(b)]. Therefore, is put to 0. (b) The long-time dynamics in the barycentric the dynamics generated by W˜ is in general not positive coordinates (cf. Fig.2): green simplex corresponds to t τ 0, [cf. Fig.3(a)]. Nevertheless, as we discuss below, when blue to t = τ 0, and red to t = τ, while the stationary state is the simplex of metastable phases is a good approximation marked by red sphere. Positive dynamics corresponds to the for the MM in the sense of the condition in Eq. (25), W˜ is simplex of metastable phases mapped onto itself, which re- well approximated by a generator of stochastic classical quires all metastable phases to be mapped inside the simplex dynamics between metastable phases. at all times. Here, ρ˜2 initially acquires a negative probability p˜3(t) at small t [red cross in the inset; cf. W˜ in panel (a)]. (c) Approximating by W alters the dynamics, with corrections increasing in time [Eq. (39)]; blue dashed simplex corresponds 2. Classical generator to t = τ 0 and orange dashed to t = τ [cf. panel (b)]. This ul- timately leads to a different stationary state (yellow sphere) Dynamics generated by W˜ conserves the probabil- [cf. Eq. (41)], which is close to the true stationary state when Pm ˜ 1 Pm ˜ Eq. (40) is fulfilled. ity, as from k=1 Pk = we have k=1(W)kl = 0 (cf. Sec.E1 in the SM). Furthermore, it can be shown to be approximately positive, with W˜ approximated by k, l = 1, ..., m, as the closest classical stochastic generator W (cf. Fig.3(a) ˜ and see Secs.E2a andE2b in the SM), kW − Wk1 p . 2 Ccl, (37) kW˜ k1 (W)kl ≡ max[(W˜ )kl, 0], k 6= l, (36) X where the norm kXk ≡ max Pm |(X) | [90] and (W)ll ≡ (W˜ )ll + min[(W˜ )kl, 0], 1 1≤l≤m k=1 kl ˜ ˜ k6=l (1 − C+ − Ccl)kLMMk . kWk1 ≤ (1 + Ccl/2)kLMMk (see 13

Sec.D2 in the SM). From Eq. (37) the normalized dis- Similarly, not only the stationary state but all eigen- tance between the generators is bounded as modes of the long-time dynamics in the MM can be approximated by those of the classical stochastic dynamics. ˜ kW − Wk1 p In particular, in Sec.E2 of the SM, we discuss approx- ∆+ ≡ . Ccl. (38) ˜ kW˜ k1 + kWk1 imation of the pseudoinverse of W in Eq. (34) by the pseudoinverse of W in Eq. (36), a result which plays an Note that columns of W sum to 0, and then negativity important role in the approximation of quantum trajec- of its diagonal terms follows from the positivity of the tory statistics that we discuss in Sec.VB. off-diagonal terms, so that dynamics generated by W is We note that the quality of the classical approxima- indeed positive and probability-conserving (cf. Sec.E1 tions for the structure of the long-time dynamics depends in the SM). For the bimodal case of m = 2, the MM is ˜ not only on the corrections Ccl within the metastability always classical with Ccl = 0, and thus W = W is exactly regime [Eq. (21)], but also on the timescale of the fi- a generator of stochastic classical dynamics [11]. nal relaxation (cf. Eq. (41) and Secs.E2d andE2e in the SM). This is due to the fact that the approximation in Eq. (37) captures the fastest among the low-lying modes, 3. Classical system dynamics while the final relaxation timescale is governed by the slowest among them. In particular, in the case of another We now discuss how the dynamics generated by W˜ is metastable regime [21], which corresponds to further sep- approximated by the classical dynamics generated by W. aration in the spectrum of the master operator in Eq. (1), We also discuss conditions for the stationary state to be the condition in Eq. (40) may generally not be valid. For approximated in terms of stationary distribution of W. example, when a classical first-order phase transition oc- In Sec.E2c of the SM, we show it follows from Eq. (37) curs at finite system size, in its proximity 1/(τkW˜ k1) is that finite when the degeneracy of m stable phases is lifted in the same order 1/(τkW˜ k ), so that time t can be chosen tW˜ tW p √1 ke − e k1 2 Ccl t kW˜ k1. (39) . τkW˜ k1 tkW˜ k1 1/ Ccl leading to k˜pss −pssk1 1. √ But when the perturbation away from the transition lifts Therefore, for times tkW˜ k 1/ C the effective dy- 1 cl the degeneracy of m phases in several different orders (so namics in the MM is well approximated by the classi- that C is of a lower order in the perturbation than 1/τ), tW˜ tW cl cal dynamics, as k˜p(t) − p(t)k1 ≤ ke − e k1k˜pk1 . Eq. (40) is no longer fulfilled (see Sec.A2 in the SM). tW˜ tW tW ke −e k1 [where p(t) = e ˜p, cf. Eq. (35); ˜p can fur- We discuss next how the approximation of the long-time ther replaced by the closest probability distribution with dynamics by classical stochastic dynamics can be refined additional corrections bounded by Ccl, which are of the to take into account the hierarchy of metastabilities. higher-order for times after the metastable regime, e.g., Finally, we note it is also possible to approximate etW˜ t ≥ 1/kW˜ k1]; see Fig.3(c). This also holds true for the corresponding density matrices (see Sec.D2 of the SM). by discrete classical dynamics. This again leads to cor- When the approximation in Eq. (39) holds for times rections scaling√ linearly in time, but proportional to Ccl after the relaxation in the MM, which requires rather than Ccl; see Sec.E2f in the SM.

1 τkW˜ k1 √ , (40) Ccl 4. Hierarchy of classical long-time dynamics the stationary state ρss described within the MM by (˜pss)k = Tr(P˜kρss) is well approximated by the stationary probability pss of the classical dynamics W [91]; When there exists a second metastable regime in cf. Fig.3(c). Indeed, 00 0 the system dynamics, t2 ≤ t ≤ t2, the corresponding metastable states of the system are simply approxi- ˜ tW˜ p ˜ k˜pss − pssk1 . kPss − e k1 + 2 Ccl t kWk1, (41) mated by the projection on the low-lying modes of the classical√ stochastic dynamics in Eq. (36) provided that where P˜ denotes the projection on ˜p . Therefore, 00 0 00 ss ss √ t2 1/ Ccl [cf. Eqs. (4) and (39)]. When, t2 ≥ 2t2 , after k˜p − p k 1 follows provided that tkW˜ k 1/ C 0 ss ss 1 1 cl the second metastable regime, t ≥ t2, the system dynam- tW˜ for t such that kP˜ss − e k1 1 [92]. As a corol- ics toward the stationary state is approximated by clas- lary of Eq. (41), the stationary probability distribution sical dynamics taking place only between m2 metastable pss of classical dynamical generator W in Eq. (36) is phases of the second MM [cf. Eqs. (4) and (39), and unique. Thus, the classical dynamics is ergodic with the see Sec.IVD]. Moreover, when that approximation holds average time spent in lth metastable phase equal (pss)l, also after the ﬁnal relaxation, the system stationary state l = 1, ..., m. Furthermore, the approximation also holds ρss is well approximated by the stationary distribution of true for the distance in the trace norm of the correspond- that classical dynamics [cf. Eq. (41)]. For further discus- ing density matrices (see Sec.D2 in the SM). sion, see Sec.F in the SM. 14

5. Classical observable dynamics dynamics generated by a classical stochastic generator. Now we argue that this relation pertains to individual ex- We now argue how at times after the initial relaxation, perimental realizations of system evolution [58]. There- the classical long-time dynamics can be observed in the fore, stochastic transitions between metastable phases behavior of expectation values or autocorrelations of gen- can be observed in continuous measurement records or eral system observables. In particular, it can be directly the system state sampled in QJMC simulations [8–12] so accessed by measuring the dual basis in Eq. (20). called quantum trajectories (see Fig.4). First, we show For times t ≥ t00, the dynamics of the average for an that statistics of quantum trajectories can be directly re- observable O depends only on the evolution of the distri- lated to the statistics of classical stochastic trajectories. bution between the metastable phases, Second, we argue how coarse-graining in time returns classical trajectories between metastable phases, which hO(t)i = o˜T etW˜ ˜p + ... = o˜T ˜p(t) + ... for metastable phases differing in activity is the mech- = o˜T etW ˜p + ... = o˜T p(t) + ..., (42) anism behind the phenomena of intermittence [24, 59] and dynamical heterogeneity [50, 51], and leads to mul- where (o˜)l = Tr(Oρ˜l), l = 1, ..., m, are the averages of timodal distribution of integrated measurement records the observable O in the metastable phases. The first line during the metastable regime. Finally, we explain how corresponds to Eq. (8), while the second line follows from system metastability can manifest itself as proximity to Eq. (39) introducing additional corrections bounded in a first-order dynamical phase transition in the ensemble √ the leading order by 2tkW˜ k1 Ccl max1≤l≤m |(o˜)l|. of quantum trajectories [59]. Similarly, the autocorrelation

2 T tW˜ ˜ T 2 1. Statistics of quantum trajectories hO(t)O(0)iss − hOiss = o˜ e O ˜pss − (o˜ ˜pss) + ... T tW ˜ T 2 = o˜ e O pss − (o˜ pss) + ...,(43) Quantum trajectories describe the system state condi- ˜ where (O)kl = Tr[P˜kO(˜ρl)] (cf. Ref. [11]). The first tioned on a continuous measurement record, e.g., count- line corresponds to Eq. (9), while the second line follows ing or homodyne measurement of photons emitted by the from Eq. (39) with the additional corrections bounded system due to action of jump operators in Eq. (1). In par- √ ˜ by max1≤l≤m |(o˜)l[|kOk1(2tkW˜ k1 Ccl + k˜pss − pssk1) + ticular, the statistics of the total number of jumps in a 2|hOiss| k˜pss − pssk1] in the leading order. quantum trajectory (total number of detected photons) is Therefore, when metastable phases differ in observable encoded by the biased or “tilted” master operator [24, 59] averages [up to the correction in Eq. (8)], the long-time −s dynamics can be observed by measuring the observable Ls(ρ) = L(ρ) + e − 1 J (ρ), (44) average or autocorrelation. For example, for an observ- P † where J (ρ) ≡ j JjρJj . That is, Θ(s, t) ≡ able chosen as a dual basis operator O = P˜l in Eq. (20), T ln(Tr{etLs [ρ(0)]}) is the cumulant generating function we simply have o˜ p(t) = [p(t)]l, l = 1, ..., m. Further- more, when the approximation in Eq. (39) holds for times for the number K(t) of jumps that occurred until time t after the final relaxation [cf. Eq. (40)], the dynamics of for quantum trajectories initialized in ρ(0). The rates of averages and autocorrelations of all system observables the asymptotic statistics are determined then by θ(s) ≡ is effectively classical [cf. Eq. (41)]. In particular, if the limt→∞ Θ(s, t)/t, which is simply the eigenvalue of Ls measurement of an observable O is noninvasive, i.e., does with the largest real part. We denote the associated not disrupt basins of attractions of metastable phases, (positive) eigenmatrix as ρss(s) and choose the normal- ˜ ization Tr[ρss(s)] = 1, so that θ(s) = Tr{Ls[ρss(s)]} = (O)kl ≈ δkl(o˜)l, k, l = 1, ..., m, the long-time dynam- −s P † ics leads to the decay of the autocorrelations exactly as (e − 1) j Tr[Jj Jjρss(s)]. Then, ρss(s) is the aver- the decay of the autocorrelation of o˜ in the classical dy- age asymptotic state of the system in trajectories with −sK(t) namics: from the observable o˜ variance in pss during the the probability biased by the factor e , while the metastable regime, toward 0 achieved after the final re- derivatives of θ(s) correspond to asymptotic rate of the laxation (cf. Ref. [93]). This is the case for the dual basis corresponding cumulants. In particular, for a unique sta- operators in Eq. (20), with the distance between the ma- tionary state ρss and the bias |s| small enough with re- ˜ spect to the gap −Re(λ2), ρss(s) = ρss + ..., while trices bounded by . 3(C+ + Ccl + Ccl/2) (see Sec.D1c in the SM). Finally, higher-order correlations, also between −s θ(s) = (e − 1) µss + ..., (45) different observables, can be analogously approximated d by correlations in classical dynamics. k(s) ≡ − θ(s) = e−s µ + ... (46) ds ss P † where µss ≡ j Tr[Jj Jjρss] (cf. Ref. [4]), so that the B. Classical characteristics of quantum trajectories asymptotic jump rate—the asymptotic activity—is determined by the stationary state. In Sec.VA, we showed that the dynamics of the aver- The nonanalyticities of θ(s) can be recognized as dy- age system state can be approximated with the classical namical phase transitions [59], in analogy to nonanalytic- 15 ities of the free energy in equilibrium statistical mechan- quantum trajectories is approximated by the asymptotic ics. In particular, a first-order dynamical phase tran- probability distribution in biased classical trajectories, sition occurs at s for which the maximal eigenvalue c m of L is not unique, so that the asymptotic activity X sc ρ (s) = [p (s)] ρ˜ + ..., (50) k(s) = −dθ(s)/ds is no longer continuous, but features a ss ss l l l=1 jump at sc [24, 51, 59, 63]. Similarly, statistics for integrated homodyne current where pss(s) is the maximal eigenmode of Ws [see and for time-integral of system observables are consid- Fig.4(a)], while the asymptotic rate of the cumulant gen- ered [12–17] (see also Secs.E3b andE3c in the SM). erating function is approximated via the asymptotic total activity in biased classical trajectories,

2. Classical tilted generator m −s X tot θ(s) = (e − 1) [pss(s)]l µ˜l + ..., (51) We now present our first result regarding classicality l=1 of quantum trajectories. We argue that the tilted mas- tot Pm in where µ˜l ≡ k=1(J)kl +µ ˜l , l = 1, ..., m, are the aver- ter operator in Eq. (44) can be approximated by a tilted age total activities in the metastable phases (cf. Sec.E3f classical generator encoding the statistics in stochastic in the SM), so that the right-hand side of Eq. (51) trajectories of the classical dynamics in Eq. (36). This is the maximal eigenvalue of Ws [see Fig.4(b)]. For leads to classical approximations for the asymptotic rate corrections from non-Hermitian perturbation theory [4], of the cumulant generating function and for the asymp- see Sec. E3a in the SM. One can consider similarly totic system state in biased quantum trajectories. approximating dθ(s)/ds and d2θ(s)/ds2, but additional The statistics of total activity [101–103] in classical dy- contributions arise from non-Poissonian fluctuations in namics is encoded by a biased or “tilted” classical genera- metastable phases (see Sec. VB3 and Sec. E3a in tor [for reviews see Refs. [104, 105]; cf. Eqs. (36) and (44)] the SM). Nevertheless, those can be neglected when the internal activities of metastable phases dominate transi- −s in Ws = W + (e − 1) (J + µ˜ ), (47) tion rates of classical dynamics [see Figs.4(c) and4(d)], making Eq. (49) the crucial result in the link between the where (J)kl ≡ (1 − δkl)(W)kl, k, l = 1, ..., m encodes system metastability and its proximity to a first-order dy- the transition rates in the classical dynamics, while namical phase transition, which we discuss in Sec.VB5. in ˜ P † (µ˜ )kl ≡ δkl[˜µl + (W)ll] with µ˜l ≡ j Tr(Jj Jj ρ˜l), In Secs.E3b andE3c of the SM, we show that in the k, l = 1, ..., m, encodes the average internal activity in presence of classical metastability generators of statis- metastable phases (which here is assumed Poissonian dis- tics of integrated homodyne current and time-integral of tributed; cf. Sec.E1 in the SM). system observables can be similarly linked to generators In Sec.E3d of the SM, we show that the tilted classi- of statistics in classical trajectories, but with respect to cal generator Ws approximates, in the metastable phase time-integrals of their average value in metastable phases. basis, the tiled master operator Ls when the latter is restricted to the low-lying modes [cf. Eq. (34)], 3. Classical cumulants (W˜ s)kl ≡ Tr[P˜kLs(˜ρl)], (48) We now discuss how the dynamics of the first and k, l = 1, ..., m, with the corrections bounded as [106] the second cumulants of the jump number, directly p kW˜ − W k 2e−skW˜ k C (49) accessible in experiments via counting measurement, s s . 1 cl are governed by the classical long-time dynamics for i −s X † p times after the initial relaxation. In particular, we argue + e − 1 m H + Jj Jj 2Ccl + 4C+. 2 max j how the asymptotic activity and fluctuation rate are approximated by the total activity and total fluctuation For dynamics of classical systems with metastability, or, rates in the classical dynamics. These results establish more generally, for the basis of metastable phases in a further correspondence between statistics of quantum Eq. (17) commuting with the dual basis in Eq. (20), and classical trajectories for times during and after the p ˜ m 2Ccl + 4C+ in Eq. (49) can be reduced to 2(Ccl + C+). metastable regime, and, even asymptotically, they do not Since the biased dynamics Ls in Eq. (44) can be con- directly follow from Eq. (51) as cumulants are encoded sidered as the perturbation of the master operator L in by derivatives of the rate function [cf. Figs.4(c) and4(d)]. Eq. (1) with (e−s−1)J , for bias |s| much smaller than the separation to the fast eigenmodes, λR −λR , the m low- 00 m m+1 Classical dynamics of√ first cumulant. For times t ≥ t lying eigenmodes and eigenvalues of Ls in Eq. (44) are such that tkW˜ k1 1/ Ccl, the rate of average jump approximated by those of PLsP [4] or, in the metastable number is approximated by the time-integral of the total phase basis, those of W˜ s in Eq. (48). From Eq. (49), it activity in classical trajectories, Kcl(t), whose statistics then follows that the average asymptotic state in biased in encoded by Ws of Eq. (47), together with the constant 16

Figure 4. Classical features of quantum trajectories: (a,b) Approximating the biased operator of jump activity Ls [Eq. (44)] by the biased classical generator Ws [Eq. (47)] gives the approximations of: ρss(s) (black solid) by the maximal eigenvector of Ws (green short-dashed) [panel (a); we plot Tr[P˜lρss(s)], l = 1, ..., m; cf. Eq. (20)], θ(s) by the cumulant generating function of total activity in classical trajectories [panel (b)], which are valid for s in the perturbative regime of the statistics captured by m = 4 slow modes (blue dashed) [Eq. (48)], as given by Eqs. (50) and (51). (c,d) Similarly, −s P † −s P † −dθ(s)/ds = e j Tr[Jj Jj ρss(s)] − (e − 1) j Tr[Jj Jj dρss(s)/ds] (black solid), with the first term being the activity of ρss(s) (red solid), is captured by the first derivative of the classical cumulant generating function (green short-dashed), up to non-Poissonian contribution to fluctuations in metastable phases [cf. Eq. (54)]. This contribution can be neglected for internal activity dominating classical dynamics, in which case the leading contribution to fluctuations is the result of long timescales of mixing between metastable phases rather than fluctuations within, as demonstrated in panel (d). (e,f) Not only asymptotically, but already for times after the initial relaxation, the average rate and the fluctuation rate of jump number K(t) (black solid) can be approximated by the constant contribution K˜ from before the metastable regime and the contribution from the dynamics within the MM (blue dashed), with the latter approximated by the corresponding rate for classical total activity Kcl(t) (green short-dashed) [cf. Eqs. (52) and (54)]. (g) Coarse-graining of jump records (top) in time gives values close to metastable phases activity (upper center) [cf. panel (c)], up to fluctuations which decrease with grain size (here δt = 0.7τ 0; cf. Sec.VB4). In turn, they capture the average activity of the conditional system state |ψ(t)i (lower center; we plot running average over δt), and the metastable phase support where |ψ(t)i is found [bottom; we plot hψ(t)|P˜l|ψ(t)i, l = 1, 3]. contribution K˜ to the jump number accumulated before When the approximation in Eq. (52) holds for time the metastable regime (cf. Fig.4(e) and see Sec. E3e t after the final relaxation [cf. Fig.3(c)], the asymp- of the SM), totic activity in quantum trajectories is approximated by the asymptotic total activity of classical trajectories hK(t)i hK (t)i K˜ = cl + + ..., (52) [cf. Fig.4(e)] t t t hK(t)i Z t m X † 1 X Tr{J SQ[ρ(0)]} µss ≡ lim = Tr(Jj Jjρss) tot t→∞ t ≡ dt1 µ˜ p(t1) l − +..., j t 0 t l=1 m hKcl(t)i X tot tot tot tW = lim + ... = µ˜ pss + ..., (53) where (µ˜ )kl ≡ δklµ˜l , p(t) = e ˜p, S is the pseudoin- t→∞ t l verse of the master operator L in Eq. (1), Q ≡ I−P is the l=1 projection on the fast-modes of the dynamics [cf. Eq. (4)]. where pss is the stationary distribution of the clas- Therefore, similarly as for averages of system observables sical dynamics W. The corrections are√ bounded ˜ [Eq. (42)], the classical dynamics can be observed by mea- by . max1≤l≤m|µ˜l|k˜pss − pssk1 + kWk Ccl [as Pm suring hK(t)i when the metastable phases differ in the Tr[J (ρss)] = l=1(µ˜˜pss)l; cf. Eqs. (41) and (37)]. total activity. 17

Classical dynamics of second cumulant. For times phases (cf. Fig.4(f) and see Secs.B1 andE3g in the SM) such that kSQkkJ k tkµ˜k1 and tkW˜ k1 2 2 √ q √ 2 hhKcl(t)i−hKcl(t)i h∆cl(t)ii min(1/ C , kR˜ k kW˜ k / C ), where R˜ denotes the σss = lim + + ... (58) cl 1 1 cl t→∞ t t pseudoinverse of the long-time-dynamics generator W˜ in m X tot in in 2 Eq. (34), the rate of fluctuations of jump number is ap- = µ˜ −2 J+µ˜ R J+µ˜ +δσ˜ pss l +... proximated by the rate of fluctuations of total activity in l=1 classical trajectories, corrected by non-Poissonian fluc- m X tot tot tot 2 tuations in metastable phases and by the contribution = µ˜ −2µ˜ R˜µ +δσ˜ pss l +..., to the average from before the metastable regime (see l=1 Fig.4(f) and Sec.E3e in the SM) where R denotes the pseudoinverse of the classical stochastic generator W in Eq. (36) and the last equality 2 2 2 2 in tot hK (t)i−hK(t)i hKcl(t)i−hKcl(t)i h∆cl(t)i follows by noting that J + µ˜ = W + µ˜ and thus = + (54) in tot t t t (J + µ˜ )pss = µ˜ pss. m (k) (l) X hK (t)i − hK (t)i − p˜ p˜ cl cl K˜ − K˜ + ..., k l t k l Other statistics. Similarly to Eqs. (52), (54), (53) k,l=1 and (58), the first and the second cumulants for integrated homodyne current or for time-integrals of system where observables can be related to the statistics in classical dynamics with respect to observables given by the corre- Z t m sponding averages for metastable phases (see Secs.E3b 2 X tot hKcl(t)i = dt1 µ˜ p(t1) l (55) andE3c in the SM). Furthermore, the integrals of av- 0 l=1 erage and autocorrelations of system measurements in m Z t Z t−t1 Eqs. (42) and (43) can be approximated analogously. X in t2W in +2 dt1 dt2 [(J+µ˜ )e (J+µ˜ )p(t1)]l, 0 0 l=1 4. Classical dynamics of quantum trajectories and we denoted For systems exhibiting metastability in the system dy- Z t m X namics, individual evolutions of the system over time h∆cl(t)i ≡ dt1 δ 2 p(t1) , (56) σ˜ l typically exhibit intermittence (distinct periods of jump 0 l=1 activity isolated in time) or dynamical heterogeneity (distinct periods of jump activity isolated both in time with (δσ˜2 )kl ≡ −δkl 2Tr[J SQJ (˜ρl)], k, l = 1, ..., m, so tot 2 tot and space) in the emission measurement record or time- that (σ˜ ) ≡ µ˜ +δ 2 are rates of total fluctuations in σ˜ integral of observables [see Fig.1(e)]. We now explain (l) metastable phases (see Sec.E3f in the SM), hKcl (t)i is that these features can be understood in terms of clas- the average of Kcl(t) for lth metastable phase in Eq. (52), sical dynamics between the metastable phases whose i.e., (˜p)k = δkl, and K˜l ≡ Tr{P˜lJSQ[ρ(0)]}/Tr[P˜lρ(0)] differences in internal (global or local) jump activity is the contribution to the jump number from before the dominate transition rates of that dynamics (see also metastable regime conditioned on the metastable phase Refs. [11, 20]). To this aim, using results of Sec.VB3, we that the system evolves into. Therefore, similarly as establish a direct relation between classical trajectories for autocorrelations of system observables [Eq. (43)], the and time-coarse-grained records of continuous measure- classical dynamics can be observed even for the station- ments. Therefore, we prove that metastability can be ary state, by measuring fluctuations of K(t) whenever observed not only on average [cf. Eqs. (42) and (43)], the metastable phases differ in the total activity. but also in individual realizations of continuous mea- When the approximation in Eq. (54) is valid for times surement experiments and individual samples of QJMC after the final relaxation, the asymptotic fluctuation simulations provided that the metastable regime is long rate [5, 107] enough. As a corollary of our results, integrated continuous 2 2 measurements can be used to distinguish metastable 2 hK (t)i−hK(t)i σss ≡ lim (57) phases during the metastable regime, as their distribu- t→∞ t tion is multimodal with distinct modes corresponding to X † = Tr(Jj Jjρss) − 2Tr[JSJ (ρss)] the metastable phases differing in the rate of the mea- j surement average. Interestingly, during the metastable regime all continuous measurements lead to negligible is approximated by the asymptotic rate of fluctuations disturbance of the system as on average they simply of total activity in classical trajectories, corrected by correspond to the system dynamics, so that the dis- non-Poissonian contribution to fluctuations in metastable turbance is bounded by 2CMM [cf. Eq. (10) and Sec.IVB]. 18

and variances inversely proportional to δt, l = 1, ..., m. Time-coarse-grained measurement records as classical This is proved in Sec.E3h of the SM, by postselecting trajectories. We focus here on the measurement of total trajectories in terms of probability of the final state in number of jumps that occur in the system, but analogous the time bin, ρcond(t + δt), evolving (on average) into arguments hold for the measurements of local jump ac- a metastable phase ρ˜l, which, formally, corresponds to tivity (see Sec.E3a in the SM) and of homodyne current performing at t + δt the measurement in Eq. (26) that (cf. Sec.E3b in the SM). approximates P˜l in Eq. (20) [cf. Fig.4(g) (bottom)]. Consider course-graining in time of a record of jump We conclude that, for long enough δt, the activity counting measurement, with the activity in time bins δt k(n), n = 0, 1, 2, ..., takes in typical measurement records in K[(n+1)δt] − K(nδt) only m values µ˜l , l = 1, ..., m, corresponding to the in- k(n) ≡ , (59) δt ternal activities of m metastable phases (approximately, up to fluctuations decaying inversely in δt). For the for n = 0, 1, 2, .... We argue that time-coarse-grained bimodal case m = 2, see also Ref. [11]. measurement records can be interpreted as classical trajectories between metastable phases when the internal ac- Dynamics of time-coarse-grained measurement records tivity dominates the long time dynamics, kµ˜k1 kW˜ k1, and δt is chosen long enough within the metastable as classical long-time dynamics. We now argue that tran- regime, as in this case the activity typically attains only sitions in coarse-grained measurement records are cap- values of the internal activities in metastable phases [see tured by the generator of the effective long-time dynam- Fig.4(g)]. ics W [Eq. (36)]. In particular, the effective lifetime of From Eq. (52), for δt ≤ t0 such that kSQkkJ k the lth metastable phase in coarse-grained trajectories is √ approximated by τ ≡ −1/(W) , l = 1, ..., m. δtkµ˜k and tkW˜ k 1/ C , the average activity in l ll 1 1 cl From the discussion above, for an initial state ρ(0), trajectories originating in ρcond(t) at time t = nδt is approximated as the distribution of activity k(0) can be approximated, up to small fluctuations, by a probability distribution m in X in over metastable phase activities µ˜l , with probabilities hk(n)iρcond(t) = µ˜l p˜l(n) + ..., (60) approximated by p˜l = Tr[P˜lρ(0)], l = 1, ..., m. Analo- l=1 gously, the distribution of the activity k(n) in a later ˜ δtW˜ n where p˜l(n) ≡ Tr[Plρcond(t)] determines the metastable nth time bin is approximated by p˜k(n) = [(e ) ˜p]k, state. Similarly, from Eq. (54), the variance where t = nδt, which is further approximated by δtW n m tot 2 [(e ) ˜p]k, k = 1, ..., m (cf. Eq. (39); corrections 2 2 X (˜σl ) hk (n)iρ (t) −hk(n)i = p˜l(n) (61) can be further reduced to nCcl by considering discrete cond ρcond(t) δt l=1 stochastic dynamics; see Sec.E2f in the SM). Therefore, m in in 2 the transition matrix, i.e., the probability of observing X (˜µk −µ˜l ) in + p˜k(n)p ˜l(n) k(n) ≈ µ˜k conditioned on the observation of the initial 2 in k,l=1 activity k(0) ≈ µ˜l , is approximated by the classical δtW m in in ˜ ˜ dynamics transition matrix (e )kl (or a discrete X (˜µ −µ˜ )(Kk −Kl) + p˜ (n)p ˜ (n) k l + ... stochastic dynamics; see Sec. E2f in the SM). This k l δt k,l=1 relation is further corroborated by Eqs. (52) and (54), with the average and variance of the integrated activity, (by assuming (1 + C˜ /2)C 1; see Sec. E3f in cl MM Pbt/δtc k(n) = K(bt/δtcδt)/δt, approximately governed the SM). n=0 by the classical long-time dynamics W [cf. Figs.4(e) When the conditional state evolves into a single and4(f)]. metastable phase P[ρcond(t)] =ρ ˜l, the average activity is in approximated by the activity µ˜l of lth metastable phase tot 2 [cf. Fig.4(e)] and its variance (˜σl ) /δt decays inversely with the increasing time-bin length δt [cf. Fig.4(f)]. Therefore, for long enough metastable regime, δt can 5. Classical metastability and dynamical phase transitions be chosen so that the fluctuations between measurement records become negligible, and the activity typi- in Finally, we explain how classical metastability can cally takes values approximately equal the average µ˜l [cf. Fig.4(g) (center)]. For ρcond(t) evolving into a mix- manifest itself as proximity to a first-order dynamical ture of metastable phases, however, a constant term is phase transition in the ensemble of quantum trajecto- present in Eq. (61) because of a multimodal distribution ries [59], i.e., to a first-order nonanalyticity of θ(s). of the activity number in nth time bin. Namely, when C˜cl 1, the distribution can be approximated, up to Metastable phases as eigenmodes of tilted generator. corrections 2(CMM + C˜cl) + Ccl, as a mixture, with prob- Building on the results of Sec.VB2, when the differences abilities approximating p˜l(n), of distributions with av- in the activity of the metastable phases dominate the in erages equal internal activities of metastable phase, µ˜l , transition rates of the classical dynamics between them, 19 we can approximate W˜ s in Eq. (48) as of long timescales of the effective classical dynamics between metastable phases which govern the intermittence ˜ in Ws = W − hsµ˜ + ... ≡ Whs + ..., (62) in emission records [20, 24], and are captured by the re- −s solvent in the second term of Eq. (66). In contrast, when where hs ≡ 1−e and Whs encodes the statistics of the in the stationary state corresponds to a single metastable time-integral of the observable µ˜ in classical trajecto- in phase (so that R˜µ pss ∝ Rpss = 0), the fluctuation rate ries, rather than their activity (cf. Sec.E1 in the SM). is finite as fluctuations originate inside that metastable The corrections in Eq. (62) additional to Eq. (49) are phase alone [up to corrections of Eq. (66)]. A large sec- ˜ in tot . hskWk1/2 [replacing µ˜ by µ˜ or µ˜ doubles them]. ond derivative of θ(s) occurs then away from s = 0 at Furthermore, for the bias large enough, so that hs is fi- intermediate (negative or positive) s values. nite, the contribution from W in Whs of Eq. (62) can be In terms of phase-transition phenomenology, the neglected. In that case, if the bias is still negligible with proximity of a first-order dynamical phase transition R R respect to the gap to the fast eigenmodes, λm − λm+1, manifests itself in a multimodal distribution of a dy- m low-lying eigenmodes of Ls are simply approximated namical quantity (i.e., the jump number) in (biased) by the metastable phases and the corresponding eigen- trajectories for times within the metastability regime, values and their derivatives approximated analogously to while at longer times in the coexistence, within individ- Eqs. (45) and (46). In particular, the maximal eigenmode ual trajectories, of active and inactive regimes that can corresponds to the metastable phase with the maximum be considered as dynamical phases [cf. Fig.4(g)]. These (for s < 0) or minimum activity (for s > 0), and dynamical phases correspond directly to metastable −s phases (cf. Sec.VB4). θ(s) = (e − 1)µ ˜l(s) + ..., (63) −s k(s) = e µ˜l(s) + ..., (64) Other statistics. Similar results to Eqs. (63) and (64), and thus the relation of metastability to dynamical phase where transitions, also follow for: individual jump activity (see ( in Sec.E3a in the SM), integrated homodyne current (see max1≤l≤m µ˜l , s < 0, µ˜l(s) ≡ in (65) Sec.E3b in the SM and cf. Ref. [12]) and time-integrals min1≤l≤m µ˜l , s > 0. of system observables (see Sec. E3c in the SM and cf. Ref. [17]). (cf. Figs.4(a),4(b), and4(c), and see Sec. E3a in the SM for corrections from non-Hermitian perturbation theory [4]). This is a key observation for the numerical VI. CLASSICAL WEAK SYMMETRIES method we introduce in Sec.VIIB to find metastable phases as well as for the relation of metastability to dynamical phase transitions we explain next. Here, we discuss how weak symmetries, i.e., symmetries of the master operator in Eq. (1), are inherited by Metastability as proximity to first-order dynamical MMs and long-time dynamics. For classical metastabil- phase transitions. For metastable phases differing in ity, we find that nontrivial symmetries are necessarily activity (or observable averages or homodyne current), discrete as they correspond to classical symmetries, i.e., Eq. (64) implies a sharp change in the derivative of θ(s), approximate permutations of metastable phases, which i.e., −k(s), close to s = 0 [see Fig.4(c)]. This sharp are inherited by the classical long-time dynamics. Since change can be interpreted as the proximity to a first- first-order dissipative phase transitions occurring in ther- order dynamical phase transition [24, 51, 59, 63]. An modynamic limit manifest themselves as metastability analogous argument was made for the classical Marko- for finite system size, our results pave a way for under- vian dynamics in Ref. [39]. standing symmetry breaking in open quantum systems A sharp change in k(s) around s = 0, implies in turn (see also Ref. [42]). a large second derivative of θ(s) [see Fig.4(d)]. In par- By a weak symmetry we refer to the generator of the ticular, d2θ(s)/ds2 at s = 0 determines the rate of fluc- system dynamics L obeying a symmetry on the master tuations in jump number, which can be approximated as operator level, [cf. Eqs. (58) and (62)] [L, U] = 0, (67) m 2 X h in 2 in in i † σss = (σ˜ ) pss − 2µ˜ R˜µ pss + ..., (66) where U(ρ) ≡ Uρ U with a unitary operator U of the l l=1 symmetry (see Refs. [1, 23, 24]). As we consider a unique stationary state, we are interested in the case when the in 2 in where (σ˜ ) ≡ µ˜ + δσ˜2 and the additional corrections symmetry operator U is not itself conserved by the dy- † are bounded by kW˜ k1(1 + 2 max1≤l≤m |µ˜l|kR˜ k1). The namics, so that in general L (U) 6= 0 (as the number of fluctuation rate is indeed large for the stationary state distinct stationary states is the same as the number of being a mixture of metastable phases with different ac- linearly independent conserved quantities [1, 24]). For tivities [cf. Figs.4(a) and4(d)]. This is a consequence example, U can describe the translation symmetry in ho- 20 mogeneous dissipative systems with periodic boundary B. Symmetry and classical metastability conditions. From Eq. (67) it follows that L is block diagonal in We now explain how weak symmetries for classical the operator basis of eigenmatrices of U. Therefore, metastability necessarily correspond to approximate per- † the eigenmatrices of L, Rk (and Lk of L ) can be si- mutations of metastable phases and thus any nontriv- multaneously chosen as eigenmatrices of U (and U †), in ial continuous weak symmetries of low-lying modes pre- iδφ † iφ which case U(Rk) = e k Rk [U (Lk) = e k Lk], where clude classical metastability. We also show the set of φk equals a difference in arguments of U eigenvalues metastable phases can be chosen invariant under the sym- (mod 2π) (cf. Fig.5 and see Sec.G1 in the SM). metry, in which case, the sets of supports and basins of attractions of metastable phases are also invariant. Fur- thermore, both the long-time dynamics and its classi- A. Symmetry and general metastability cal approximation are then symmetric with respect to the corresponding permutation. This restricts the structure of the low-lying eigenmodes, including the station- We first discuss how a weak symmetry in Eq. (67) af- ary state, and, in turn, simplifies the test of classicality fects the structure of a general MM and the long-time introduced in Sec.III. dynamics within it.

1. Approximate symmetry of metastable phases 1. Symmetry of metastable manifolds The symmetry U in Eq. (67) transforms the projections As the set of all density matrices is invariant under ρ˜1, ..., ρ˜m in Eq. (17) into U(˜ρ1), ..., U(˜ρ1), which are any symmetry, its image under the dynamics featur- also projections of system states [e.g., U(ρ1), ..., U(ρm) ing a weak symmetry is also symmetric at any time t, for states in Eq. (14)]. In the space of coefficients, the U{etL[ρ(0)]} = etL{U[ρ(0)]}. In particular, a unique sta- symmetry transformation is unitary, and does not change tionary state achieved asymptotically is necessarily sym- distances. Therefore, as the simplex with vertices corre- metric U(ρss) = ρss (or, in the case of degeneracy, the sponding to ρ˜1, ..., ρ˜m approximates well the MM in the manifold of stationary states is invariant). Similarly, the space of coefficients, so does the simplex of the trans- set of system states during the metastable regime, i.e., formed new vertices. In fact, it can be shown that the the set of metastable states, is invariant under the sym- corrections of classicality in Eq. (21) are the same for metry U. This can be seen from the MM being deter- both choices (see Sec. G2a of the SM). We thus ex- mined by the projection P on the low-lying modes in pect that the new vertices to be close to the those of Eq. (4), which in the presence of the weak symmetry ful- metastable phases. fills Indeed, it can be shown that the set of metastable phases is approximately invariant under the symmetries [P, U] = 0 (68) of the dynamics. In Sec.G2a of the SM, we prove that the action of the symmetry on the metastable phases [cf. Eq. (67)]. This is a direct consequence of ˜ ˜ the modes of L being eigenmatrices of the sym- (U)kl ≡ Tr[Pk U(˜ρl)] = Tr[Pk UMM(˜ρl)], (70) metry, so that the coefficients gain a phase under † k, l = 1, ..., m, can be understood as an approximate per- the symmetry, Tr{LkU[ρ(0)]} = Tr[U (Lk)ρ(0)] = iφk iφk mutation of metastable phases, that is, e Tr[Lkρ(0)] = e ck, and thus P{U[ρ(0)]} = Pm iφk Pm k=1 e Tr[Lkρ(0)] Rk = k=1 Tr[Lkρ(0)] U(Rk) = n n U{P[ρ(0)]}; see Fig.5(a). kU − Π k1 . 3 Ccl, (71) where Π is a permutation matrix and n = 1, 2, ... are powers of the transformation [109]. Therefore, from 2. Symmetry of long-time dynamics Eq. (71) we obtain that ρ˜l is approximately transformed into πn(l) under symmetry applied n times, n The weak symmetry in Eq. (67) together with the sym- U (˜ρl) − ρ˜πn(l) . 3Ccl, where π is the permutation cor- metry of the MM in Eq. (68) yields the symmetry of the responding to Π. Similarly for ρl being the closest state n long-time dynamics in the MM in Eq. (6) as to ρ˜l we have U (ρl) − ρπn(l) . 3Ccl+2C+ [cf. Eq. (12)].

[LMM, U] = 0 = [LMM, UMM], (69) 2. No continuous symmetries where UMM ≡ PUP; see Fig.5(b). This follows since [PLP, U] = P[L, U]P = 0 and [PLP, PUP] = We now argue that any continuous weak symmetry acts P[L, U]P = 0 from Eq. (68) and [L, P] = 0. trivially on the low-lying modes of the master operator 21

choose an element l and deﬁne

D −1 dl dl X ρ˜0 ≡ U ndl (˜ρ ), (72) l D l n=0

d where dl is the length of the cycle π l (l) = l (and thus D is divisible by dl), while for the other elements of that cycle we deﬁne

0 n 0 ρ˜πn(l) ≡ U (˜ρl), n = 1, ..., dl − 1, (73) and denote the transformation from the eigenmodes to this basis as C0 [cf. Eq. (18)]. This gives a symmetric set of metastable states,

0 0 U(˜ρl) ≡ ρ˜π(l), l = 1, ..., m, (74) Figure 5. Symmetry of dynamics and classical metastability: (a) A weak symmetry leads to the MM being sym- for which the distance to system states is again bounded metric under the corresponding transformation of coefficients, by C+ in Eq. (12). Furthermore, from Eq. (71) it can 0 here c3 7→ −c3 [cf. UMM in panel (b)], which is also preserved be shown that kρ˜l − ρ˜lk . 6Ccl, l = 1, ..., m, and the 0 0 by the long-time dynamics (blue simplex at t τ , red sim- corresponding corrections to the classicality, Ccl, defined plex at t = τ 0). States invariant under the symmetry, e.g., analogously to Eq. (21), can increase at most by . 6Ccl metastable phases ρ˜2 and ρ˜4 and the stationary state ρss (red (see Sec.G2c in the SM for the proofs). Therefore, with- sphere), necessarily feature c3 = 0. (b) The transformation 0 out loss of generality, the set of metastable phases can be C to the symmetric set of metastable phases in Eqs. (72) considered invariant under the symmetry. In Sec. VIIA, and (73) yields the classical long-time dynamics W0 symmet- we show how symmetric sets of m candidate sets can be ric with respect to the permutation Π that corresponds to the action of the symmetry on metastable phases, and in this generated efficiently. In the invariant basis of metastable phases, the action case swaps ρ˜1 and ρ˜3 [cf. Eqs. (69) and (77)]. of the symmetry is exactly the permutation [see Fig.5(b)] when metastability is classical. A continuous weak sym- U0 ≡ Π, (75) metry is a symmetry [Uφ, L] = 0 [cf. Eq. (67)] for all φ, φG 0 ˜0 0 ˜0 0 where Uφ ≡ e with G(ρ) ≡ i[G, ρ] for a Hermitian oper- where (U )kl ≡ Tr[Pk U(˜ρl)] = Tr[Pk UMM(˜ρl)], k, l = ˜0 0 ator G. For a small enough φ, Uφ is approximated by the 1, ..., m, and Pl is the dual basis to ρ˜l in Eqs. (72) 0 0 identity transformation, and therefore for such values of φ and (73), l = 1, ..., m; that is, Tr(P˜ ρ˜ ) = δkl, from which n k l we have Π = I in Eq. (71) with n = 1. Since Uφ = Unφ, ˜0 ˜0 it follows U(Pl ) = Pπ(l) [cf. Eq. (74)]. from Eq. (71) the symmetry Uφ is approximated by I for Finally, the set of corresponding basins of attraction is any φ. But this is only possible when Uφ = I, i.e., the symmetric. That is, not only supports of the metastable symmetry leaves each metastable phase invariant, other- phases in a cycle are connected by the symmetry op- wise the corrections, as given by the Taylor series, could erator U, but also their basins of attractions. Indeed, accumulate beyond 3Ccl 1 (see Sec.G2b in the SM 0 ˜0 0 ˜0 for p˜l = Tr[Pl ρ(0)], we have p˜l = Tr{U(Pl )U[ρ(0)]} = for a formal proof). Therefore, all slow eigenmodes of the ˜0 Tr{Pπ(l)U[ρ(0)]}, l = 1, ..., m. Thus, when ρ(0) belongs dynamics must be invariant as well. As a corollary, we 0 obtain that any nontrivial continuous symmetry of slow to the basis of attraction of ρ˜l, i.e., |1 − p˜l| 1, U[ρ(0)] eigenmodes precludes classical metastability. belongs to the basis of attraction of ρ˜π(l). Similarly, 0 0 U(1 0 ) = 1 0 for H defined for P˜ as in Eqs. (31)– Hl Hπ(l) l l (33).

3. Symmetric set of metastable phases 4. Symmetry of classical long-time dynamics

Permutation symmetry. The weak symmetry of the We now show that the set of metastable phases can be long-time dynamics in Eq. (69) in the basis of the chosen invariant under the action of a weak symmetry. metastable phases reads For a discrete symmetry, there exist a smallest nonzero integer D such that U DP = P. We then have UD = I, [W˜ , U] = 0 (76) and thus from Eq. (71) also ΠD = I. Let be π be a permutation associated with Π, that is, (Π)kl = δkπ(l), [cf. Eqs. (34) and (70)]. For the set of m metastable k, l = 1, ..., m. For each cycle in the permutation, we phases chosen invariant under the symmetry [Eqs. (72) 22 and (73)], the classical stochastic dynamics between to the eigenmodes of UMM [cf. Eq. (79)], metastable phases W0 that approximates the long-time- 0 dynamics W˜ [Eq. (36)] also features the weak symmetry dl−1 1 j n 0 X −i2π d 0 with respect to the permutation Π, R j ≡ e l ρ˜ n , (80a) π (l) d π (l) l n=0

˜ 0 0 dl−1 n [W , Π] = 0 = [W , Π] (77) i2π j 0 X d ˜0 Lπj (l) ≡ e l Pπn(l), (80b) n=0 [cf. Eq. (75) and Fig.5(b) and see Sec.G3a in the SM for the proof]. with j = 0, 1, ..., dl − 1, dl being the length of the considered cycle, and ei2πj/dl the corresponding symmetry Structure of low-lying eigenmodes. We now show that, eigenvalue [cf. Eqs. (72) and (73)]. Therefore, the low- as a consequence of the symmetry in Eq. (77), the long- lying modes are their linear combinations, time dynamics may couple only plane waves over cycles of metastable phases with the same momentum, which m m X −1 0 X 0 results in the low-lying eigenmodes being their linear Rk = (CU )kl Rl,Lk = (CU)klLl, (81) combinations. In particular, a unique stationary state l=1 l=1 is composed of uniform mixtures of states in cycles, where CU is the transformation from the basis of the low-lying eigenmodes to the basis of Eq. (80), dl−1 X 0 1 X 0 ρ = (˜p ) ρ˜ n , (78) j ss ss l π (l) 0(l) iφk i2π d dl (CU) j = c if e = e l (82) l n=0 kπ (l) k (CU)kπj (l) = 0 otherwise, where l runs over cycles representatives with dl denoting 0(l) 0 the length of the corresponding cycle [cf. Eqs. (72)–(74)], with k = 1, 2, ..., m, j = 0, ..., dl − 1, and ck = Tr(Lkρl) 0 ˜0 [cf. Eq. (18) and Fig.5(a)]. Importantly, C is block and (˜pss)l = Tr(Pl ρss) corresponds to the stationary dis- U tribution of approximately classical dynamics of the sym- diagonal in the eigenspaces of Π, so that Rk and Lk are metric degrees of freedom (see Fig.5(a) and cf. Secs.E1c only linear combinations of the eigenmatrices in Eq. (80) andG3b in the SM). that fulfill ei2πj/dl = eiφk . In particular, the number In the presence of the weak symmetry in Eq. (77), the of symmetric low-lying modes equals the number of cycles in the permutation, the corresponding block of long-time dynamics generator W˜ 0 is block diagonal in an C is determined by coefficients for uniform mixtures eigenbasis of Π, which we can choose as plane waves over U of metastable phases in each cycle, and the symmetric the cycles in the corresponding permutation π. Thus, the stationary state is given by Eq. (78). Furthermore, weak symmetry limits the number of free parameters of when the symmetry eigenvalue eiφk is unique among W˜ 0 to the sum of squared degeneracies of the symmetry low-lying spectrum modes, R and L are necessarily eigenvalues, i.e., the plane-wave momenta (less 1 from the k k proportional to Eqs. (80a) and (80b), but this is not the trace-preservation condition), and results in the eigenvec- case for multiple cycles in general, e.g., for symmetric ˜ 0 tors of W being linear combination of the plane waves eigenmodes. For a single cycle, however, all symmetry with the same momenta. In particular, W˜ 0 restricted to eigenvalues are unique, so that CU is diagonal and the symmetric plane waves, i.e., the uniform mixtures of determined by the coefficients of a single candidate metastable phases in each cycle, governs the long-time phase. Thus, all low-lying eigenmodes are determined dynamics of symmetric states, which is trace-preserving uniquely and the stationary state is an equal mixture and approximately positive with the corrections 2pC0 . cl of m metastable phases [cf. Eq. (78)], as discussed in (see Sec.G3b in the SM). Ref. [42]. Eigenvectors of W˜ 0 correspond directly to the low- lying eigenmodes of the master operator L, as they deter- (No) conservation of symmetry. Finally, note that the mine the coefficients in the basis of the metastable phases symmetry of the MM can take place without the unitary [cf. Eqs. (72)–(74)], operator U being conserved during the initial relaxation and the metastable regime U, P†(U) = U, analogously m m as is the case for the symmetric manifold of stationary X 0−1 0 X 0 ˜0 Rk = (C )kl ρ˜l,Lk = (C )kl Pl , (79) states [1, 24, 110]. Indeed, U itself is a symmetric oper- † l=1 l=1 ator, U (U) = U, and thus, if conserved, it is spanned P Pdl−1 ˜ by the symmetric eigenmodes, U = l ul n=0 Pπn(l), where k = 1, ..., m. In particular, the left eigenvec- where ul ≡ Tr(Uρ˜l) and l runs over cycles representa- 0 tor of W˜ corresponding to the eigenmode Lk is simply tives. For a MM with a single cycle, however, the conser- the vector of kth coefficient for the metastable phases vation would lead to contradiction, as it would imply a [cf. Eq. (18)]. Analogously, the plane waves correspond trivial symmetry, U ∝ 1, with m, rather than one, cycles. 23

5. Symmetric test of classicality

We now use the structure of eigenmodes of the dynamics in the presence of symmetry in Eqs. (81) and (82) to simplify the test of classicality introduced in Sec.IIIB. It is important to note beforehand that Eq. (80) forms a valid basis for any symmetric set of m candidate states that are linearly independent. Thus, to verify whether candidate states indeed correspond to m metastable phases, the test of classicality is necessary even in the case of a single cycle (see Sec.G4 in the SM for an example). Exploiting the structure of the eigenmodes, the test of classicality can be simplified as follows. First, only coefficients of cycle representatives are needed to construct CU in Eq. (82). Second, as CU is block diagonal, to find the dual basis to the plane waves [Eq. (80b)], only matrices of the size of the permutation eigenspaces need to Figure 6. Metastable phases construction. We sketch be inverted [111]. The dual basis to metastable phases the approach to construct a candidate set of m metastable in Eq. (20) can then be found by the inverse transfor- phases within a given MM that provides the best classical mation to Eq. (80b), that is, with the coefficients as in approximation in terms of corrections in Eq. (21). The construction can be refined by considering symmetries of the dy- Eq. (80a). Finally, to estimate the corrections to the namics. classicality as in Eq. (23), it is enough to consider the elements of the dual basis corresponding to the chosen states. This algebraic structure, however, is not gener- ˜0 cycle representatives, Pl [Eq. (80b) averaged over j], and ally present for the low-lying left eigenmodes, as visible, multiply their contribution by dl [112]. e.g., in first-order corrections to the formerly stationary modes when they are perturbed away from a dissipative phase transition at a finite system size by excitation of VII. UNFOLDING CLASSICAL decaying modes (see Supplemental Material of Refs. [5]). METASTABILITY NUMERICALLY Here, we introduce a general approach which delivers the set metastable phases and the structure of the long-time dynamics when the metastability is classical. With the theory of classical metastability now estab- Similarly to the algorithm in Refs. [113, 114], it is based lished, we turn to the question of how to efficiently un- on the low-lying left eigenmodes of the master opera- cover the structure of a MM and long-time dynamics in a tor in Eq. (1), but their connection to basins of attrac- given open quantum system governed by a master equations is utilized by the observation that extreme values of tion. We introduce two numerical methods to analyze the corresponding coefficients are achieved by metastable the classical metastability in such systems with or with- phases (cf. Fig.2), which thus correspond to the pro- out weak symmetries. The first approach in Sec.VIIA jections on the MM of the extreme eigenstates of left requires diagonalizing the master operator and its low- eigenmodes. To guarantee that all metastable phases lying eigenmodes. It verifies the presence of classical are found, random rotations of low-lying modes are em- metastability, delivers the set of metastable phases, and ployed until the corrections to the classicality are small. uncovers the structure of the long-time dynamics. The This way, the algorithm remains efficient, as it does second approach in Sec. VIIB instead utilizes quantum not simply probe the whole space system space of pure trajectories with probabilities biased according to their states (cf., e.g., Ref. [117]). Furthermore, symmetries of activity, so that the metastable phases with the extreme the dynamics can be exploited to simplify the method. activity are found. We also consider how observable averages distinguishing metastable phases (i.e., order parameters) can be utilized. Finally, we note that this method has been recently A. Metastable phases from master operator successfully applied in Ref. [54] to the open quantum East spectrum model [50] featuring a hierarchy of metastabilities and translation symmetry. Efficient algorithms to uncover the structure of the stationary state manifold [113, 114] (which utilize Ref. [115]; see also Ref. [116]) rely on the exact diagonalization of 1. Metastable phases construction the master operator and the von Neumann algebra structure of the stationary left eigenmodes which arises when Our approach consists of the following steps (see also they are restricted to the maximal support of stationary Fig.6): 24

1. Diagonalize L to find the left eigenmatrices Lk be- of metastable phase is approximately the largest simplex low the gap in the spectrum, k = 2, ..., m. inside the MM; cf. Fig.2). However, such mixtures may cause less than m candidates to remain after clustering, 2. Construct candidate metastable states: or result in a set of phases which provides a poor approximation to the true MM; even without degeneracy, - diagonalize the (rotated) eigenmatrices L , k it is possible that some metastable phases may reside on - choose the eigenstates associated to their ex- the interior of the hypercube defined by the extreme val- treme eigenvalues as initial states for candi- ues of the coefficients, and as such will not appear in date metastable states, the set of candidate states taken from extreme eigenval- - discard repetitions in candidate metastable ues of the eigenmodes. Nevertheless, random rotations states—cluster in the coefficients space. in Step 3ii ensure that each metastable phase is eventually exposed, i.e., a basis in which that metastable phase 3. Find best candidate metastable states: achieves an extreme coefficient value without degeneracy is eventually considered (with the probability 1 achieved i. If the number of candidate states ≥ m, then exponentially in the number of rotations; cf. Sec.H2 in choose the set of m states providing the sim- the SM). When the set of metastable phases leading to plex with the largest volume, i.e., the largest small corrections to the classicality is found, the dual |detC| [cf. Eq. (18)] and calculate the cor- basis in Eq. (20) can be constructed, and the long-time responding corrections to the classicality in dynamics decomposed as in Eq. (34). Eq. (21), as can be easily bounded by Eq. (23). ii. If the number of candidate states < m, or the Naturally, instead of considering distances between corrections to the classicality in Step 3i are not candidate states in the space of coefficients, as used in negligible, then enlarge the set of candidates Steps 2 and 3i, candidate states can be clustered with obtained from Step 3 by considering a random respect to the trace distance in the space of density ma- rotation of the basis of the left eigenmatrices trices, while the best m candidate states can be chosen in Step 2. to achieve minimal corrections to the classicality instead of the maximal simplex volume. These modifications, Step 1 in the above construction provides the low- however, require working with operators on the system dimensional description of the MM, and, as explained in Hilbert space, rather than on the space of coefficients, Sec.III, allows for testing the approximation of the MM and thus are in general more expensive numerically [for as mixtures of m candidate states. We choose Hermitian † classical MMs, m ≤ dim(H)], while not necessary for Lk replacing conjugate pairs of eigenmodes Lk, Lk [18] MMs with nonnegligible volumes (cf. Secs.H2 andH3 by in the SM). −iϕ iϕ † −iϕ iϕ † e k Lk + e k L e k Lk − e k L LR ≡ √ k ,LI ≡ √ k , (83) Our approach will not deliver a set of metastable k 2 k 2i phases with negligible corrections to the classicality where e−iϕk is an arbitrary phase. Step 2 relies on the whenever the MM is not classical. In particular, quan- result in Sec.H1 of the SM, that metastable states aris- tum MMs [5] featuring decoherence free subspaces [73– ing from extreme eigenstates of the dynamics eigenmodes 75] or noiseless subsystems [76, 77]: which, e.g., at m = 4 can be used to approximate metastable phases in clas- amount to Bloch-sphere in the coefficient space, rather sical MMs, as long as only a single metastable phase than a tetrahedron (see also Ref. [118]), cannot be ap- is close to the extreme value of the corresponding co- proximated as probabilistic mixtures of m metastable efficient ck (which we refer to as the case without de- phases. Even for classical metastability emerging in generacy); cf. Figs.2 and5(a). We then discard any many-body open quantum systems, the approach relies repetitions in the set of candidate states (to treat all on the condition in Eq. (25), which may be fulfilled only coefficients on equal footing, we set the normalization at larger system sizes, when the low-lying part of the max min max min master operator spectrum to be sufficiently separated ck − ck = 1, where ck and ck are extreme eigenvalues of Lk). Indeed, for a given left basis we obtain from the fast modes. In this case, our approach may 2(m − 1) candidate metastable states corresponding to not succeed for smaller system sizes with less pronounced 2(m−1) extreme eigenvalues of the basis elements, which metastability. may provide up to m metastable phases. In the case without degeneracy, each candidate corresponds to one While degeneracies are unlikely to occur in a generic of m metastable phases. In the case with degeneracy, model, they typically appear in the presence a hierarchy some of extreme eigenstates may correspond to mixtures of metastabilities or as a consequence of symmetries of of metastable phases: provided that the set of candi- the dynamics, which we discuss in detail below. In par- date states features all metastable phases, such a can- ticular, in the presence of a weak symmetry, not only didate state should be discarded in Step 3i (this relies the degeneracy can be efficiently remedied, but also the on the result from Sec.H3 in the SM, that the simplex search for candidates states made even more efficient. 25

2. Construction for hierarchy of metastable manifolds metastable phase from each cycle can be obtained by R I considering extreme eigenstates of both Lk and Lk [119]. In the presence of hierarchy of metastabilities with a Other metastable phases in the considered cycles can be recovered by applying the symmetry n − 1 times, so further separation at m2 < m in the spectrum of the mas- k ter operator, the degeneracy appears as a consequence there is no need to consider eigenmodes Ll with a differ- of the fact that the simplex of m metastable phases ent symmetry eigenvalue supported on the same cycles iφ iφ of the first MM, when projected onto the coefficients (i.e., Ll with e k 6= e k but nl = nk). Furthermore, metastable phases in cycles with the (c2, ..., cm2 ), is approximated by a simplex with m2 vertices corresponding to m2 metastable phases of the sec- length corresponding to subcycles, e.g., invariant ond MM. This requires (at least) m2 metastable phases in metastable phases, can also be found. For Ll such that the first MM to evolve directly into m2 metastable phases nl is divides only nk > nl for the above considered eigen- iφl of the second MM. Each of other m − m2 metastable modes Lk, when the degeneracy of e equals the num- phases of the first MM either evolves into a single phase ber of already considered cycles with nk divisible by nl of the second MM, or it belongs to the decay subspace (i.e., the sum of the corresponding symmetry eigenvalue in which case it in general evolves into a mixture of m2 degeneracy for all such nk values), Ll is supported on metastable phases of the second MM. In the former case, the already considered cycles. Otherwise, Ll features the metastable phases in the first MM that evolve into new cycles with the length nl, which can be unfolded, the same phase in the second MM are degenerate in the as before, by rotations of all eigenmodes with the same iφl R coefficients (c2, ..., cm ). In the latter case, they do can- symmetry eigenvalue as e and considering both Ll 2 I not take extreme values of those coefficients, even after and Ll [119]. Here, equal mixtures of already consid- nk/nl a rotation of the first m2 modes (see Secs.F1 andH4 ered phases connected by U will also be found, but in the SM). Nevertheless, rotations of all m modes al- such candidate states will not lead to the maximal vol- low both for the degeneracy to be lifted and for every ume simplex. Again, by applying symmetry U the full metastable phase to take an extreme value in one of the (sub)cycles can be recovered, and other eigenmodes Lj coefficients. with nj = nl, but a different symmetry eigenvalue can be discarded. Analogous results hold for the remaining iφ eigenmodes, but with respect to Ll and e l degeneracy. 3. Construction for metastable manifolds with symmetries In summary, the set of eigenmodes considered in Step 2 is significantly reduced, with its size equal to the num- In the presence of a weak symmetry U [Eq. (67)], the ber of cycles and the subcycles with other cycle’s length. master operator L is block diagonal in the eigenspaces Furthermore, only rotations of eigenmodes with the same of U, which simplifies its diagonalization [1, 23, 24]. symmetry eigenvalue are necessary in Step 3ii. Impor- R I As a consequence, the low-lying eigenmodes of the dy- tantly, as the eigenstates of Lk and Lk and the corre- namics are chosen as linear combinations of the plane sponding metastable states are invariant under U nk , that waves over the cycles induced by the symmetry on m is, generate cycles of candidate states with the length di- metastable phases [cf. Eq. (81)]. In particular, for an viding nk, following the prescription above, we arrive at eigenmode Lk, k = 2, ..., m with a symmetry eigenvalue an invariant set of candidate states. This invariance can iφ † iφ e k , U (Lk) = e k Lk, and the minimal integer nk > 0 be maintained by clustering whole cycles of candidate in φ leading to e k k = 1, Lk is supported on cycles with states rather than individual states in Step 2. Then, the length equal nk or larger but divisible by nk. The without loss of generality, in Step 3i, instead of consider- latter case, of subcycles with length nk, leads to degen- ing subsets of all candidate states, we can choose candi- eracy of the coefficient ck for the metastable phases con- date states as sets of cycles with their lengths summing nected by U nk , with its extreme eigenstates generically to m. In that case, the volume of the simplex can be effi- Q √ projecting onto their uniform mixture [cf. Eq. (80b) and ciently calculated as |det CU|/(m−1)! l dl, where CU Fig.5(a), where ρ˜1 and ρ˜3 are degenerate in c2 and c4 is the block-diagonal matrix in Eq. (82) and the product, (n2 = n4 = 1)]. Nevertheless, coefficient degeneracy can which runs over cycle representatives, is the same for all be remedied and the structure of the low-lying eigen- sets of linearly independent candidates [111], while the modes can be used to actually enhance the introduced corrections to the classicality can be efficiently calculated approach, as we now explain (see also Sec.H5 in the SM). with the symmetric test of classicality of Sec. VI B 5. We note that when GCF(nk, nl) < nk for all nl 6= nk, k, l = 2, ..., m, the eigenmode Lk is supported only on cycles of length nk, as there are no longer cycles with n 4. Construction utilizing order parameters the length divisible by nk. The symmetry U k then acts trivially on the corresponding metastable phases and the above discussed degeneracy is absent. Analogously to Instead of considering the eigenmodes of the dynam- the general case, any plane wave in Lk can then be ex- ics, we can choose a left eigenbasis formed by a set of posed by random rotations of the eigenmodes with the m observables Ol, l = 1, ..., m projected onto the low- iφk † Pm (l) same symmetry eigenvalue e , while at least a single lying eigenmodes, i.e., P (Ol) = k=1 bk Lk, where 26

(l) bk ≡ Tr(OlRk) [cf. Eq. (4)], provided that those projec- responding to the extreme values of jump activity. tions are linearly independent. In this case, the extreme † eigenstates of P (Ok) will give metastable states attain- ing extreme values in the average of the observable Ok. VIII. CONCLUSIONS AND OUTLOOK Those metastable states will correspond to metastable phases, up to degeneracy of metastable phase averages of In this paper, we formulated a comprehensive the- Ok (in particular, in the presence of nontrivial weak sym- ory for the emergence of classical metastability for open metry of low-lying eigenmodes, it is necessary to consider quantum systems whose dynamics is governed by a mas- observables breaking the symmetry). Among others, this ter operator. We showed that classical metastability is can be helpful when the volume of MM in the space of characterized by the approximation of metastable states coefficients is negligible. In the next section, we extend as probabilistic mixtures of m metastable phases, where this approach by considering continuous measurements m is the number of low-lying modes of the master oper- instead of system observables. ator. Namely, in terms of the corresponding corrections, metastable phases are approximately disjoint, while the long-time dynamics, both on average and in individual B. Metastable phases from biased quantum quantum trajectories, is approximately governed by an trajectories effective classical stochastic generator. Furthermore, any nontrivial weak symmetries present at long times are nec- In some systems, the metastability can be a collec- essarily discrete as they correspond to approximate per- tive effect emerging as the system size increases [11]. If mutations of metastable phases, under which the classical large system sizes are required for prominent metasta- dynamics is invariant. To investigate metastability for a bility, then exact diagonalization may not be feasible. given open quantum system, we introduced the test of Therefore, we now introduce an alternative numerical ap- classicality—an approach to verify the approximation of proach to finding metastable phases in classical MMs us- the MM by a set of candidate metastable phases. We also ing QJMC simulations [8–12] and biased sampling in the developed a complementary numerical approach to de- framework of large deviation theory (see Ref. [120] for liver sets of candidate metastable phases. Since that ap- a review). In classical stochastic dynamics biased sam- proach requires diagonalization of the master operator— pling can be efficiently incorporated into the generation a difficult task in systems of larger size—we also discussed of trajectories, with techniques such as transition path an alternative based on the concept of biased trajectory sampling [121] and cloning [122]. sampling. Trajectories of the biased master equation Ls in The techniques we introduced here allow us to achieve Eq. (44) can be viewed as trajectories of L with their a complete understanding of classical metastability probability multiplied by e−sK(t), where K(t) is the to- emerging in an open quantum system. A concrete ap- tal number of jumps occurring in a quantum trajectory plication of the methods described here to a many-body of length t. The maximal eigenmode ρss(s) of Ls cor- system of an open quantum East model [50] is given in responds then to the asymptotic system state in quan- Ref. [54], where despite the stationary state being an- tum trajectories averaged with the biased probability. alytic in dynamics parameters, the dynamics is found In Sec. VB5, we argued that ρss(s) can approximate to feature a hierarchy of classical metastabilities, with metastable phases of extreme activity for appropriately metastable phases breaking the translation symmetry of chosen s when the activity dominates the transition rates the model and their number increasing with the system of long-time dynamics [cf. Figs.4(a) and4(c)]. Thus, size. This structure is then shown to be analogous to if the efficient biased sampling could be generalized to metastability in the classical East model, but with an QJMC sampling, metastable phases with extreme activ- effective temperature and coherent excitations. ity could be accessed via time-average of a biased trajec- While our general approach relies only on the Marko- tory over time-length within the metastable regime [123]. vian approximation of system dynamics, non-Markovian Similarly as in the case of degeneracy of coefficients, effects could be includedvia the chain partial diagonal- when more than a single metastable phase corresponds ization [124–126] (see also Ref. [127]). However, an open to the maximum or the minimum activity, ρss(s) corre- quantum system and its environment form an isolated sponds to a mixture of the metastable phases with the system, which raises a question about the relation of clas- extreme value (e.g., when both L and J obey a symme- sical metastability to separation of timescales in closed try that is broken in the MM, the mixture is symmetric). system dynamics, which generically leads to prethermal- Nevertheless, the discussion in Sec.VB5 is analogous for ization of subsystem states [128]. Furthermore, as in the activity of individual jumps, and thus a further dis- this work no assumptions were made on the structure tinction between metastable phases can be enabled this of system Hamiltonian or jump operators, it should be way, e.g., by breaking the translation symmetry of Ls in explored how metastable phases and their basins of at- the case of identical local jumps (see Sec.E3a in the SM). traction can be further characterized in systems with lo- Finally, we note that in the case of general metastabil- cal interactions and local dissipation (see also Ref. [25]). ity, this approach will return the metastable states cor- Finally, it remains an open question what is the struc- 27 ture of general quantum metastability and how it can be search Fellowship. We acknowledge financial sup- efficiently investigated, e.g., to uncover metastable coher- port from EPSRC Grant No. EP/R04421X/1, from ences [5]. Extending the methods described here to this the H2020-FETPROACT-2014 Grant No. 640378 general case would inform, among others, the study of a (RYSQ), and by University of Nottingham Grant general structure of first-order dissipative phase transi- No. FiF1/3. I.L. acknowledges support by the tions in open quantum systems. “Wissenschaftler-Rückkehrprogramm GSO/CZS” of the Carl-Zeiss-Stiftung, through the Deutsche Forschungsge- meinsschaft (DFG, German Research Foundation) under Project No. 435696605 and through the Euro- ACKNOWLEDGMENTS pean Union’s H2020 research and innovation programme [Grant Agreement No. 800942 (ErBeStA)]. We are grate- K.M. thanks M. Guţă for helpful discussions. K.M. ful for access to the University of Nottingham Augusta gratefully acknowledges support from a Henslow Re- HPC service.

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Theory of classical metastability in open quantum systems Supplemental Material

In this Supplemental Material, we provide proofs of the results in "Theory of classical metastability in open quantum systems". The open quantum system which serves as the illustration of our general results in Figs.1–5 of the main text is discussed in Sec.A, together with classical metastability arising in proximity to dissipative phase transitions at ﬁnite size. The derivations for the quantitative approach to metastability are given in Sec.B. Then proofs are given of the following results: classical metastability in Sec.C, classical metastable phases in Sec.D, classical long-time dynamics in Sec.E, classical hierarchy of metastabilities in Sec.F, and classical weak symmetries in Sec.G. Finally, the correctness of the new numerical approach introduced in the main text is argued in Sec.H.

A. CLASSICAL METASTABILITY IN PROXIMITY TO DISSIPATIVE PHASE TRANSITION AT FINITE SIZE

Here, we first describe the model we illustrate our general theory of metastability in Markovian open quantum systems with in Figs.1–5 of the main text. This example belongs to a class of systems where classical metastability is a consequence of perturbing dynamical parameters away from effectively classical dissipative phase transitions occurring at a finite size. Second, we find and discuss the general structure of metastable phases and long-time dynamics for this class. Figure A1. Example of open quantum dynamics defined in Eqs. (A1), (A2) and (A3). Four phases with distinct support arise in the perturbative limit of Eq. (A4) when dy- 1. Example in Figs.1–5 of the main text namics associated with ω1 and ω2 (green dashed arrows) can be neglected. When the jump from |6i to |4i (red arrow) a. Model is reverted, the dynamics features a discrete swap symmetry [cf. J4 in Eq. (A3)].

In Figs.1–4 of the main text, we choose a system of the dynamics is in the proximity of a dissipative phase 6 levels |ji, j = 1, 2, ..., 6, connected in the Markovian transition with four independent sectors corresponding dynamics governed by the master operator in Eq. (1), to: two decoupled pure states |1i and |2i, whose coher- with the Hamiltonian ences between two decoupled states are not stable be- x x x x x x cause of dephasing caused by J1 and J2 in Eq. (A3), and H = ω1(σ13 + σ14) + ω2(σ25 + σ26) + Ω (σ35 + σ46),(A1) two eﬀective two-level atoms, which are formed by |3i x where σjk ≡ |jihk| + |kihj|, and the dissipative jumps and |5i, and by |4i and |6i (see Fig. A1). Note that √ √ there is no decay subspace. Due to the perturbation by J1 = γ σ11,J2 = γ σ22, (A2) x x x x the Hamiltonian δH = ω1(σ13 + σ14) + ω2(σ25 + σ26), √ † √ J3 = κ σ35,J4 = κ σ46, (A3) the long-time dynamics arises in the second order in ω1 and ω2, while metastable phases arise from the sta- where σjk = |jihk|, as depicted in Fig. A1. We assume tionary states, perturbed already in the ﬁrst order (see J3 and J4 correspond to emission of quanta [cf. Fig.1(e) Sec.A2b). Therefore, the condition in Eq. (A4) leads to in the main text]. the classical metastability with m = 4. This is the limit In Fig.5 of the main text, we replace J4 in Eq. (A3) by considered in Figs.1–5 of the main text (see below for † J4 . This introduces a discrete weak symmetry under the parameter values). simultaneous swap |3i ↔ |4i and |5i ↔ |6i (cf. Fig. A1 and see Sec.VI in the main text). c. Plot parameters and numerical results b. Dissipative phase transition In Figs.1–4 of the main text, we choose ω1 = 2.5ω2 In the limit and ω2 = 0.2κ, while Ω = 1.5κ and γ = 0.75κ. The low- lying eigenvalues are λ2 = −0.0785κ, λ3 = −0.1564κ and |ω1|, |ω2| |Ω|, κ, γ, (A4) λ4 = −0.2886κ with the next eigenvalue λ5 = −0.4297κ. 32

Although the separation is not significant, the choice of −0.0181κ. Here, the stationary state decomposes as T such set of parameters is motivated by clearly illustrat- ˜pss = (0.2605, 0.1947, 0.2605, 0.2843) . We observe that ing approximations in the theory of classical metasta- the symmetry increases the separation in the master op- bility. The corrections to the classicality are bounded erator spectrum and reduces the corrections to the clas- by C˜cl = 0.2369 in Eq. (23), which we get for the sicality. We find it also leads to W˜ = W [cf. Fig.5(b) in metastable phases obtained from initial states chosen as the main text]. extreme eigenstates of L2, L3 and L4 (normalized so that Finally, in Figs.1–5, due to the simplicity of the max min h1|Lk|1i > 0 and ck −ck = 1, k = 2, 3, 4) that give the model, we approximate the timescales as τ = −1/λ2, 0 00 maximal simplex (cf. Fig.2 in the main text). In particu- τ = −1/λ4, and τ = −1/λ5; cf. Sec.B3. Furthermore, 0 lar, in Figs.2–4 of the main text, ρ˜1 is supported mostly in Figs.1(b),3(b), and5(a), for t τ we choose t = 0. on |3i and |5i, ρ˜2 on |1i, ρ˜3 on |4i and |6i, while ρ˜4 on |2i (cf. Fig. A1), and the stationary state decomposes as T ˜pss = (0.2763, 0.1689, 0.4021, 0.1528) . The initial state 2. General case considered in Figs.1(c),1(d),1(e),4(e),4(f), and4(g) of the main text, is ρ(0) = |3ih3|. In Figs.1(c) and1(d) Here, we consider perturbing dynamics of an open of the main text, the chosen observable is the activity of quantum system that features multiple disjoint station- jumps in Eq. (A2), O = J †J + J †J = κ(σ + σ ), 3 3 4 4 33 44 ary states. First, we show that the corrections to the which in trajectories in Figs.1(e) and4(g) of the main positivity in Eq. (12) and the corrections to the classical- text are represented by blue (J ) and red (J ). There- 3 4 ity in Eq. (21) are of the second order in the perturbation. fore, hK(t)i in Fig.4(e) of the main text corresponds to Second, we find that the long dynamics arises in the sec- the time-integral of hO(t)i. In terms of the asymptotic ond order, with corrections to its positivity at most of rates, we have µ = 0.3115κ and σ2 = 1.057κ, while the ss ss the forth order. We also discuss the effectively classi- metastable phase rates are µ˜ = 0.5129κ, µ˜ = 0.01832κ, 1 2 cal statistics of quantum trajectories and explain how a µ˜ = 0.3971κ, µ˜ = 0.0458κ, and σ˜2 = 0.3868κ, σ˜2 = 3 4 1 2 hierarchy of classical metastabilities arises when degener- 0.01876κ, σ˜2 = 0.4026κ, σ˜2 = 0.03852κ. 3 4 acy of zero eigenvalues of the master equation is partially In Figs.1(a),1(c),1(d),4(e), and4(f) of the main lifted in the second order. text, we choose reduced ω2 = 0.025κ, to obtain a more pronounced separation of the eigenvalues: λ2 = −0.0008522κ, λ3 = −0.001114κ, λ4 = −0.004682κ a. Dissipative phase transition and its proximity and λ5 = −0.5000κ. In this case, the activity of the stationary state is characterized by µss = 0.3137κ 2 and σss = 69.03κ. Considering the extreme eigen- Unperturbed dynamics. We consider dynamics with m states of (L2 + L3)/2, (L2 − L3)/2 (rotation of coef- disjoint stationary states ρ1, ..., ρm. Let H denote the ficients by π/4) and L4 as initial states for candidate system Hilbert space and Hl, l = 1, ..., m, the respective metastable phases and the maximal simplex, we obtain orthogonal supports of the stationary states, so that H = m C˜cl = 0.001703, while µ˜1 = 0.4739κ, µ˜2 = 0.00004521κ, ⊕l=1Hl ⊕ D, where D is the decay subspace. The dual 2 µ˜3 = 0.4732κ, µ˜4 = 0.0002876κ, and σ˜1 = 0.4034κ, basis Pl, l = 1, ..., m to the stationary states then fulfills 2 2 2 Pm 1 σ˜2 = 0.00003849κ, σ˜3 = 0.4027κ, σ˜4 = 0.0002447κ. 0 ≤ Pl, l=1 Pl = , and T Here, ˜pss = (0.1833, 0.4792, 0.1907, 0.1467) and, thus, 2 1 1 1 a significant increase in σss = 1.057κ is due to longer Pl = Hl + DPl D. (A5) timescales of the dynamics within the metastable manifold (MM) (cf. Sec.VB5 in the main text). In this case It also follows that [S1–S3] ˜ W = W and, to compare the classical approximation to m (0) X h (0) (0) (0) i the rate of jump average and fluctuations in Figs.4(e) J = 1 J 1 + 1 J 1 + 1 J 1 , (A6) ˜ j Hl j Hl Hl j D D j D and4(f) of the main text, we rescale W obtained for l=1 ω = 0.025κ by the square of the frequency ω ratio 2 2 m h i (= 0.1252). X 1 (0)1 1 (0)1 1 (0)1 Heff = Hl Heff Hl + Hl Heff D + DHeff D , † In Fig.5 of the main text, we replace J4 by J4 and l=1 again choose ω2 = 0.2κ leading to λ2 = −0.10099κ, λ3 = (0) (0) P (0)† (0) −0.14641κ, λ4 = −0.22028κ and λ5 = −0.458922κ and where Heff = H − i j Jj Jj /2 is referred to as C˜cl = 0.1597, and the stationary state with µss = 0.2499κ the effective Hamiltonian. We denote the corresponding 2 (0) and σss = 1.1790κ. Considering again the extreme eigen- master operator as L , the projection on the stationary (0) (0) states of L2, L3 and L4 and the maximal simplex, in states as P , and the reduced resolvent at 0 as S (0) (0) (0) (0) (0) (0) (0) Fig.5(a) of the main text, we again have that ρ˜1 is sup- [S L = I − P = L S and S P = 0 = (0) (0) ported mostly on |3i and |5i, ρ˜2 on |1i, ρ˜3 on |4i and P S ]. |6i, while ρ˜4 on |2i (cf. Fig. A1), with µ˜1 = 0.4781κ, In the discussion of classical metastability, we µ˜2 = 0.0297κ, µ˜3 = 0.4781κ, µ˜4 = −0.01771κ, and will consider states that evolve into ρl with prob- 2 2 2 2 σ˜1 = 0.4187κ, σ˜2 = 0.0284κ, σ˜3 = 0.4802κ, σ˜4 = ability 0, l = 1, ..., m. Those states are supported 33 in the eigenspace of Pl corresponding to the eigen- where l = 1, ..., m and ... denotes corrections of the third value 0 , which consists of ⊕k6=lHk and a sub- and higher order. The corresponding (normalised) dual space in D, which we denote Dl,0. It follows that basis is given by [cf. Eq. (20)] (0) (0) 1 1 ⊥ 1 1 1 ⊥ 1 ( Hl + D )Jj Dl,0 = 0 = ( Hl + D )H Dl,0 , (0)† (1)† (0)† (2)† l,0 l,0 eff P˜ = P − S L (P ) − S L (P ) (A12) ⊥ l l l l where Dl,0 denotes the orthogonal complement of (0)† (1)† (0)† (1)† +S L S L (Pl) + .... Dl,0 within D. We will also consider states that evolve into ρl with probability 1, l = 1, ..., m. Those From Eq. (A11), we have Tr(˜ρl) = 1+... [actually, this states are analogously supported in the eigenspace holds to all orders as 1 is a conserved quantity for L, of Pl corresponding to the eigenvalue 1, which con- and thus P†(1) = 1]. Furthermore, as L(1) in Eq. (A9) sists of Hl and a subspace in D, which we denote can only create coherences between H and ⊕ H ⊕ D P (0) l k6=l k Dl,1. We have ( 1H + 1 ⊥ )J 1D = 0 = (0) k6=l k Dl,1 j l,1 [cf. Eqs. (A6) and (A7)], while S maps such coher- P (0) 1 1 ⊥ 1 ences onto themselves and operators on Hl (the latter ( Hk + D )H Dl,1 , and Dl,1 for l = 1, ..., m k6=l l,1 eff ⊥ are orthogonal subspaces. only from coherences between Hl and ⊕k6=lHk ⊕ Dl,0), where eigenvalues of ρl are strictly larger than 0, it fol- Perturbation. We consider perturbing the Hamiltonian lows that kρ˜lk1 −1 is at most of the second order. There- fore, the corrections to the positivity in Eq. (12) are at and jumps as H = H(0)+H(1) and J = J (0)+J (1), where j j j most of the second order [cf. Eq. (A13) below]. Furthermore, it also follows that Tr[(1H + 1D )˜ρl] − 1 (1)1 k k,1 Hl Jj Hl = 0 (A7) δkl is at most of the second order, so that the correc- (1) tions to disjointness of the supports and the basins of 1H H 1H = 0 (A8) l l attraction of the metastable phases are of the second for l = 1, ..., m, as other perturbations can be included order. Similarly, since subspaces corresponding to the ˜ in H(0) and J (0) without changing their structure with sum of eigenspaces of Pk with the eigenvalues > 1/2, j considered in Sec.IV of the main text are included respect to the subspaces Hl, l = 1, ..., m, and D, as in the perturbed sum of eigenspaces of P˜k with the well as Dl, l = 1, ..., m [here, we could also assume (1) (1) eigenvalues ≥ 1/2, their overlap with ρl changes from δkl J 1 m ⊥ = 0 = H 1 m ⊥ ]. j ∩l=1Dl ∩l=1Dl at most in the second order. Finally, if the corrections This leads to the first- and second-order perturbation are of the second order, the bounds in Eqs. (31)–(33) are (0) (1) (2) of the master operator L = L + L + L , where saturated in terms of the figure of merit, since we show [cf. Eq. (1)] next that the corrections to the classicality, like the corrections to the positivity, are at most of the second order. (1) (1) X (1) (0)† (0) (1)† L ≡ −i[H , ρ] + Jj ρJj + Jj ρJj (A9) j Corrections to classicality. We now argue that the corrections to the classicality arise at most in the second 1 n o − J (1)†J (0) + J (0)†J (1), ρ , order of the perturbation. We have [cf. Eq. (A12)] 2 j j j j ˜ X (1) (1)† 1 n (1)† (1) o p˜l ≡ Tr[Plρ(0)] (A13) L(2) ≡ J ρ J − J J , ρ . (A10) j j 2 j j (1) (0) (2) (0) j = pl − Tr{PlL S [ρ(0)]} − Tr{PlL S [ρ(0)]} (1) (0) (1) (0) +Tr{PlL S L S [ρ(0)]} + ...,

b. Perturbation theory where pl ≡ Tr[Plρ(0)], so that the barycentric coordinates for the metastable phases in Eq. (A11) in general feature first-order corrections. The corrections To discuss the classical metastability arising due to classicality in Eq. (21), however, are determined to perturbations in Eqs. (A9) and (A10), we consider by the corrections in Eq. (A13) for states projected the non-Hermitian perturbation theory in finitely- by P(0) onto the surface of the simplex of probability dimensional spaces (see Chapter 2 in [S4]). In particular, distributions, that is, featuring pl = 0 for some l. This we exploit the perturbation series for the projection on requires ρ(0) to be supported inside the 0-eigenspace the low-lying eigenmodes arising from the stationary of Pl, i.e., ⊕k6=lHk ⊕ Dl,0. From the structure of the states. dynamics discussed in Sec.A2a, this leads to S(0)[ρ(0)] supported within the same subspace. Furthermore, Metastable phases. We choose the candidate states as as L(1) in Eq. (A9) can only create coherences from the stationary states of the unperturbed dynamics lead- ⊥ ⊕k6=lHk ⊕ Dl,0 to Hl ⊕ Dl,0 [cf. Eqs. (A6) and (A7)], ing to (cf. Eq. (17) and see Eq. (2.14) in Ref. [S4]) the first-order corrections are not present in this case (1) (0) (0) (1) (0) (2) Tr{PlL S [ρ(0)]} = 0 [cf. Eq. (A12)]. We conclude ρ˜l = ρl − S L (ρl) − S L (ρl) (A11) that Ccl > 0 arises, at earliest, in the second order (0) (1) (0) (1) (0) (1) (0) 2 (1) +S L S L (ρl) − P L [S ] L (ρl) + ..., (for example, see Sec. A1b). This also holds for C˜cl 34

(2) in Eq. (23) [cf. Eq. (24)]. Finally, we note that the (1 − δkl)[W ]kl, and ... denotes corrections of the third (1) Pm 1 (1)1 and higher order. perturbations restricted to Jj = l=1 Hl Jj D and H(1) = Pm 1 H(1)1 , do not lift the degeneracy of Similarly, for the statistics of the homodyne measure- l=1 Hl D ment we have (see Sec.E3b) m stationary states, but lead to reduced subspaces Dl,0 and thus p˜ ≥ 0, so that C = 0 = C˜ . r2 l cl cl (W˜ ) = [W(2)] − r δ x˜ + (A16) r kl kl kl l 8 Long-time dynamics. For the generator of the long- −r((1−δ )Tr{P X (0)[˜ρ(2)]}+δ Tr{(P −1)X (0)[˜ρ(2)]} time dynamics in Eq. (34), we have [cf. Eqs. (A11) kl k l kl l l (1) (1) 1 (1) (1) and (A12)] +(1−δkl)Tr{PkX [˜ρl ]}+δklTr{(Pl − )X [˜ρl ]} (2) (1) (0) (1) ˜(2) (0) ˜(1) (0) (1) ˜(1) (1) (W˜ )kl = Tr[PkL (ρl)] − Tr[PkL S L (ρl)] (A14) +Tr[Pk X (ρl)]+Tr{Pk X [˜ρl ]}+Tr[Pk X (˜ρl)]) (1) (0) (2) (2) (0) (1) +..., −Tr[PkL S L (ρl)] − Tr[PkL S L (ρl)] (1) (0) (1) (0) (1) (0) P −iϕ (0) iϕ (0)† (1) +Tr[PkL S L S L (ρl)] + ..., where X (ρ) ≡ j[e Jj ρ + e ρJj ], X (ρ) ≡ P −iϕ (1) iϕ (1)† (1) (2) where ... denotes corrections of the fourth and higher j[e Jj ρ + e ρJj ], and x˜l = xl +x ˜l +x ˜l + (2) order. The second-order dynamics in the first line, W˜ , P −iϕ (0) iϕ (0)† −iϕ (0) ... ≡ Tr{[e J + e J ]ρl}/2 + Tr{[e J + is trace-preserving and and can be shown to be positive j j j j iϕ (0)† (1) −iϕ (0) iϕ (0)† (2) (see below and cf. Supplemental Material of Ref. [S5]), e Jj ]˜ρl }/2 + (Tr{[e Jj + e Jj ]˜ρl }/2 + (2) −iϕ (1) iϕ (1)† (1) and thus we denote them as W . Tr{[e Jj + e Jj ]˜ρl }) + ... with xl being the While the third-order corrections are generally asymptotic rate of integrated homodyne current in the present, breaking the positivity of the dynamics requires unperturbed dynamics inside Hl. (3) (2) [W˜ ]kl 6= 0 when [W ]kl = 0. This is not possible, Finally, for the statistics of time-integrals of system and, thus, the corrections to the positivity of W˜ are observables (see Sec.E3c) at most of forth order. Indeed, let us rescale the (W˜ ) = [W(2)] − h δ m˜ (A17) perturbations J (1) and H(1) by a parameter δ. It can h kl kl kl l j (1) (1) tL (1) −h((1−δkl)Tr{PkM[˜ρ ]}+δklTr{(Pl −1)M[˜ρ ]} be shown that limt→∞[∂δ|δ=0Tr(Pke ρl)]/t = [W˜ ]kl, l l 2 tL ˜ (2) ˜(1) limt→∞[∂δ |δ=0Tr(Pke ρl)]/t = 2[W ]kl and +Tr[Pk M(ρl)]) 3 tL ˜ (3) (2) (2) limt→∞[∂δ |δ=0Tr(Pke ρl)]/t = 6[W ]kl. Since 1 tL −h((1−δkl)Tr{PkM[˜ρl ]}+δklTr{(Pl − )M[˜ρl ]} Tr(Pke ρl) ≥ 0 from Pk ≥ 0 and ρl ≥ 0, while (2) (1) (1) tL (1) ˜ ˜ Tr(Pke ρl)|δ=0 = δkl, it follows that [W˜ ]kl = 0 +Tr[Pk M(ρl)]+Tr{Pk M[˜ρl ]}) + ..., (there are no first-order corrections to the dynamics), where M(ρ) ≡ Mρ + ρM, m˜ = m +m ˜ (1) +m ˜ (2) + ˜ (2) ˜ (2) l l l l [W ]ll ≤ 0 and [W ]kl ≥ 0 for k 6= l (the second-order (1) (2) (2) ... ≡ Tr(Mρl) + Tr[Mρ˜l ] + Tr[Mρ˜l ] + ... and ml is dynamics is positive). Analogously, when [W˜ ]kl = 0, (3) the asymptotic observable average in the unperturbed it follows that [W˜ ]kl = 0. dynamics inside Hl. Note that there are no first-order corrections to the Statistics of quantum trajectories. For the tilted gener- difference to corresponding generator for the classical ator in Eq. (48), which encodes the statistics of jumps in dynamics for continuous measurements [Eqs. (A15) quantum trajectories, we have [cf. Eqs. (A11) and (A12) and (A16); cf. Eqs. (47) and (E212)]. Furthermore, if and see Eq. (49)] there is no decay subspace, the first-order corrections ˜ (2) −s (2) in (Ws)kl = [W ]kl +(e −1)[J +δklµ˜l ] (A15) are not present in the rates µl and xl. Finally, if −(e−s −1) × the first-order corrections in second and third line of Eq. (A17) are present, the corrections to the classicality (0) (2) 1 (0) (2) ((1−δkl)Tr{PkHeff [˜ρl ]}+δklTr{(Pl − )Heff [˜ρl ]} are necessarily of the second order and the corresponding (1) (1) 1 (1) (1) bound derived in Sec.E3d is saturated in terms of the +(1−δkl)Tr{PkHeff [˜ρl ]}+δklTr{(Pl − )Heff [˜ρl ]} order of the figure of merit [this requires 1H M1H 6= 0 ˜(2) (0) ˜(1) (0) (1) ˜(1) (1) l l +Tr[Pk Heff (ρl)]+Tr{Pk Heff [˜ρl ]}+Tr[Pk Heff (˜ρl)]) for some l; cf. Eqs. (A7), (A15), and (A16)]. +..., Hierarchy of metastabilities. When the classical where H(0)(ρ) ≡ −iH(0)ρ + ρH(0)†, H(1)(ρ) ≡ eff eff eff eff second-order dynamics in Eq. (A14) features m2 station- (1) (1)† (0) (0)† (1) (1)† (0) −i[H , ρ] − [Jj Jj + Jj Jj ]ρ + ρ[Jj Jj + ary probability distributions pl2 , l2 = 1, ..., m, we have (0)† (1) in (2) J J ]}/2, µ˜ =µ ˜l + [W ]ll and µ˜l = µl + X (2) j j l [W ]kl = 0, k2 6= l2 (A18) (1) (2) (0)† (0) (0)† (0) (1) µ˜l +µ ˜l + ... ≡ Tr[Jj Jj ρl] + Tr[Jj Jj ρ˜l ] + k∈Sk2 (0)† (0) (2) (0)† (1) (1) (1)† (0) (1) l∈Sl2 {Tr[Jj Jj ρ˜l ] + Tr[Jj Jj ρ˜l ] + Tr[Jj Jj ρ˜l ] + X (2) (1)† (1) [W ]kl = 0, Tr[Jj Jj ρl]} + ... with µl encoding the stationary ac- (2) k∈D tivity in the unperturbed dynamics inside Hl, (J )kl ≡ l∈Sl2 35

where sets Sl2 consist of the labels l = 1, ..., m for [cf. the discussion in Sec.C2a, in particular Eqs.(C42)- which (p ) > 0 and D is the complement of ∪m2 S (C44)], leading to the distance in the trace norm equal l2 l l2=1 l2 [cf. Eq. (A6)]. We denote the corresponding projection kP[ρ(0)]k − 1. As for any density matrix ρ from the (0) (2) (0) triangle inequality we have as P2 and the reduced resolvent for W at 0 as R2 . ˜ (3) Note that the third-order corrections W do not change kP[ρ(0)]k − 1 ≤ kP[ρ(0)] − ρk + kρk − 1 (B20) the structure above or the positivity of the dynamics = kP[ρ(0)] − ρk, [cf. Eq. (A7)]. The metastable phases in the second metastable man- the choice of ρ as diagonal in the eigenbasis of P[ρ(0)] P ifold arise from ρ2,l2 ≡ l=1m (pl2 )lρl, l2 = 1, ..., m2. with the eigenvalues given as in Eq. (C43) is optimal. The first-order corrections are as given in Eq. (A11) P (1) (1) (0) (3) and by m [˜p ] ρ , where ˜p = −R W˜ p . l=1 l2 l l l2 2 l2 The latter contribution does not affect the positivity of b. Bound on metastability of states closest to Eq. (4) in the metastable phases since it only redistributes the prob- main text ability within the supports Sl2 of pl2 , l2 = 1, ..., m2. Therefore, the corrections to positivity are at most of We now show that for state ρ chosen as the closest the second order. Similarly, the projections on ρ2,l are state to P[ρ(0)] we have P 2 P2,l2 ≡ l=1m (vl2 )lPl, l2 = 1, ..., m2, where are the dual vectors to m2 stationary probability distributions, kρ − P(ρ)k ≤ (2 + C+)C+ . 2C+. (B21) T Pm2 v pl = δk l , so that vl ≥ 0, vl = 1, and k2 2 2 2 2 l2=1 2 Indeed, we have [cf. Eq. (12)] (vl2 )l = 1 for l ∈ Sl2 [cf. Eq. (A5)]. Their first-order per- P (1) turbation is as given in Eq. (A12) and by m [˜v ]lPl, l=1 l2 kρ − P[ρ(0)]k ≤ C+ (B22) (1) (0)T ˜ (3)T where ˜vl = −R2 W vl2 does not affect the the 2 and positivity of ˜vl2 . We conclude that the corrections to the classicality are at most of the second order. kP(ρ) − P[ρ(0)]k ≤ kPk kρ − P[ρ(0)]k (B23) The dynamics after the second metastable regime is ≤ (1 + C )C , governed by a generator W˜ 2 at most of the fourth order + + [cf. Eq. (A14)], where we used P2 = P. These lead to Eq. (B21) thanks ˜ (0) ˜ (4) (0) to the triangle inequality. W2 = P2 W P2 + ..., (A19) where W˜ (4) is the fourth-order correction to W˜ and ... 2. Metastable regime denotes the corrections of the fifth and higher order. This (0) ˜ (3) (0) ˜ (3) (0) is due to the fact that P2 W R2 W P2 = 0, as the third-order corrections do not facilitate the dynamics We now relate the metastable regime directly to the between supports of ρ , l = 1, ..., m. spectrum of the master operator (cf. Sec.IIB). For the 2,l2 2 metastable regime, we have [cf. Eq. (10) and see below]

t00λR t00L B. METASTABILITY IN OPEN QUANTUM e m+1 ≤ ke − Pk ≤ CMM, (B24a) SYSTEMS t0λR t0L 1 − e m ≤ ke − Pk ≤ CMM, (B24b) t0L In this section, we first discuss the properties of the 0 I ke − Pk | sin(t λk)| ≤ 0 . CMM. (B24c) projection on the low-lying modes in Eq. (4). We then 1 − ket L − Pk discuss a quantitative relation of the metastable regime to the spectrum of the master operator in the context of Therefore, corrections in Eq. (10). Finally, we consider timescales of 00 ln(CMM) the initial relaxation and the long-time dynamics defined t ≥ R , (B25a) in analogy to Eq. (10). λm+1 0 −CMM t . R , (B25b) λm 1. Projection on low-lying modes C t0 MM , k ≤ m. (B25c) . |λI | a. Derivation of Eq. (12) in the main text k where the last inequality holds for t0 − t00 ≥ t00. Indeed, I As the projection P on the MM in Eq. (4) preserves the note that | sin(tλk)| . CMM for k ≤ m is valid for trace, the closest state (density matrix) diagonal in the all times within the metastable regime, so that there eigenbasis of P[ρ(0)] is that with the spectrum given by a exists an integer l such that |tλk − lπ| . CMM. When closest probability distribution to the spectrum of P[ρ(0)] t0 − t00 ≥ t00 this implies l = 0 [this also holds when 36 t00/n ≤ t0 − t00 ≤ t00/(n − 1) with n ≥ 2 provided that a. Definitions nCMM 1]. We define the final relaxation time τ in terms of the Derivation. The results in Eq. (B24) follow directly closeness to the stationary state from the following bounds, τL sup kρ(t) − ρssk = ke − Pssk = c, (B27) tλR tL ρ(0) e k ≤ ke − Pssk, (B26a) tλR tL e k ≤ ke − Pk, k > m, (B26b) where Pss is the projection on the stationary state and c < 1 is a constant typically chosen as 1/e. Due to the tλR tL 1 − e k ≤ ke − Pk, k ≤ m, (B26c) contractivity of the dynamics, this timescale is uniquely ketL − Pk defined and for t ≤ τ we also have kρ(t) − ρssk ≤ c [we | sin(tλI )| ≤ , k ≤ m. (B26d) have ke(t−τ)Lk ≤ 1]. k 1 − ketL − Pk We can analogously define the initial relaxation time τ 00 as Moreover, in Eq. (B26b) can be also replaced by k(I − tL 00 P)e k. keτ L − Pk = c00, τ 00 ≤ t00, (B28) To derive Eq. (B26) we repeatedly use the fact that 00 00 |Tr{OX [ρ(0)]}| ≤ kOkmaxkX k for an observable O and a where we require c > CMM for τ to exist [e.g., valid superoperator X . when c00 = 1/e; cf. Eq. (11)]. ∗ First, for a real eigenvalue λk = λk, Lk can be chosen Similarly, the smallest timescale of the long-time dy- † namics τ 0 can be defined as Hermitian (Lk = Lk) and we consider O = Lk and choose ρ(0) as the pure state corresponding to the eigenvalue of 0 keτ L − Pk = 1 − c0, t0 ≤ τ 0 ≤ τ, (B29) Lk with the maximal eigenvalue. By considering then tL tL X = e − P and X = e − P, we obtain Eq. (B26). 0 ss where we require kP − Pk ≥ 1 − c0 + ket L − Pk for Second, for a complex eigenvalue λ with a correspond- ss k τ 0 > t0 to exist, and c + c0 ≤ 1 to also get τ 0 ≤ τ. ing non-Hermitian eigenmode Lk, we consider ρ(0) corresponding to the maximum |c | = |Tr[L ρ(0)]|. For k k Other definitions. We note that when the relaxation iφ −iφ † an observable Lk(φ) ≡ (e Lk + e Lk)/2, we have time is instead of Eq. (B27) defined as the inverse of R kLk kmax ≤ |ck|. Indeed, otherwise there would ex- the maximal rate of an upper exponential bound on ap- 0 0 0 ist an initial state ρ (0) with ck = Tr[Lkρ (0)] such proaching the stationary state R 0 0 iφ that |Tr[Lk ρ (0)]| = |Re(cke )| > |ck|, but this re- 0 iφk tL − t quires |ck| > |ck|. Let φk be such that e ck = ke − P k ≤ e τ , (B30) R ss I tλk |ck|. We then have Tr[Lk(φk − tλk)ρ(t)] = |ck|e , which leads to Eqs. (B26a) and (B26b) [as well as where Pss is the projection on the stationary state, we R I tλk tL can define the initial relaxation analogously with respect | cos(tλk) − e | ≤ ke − Pk for k ≤ m]. Further- R to times only before the metastable regime as I tλk more, Tr[Lk(φk)ρ(t)] = |ck| cos(tλk)e , which leads to R R tλk I tλk tL t Eq. (B26c) as 1 − e ≤ |1 − cos(tλ )e | ≤ ke − Pk tL − 00 00 k ke − Pk ≤ e τ , t ≤ t . (B31) R I tλk tL for k ≤ m [as well as | cos(tλk)e | ≤ ke − Pssk and R 0 I tλk tL The smallest timescale of the long-time dynamics τ can | cos(tλk)e | ≤ ke − Pk for k > m]. Finally, from R be instead defined as the inverse of the minimal rate of a I tλk Tr[Lk(φk − π/2)ρ(t)] = |ck| sin(tλk)e we arrive at R lower exponential bound on departing from a metastable I tλk tL tL | sin(tλk)|e ≤ min[ke − Pssk, ke − Pk]. For k ≤ m state, so that this together with Eq. (B26c) leads to Eq. (B26d). tL − t 0 Third, for an eigenvalue λk corresponding to a Jordan ke − Pk ≤ 1 − e τ0 , t ≤ t. (B32) block, we consider Lk corresponding to its last mode, 0 00 so that Tr{LkL[ρ(0)]} = ckλk (i.e., in this case Lk is We have that for c = c = c = 1/e the timescales defined an eigenmode of L). Then, the above discussion applies in Eqs. (B27), (B28), and (B29) are lower bounds on the directly. definitions in Eqs. (B30), (B31), and (B32). tL tL We note that, as e − Pss = (I − Pss)e , another definition of the initial relaxation time τ 00 that correspond to Eq. (B27) is 3. Relaxation times 00 k(I − P)eτ Lk = c00, τ 00 ≤ t00, (B33) Here, we first discuss the definitions of the initial relax- 00 0 00 00 ation time τ and the smallest timescale τ of the long- where we assume c ≥ (2 + C+)CMM for τ to exist. Note tL tL tL time consistent with Eq. (10). We then consider their that, since k(I − P)e k ≤ (2 + C+)ke − Pk and ke − tL tL tL relation to the master operator eigenvalues. Pk ≤ k(I − P)e k + kPe − Pk . k(I − P)e k + CMM, 37 for the initial relaxation time defined in Eq. (B28) can be First, we consider the speed of dynamics in the trace bounded from below and above by the relaxation time in norm given by kLetLk. In particular, we make use of the 00 00 t L t L R t1 tL Eq. (B33) corresponding to c /(2 + C+) and c − CMM, fact that ke 1 − e 2 k ≤ dtkLe k follows from the t2 respectively, rather than c00. triangle inequality. Finally, the smallest timescale of the dynamics can be At any time, the speed is bounded as kLetLk ≤ kLk tL tkLk considered as 1/kLk, as ke − Ik ≤ e − 1 . tkLk since ketLk = 1. Therefore, for τ 00 in Eq. (B28), from the Taylor series expansion. Analogously, we can 00 define c00 − ket L − Pk c00 − C t00 − τ 00 ≥ ≥ MM . (B37) 1 kLk kLk τ 0 = . (B34) 00 kLMMk Analogously, by replacing c with 1/e, we obtain a bound for τ 00 in Eq. (B31). For times after the metastable regime, the speed is b. Relation to master operator spectrum t0L 2 0 00 bounded by kLe k . kLMMk+2CMMkLk when t ≥ 2t 0 0 0 t0L [? ]. Therefore, τ − t ≥ (c − ke − Pk)/kLMMk ≥ 0 00 0 2 0 For the definitions of the timescales τ, τ , and τ (c − CMM)/(kLMMk + 2CMMkLk) for τ in Eq. (B29). in terms of the projections on the low-lying modes in Using Eq. (B39) below we then obtain Eqs. (B28) and (B29), from Eq. (B26) we have 0 0 2 0 τ c + 2CMMt kLk ln(c) 0 & 0 , (B38) t CMM(1 + 2CMMt kLk) τ ≥ R , (B35a) λ2 00 where the right-hand side is 1 when C2 t0kLk c0. 00 ln(c ) MM τ ≥ , (B35b) The bound also holds for τ 0 defined in Eq. (B32) when c0 λR m+1 is replaced by 1/e. Finally, for the definition in Eq. (B32), ln(c0) 0 0 0 we simply have τ /t & 1/CMM as τ ≤ R , (B35c) λm 0 0 t kLMMk CMM, (B39) 0 − ln(c ) . τ ≤ I , k ≤ m, (B35d) |λk| which we prove below for the metastable regime such that t0 − t00 ≥ t00 [or t00/n ≤ t0 − t00 ≤ t00/(n − 1), where where the last inequality holds for t0 − t00 ≥ t00 or t00/n ≤ 0 00 00 n ≥ 2 allows for nCMM 1]. t − t ≤ t /(n − 1) with n ≥ 2 provided that nCMM 1; cf. Eq. (B25c). We can also find a quantitative relation between the Other definitions. For τ, τ 00 and τ 0 defined in initial relaxation time in Eq. (B33) and the metastable Eqs. (B30), (B31), and (B32), respectively ln(c), ln(c00) regime. By noting that k(I − P)entLk ≤ k(I − P)etLkn. and ln(c00) are all replaced by −1 in Eq. (B35). For τ 00 Therefore, for τ 00 defined in Eq. (B33) with c00 < 1 we defined in Eq. (B33), Eq. (B35b) holds directly. Finally, have below we show that 00 t t00L R ≤ dlogk(I−P)eτ00Lkk(I − P)e ke (B40) 1/kLMMk ≤ 1/ max |λk| ≤ −1/λm, (B36) τ 00 k≤m t00L ≤ logc00 k(I − P)e k, which provides an upper bounds for the definition in 00 00 Eq. (B34). where we assumed c < 1. For τ defined in Eq. (B28), 00 00 the bound holds as well with c replaced by c (2 + C+) 00 Derivation of Eq. (B36). The lower bound on the norm (assumed < 1), while for τ in Eq. (B31) by (2 + C+)/e. of the master operator projected on the low-lying modes, We note that an analogous result holds for times after the final relaxation, that is, max2≤k≤m |λk| ≤ kLMMk, can be obtained by considering system states and observables as in the derivation of t tL iϕk ≤ dlog τL ke − Psske (B41) Eq. (B26). Indeed, let ϕk such that that e λk = |λk|. τ ke −Pssk This gives Tr{Lk(φk + ϕk)L[ρ(0)]} = |ck||λk|, and thus tL = dlogcke − Psske |ck||λk| ≤ kLk(φk + ϕk)kmaxkLMMρ(0)k ≤ |ck|kLMMk. for t ≥ τ, where τ is defined in Eq. (B27) and we assume c < 1. For τ defined in Eq. (B30), c above is instead c. Relation to metastable regime replaced by 1/e and the equality by ≤.

Here, we discuss the relation between timescales τ 00 Derivation of Eq. (B39). We have ketLMM P − Pk ≤ 0 tL 00 0 and τ and the metastable regime. For all deﬁnitions, kPkke − Pk ≤ (1 + C+)CMM for t ≤ t ≤ t and a we obtain τ 00 < t00 and t0 τ 0. similar bound exists for time t ≤ t0 − t00, as follows. We 00 00 have ke(t+t )LMM P − etLMM Pk ≤ ketLkket LMM P − Pk ≤ 38

tL (t+t00)L (1 + C+)CMM, so that ke MM P − Pk ≤ ke MM P − by the states closest to their projections in Eq. (17) tL t00+tL e MM Pk + ke MM P − Pk ≤ 2(1 + C+)CMM for t ≤ [with an increase of corrections in Eq. (14) by at most 0 00 0 00 00 t −t (in particular, when t −t ≥ t this holds for all t ≤ CMM + C˜+ + C+; see Eq. (C59) below], due to linear de- 00 t ). Next, let us consider time t such that (2 + c)CMM = pendency in the space of coefficients, this again will lead tL tkLMMk, where CMM c ≤ 1, so that ke MM P − Pk ≈ to a nonunique decomposition. 0 tkLMMk = (2+c)CMM. It then follows that t fulfills t > t A lower number of states than m in Eq. (14) indicates or t0 − t00 < t < t00. The former case directly implies degeneracy of the description of the MM in the space of Eq. (B39), while the latter case leads to a contradiction, coefficients (see e.g., Ref. [S7]), leading to the effective as when t0 − t00 ≥ t00 it is not possible, but for t00/n ≤ lower dimension of the metastable state manifold than t0 − t00 ≤ t00/(n − 1), we have 0 < t00 − t < (n − 1)t so m − 1. This case will be discussed elsewhere. 00 that (2+c)CMM < t kLMMk < n(2+c)CMM which, when t00L nCMM 1, is in the contradiction with ke MM P−Pk . CMM. This ends the proof of Eq. (B39). 2. Test of classicality

a. Distance of barycentric coordinates to probability C. CLASSICAL METASTABILITY IN OPEN distributions QUANTUM SYSTEMS We now show that the distance of barycentric coordi- In this section, we ﬁrst discuss the correctness of the T nates ˜p = (˜p1, ..., p˜m) to the set of probability distribu- deﬁnition of classical metastability in Eq. (14) in terms tions is given by k˜pk1 − 1. of the number of metastable phases. We then consider T For any probability distribution p = (p1, ..., pm) , we the test of classicality and prove that Ccl in Eq. (21) have in the L1-norm is the maximal distance of the barycentric coordinates m of a metastable state from a given simplex of candidate X k˜p − pk ≡ |p˜ − p | (C42) metastable phases when measured by L1-norm. We fur- 1 l l ther prove the bound on the corrections in Eq. (22). We l=1 X X also derive a similar bound on the average distance. Fi- = (−p˜l + pl) + |p˜l − pl| nally, we consider the optimality of the metastable phase l:p ˜l<0 l:p ˜l≥0 construction in Eq. (22) in the context of corrections Eq. (14). As a corollary of derivations presented in this X X ≥ (−p˜l + pl) + (˜pl − pl) section, the second line of Eq. (12) in the main text fol- l:p ˜l<0 l:p ˜l≥0 lows. X = 2 (−p˜l + pl)

l:p ˜l<0 m 1. Definition of classical metastability X ≥ 2 max (−p˜l, 0) ≡ k˜pk1 − 1, In Eq. (14) we assumed the number of states to be l=1 equal to the number m of low-lying modes in the spec- where in the second and last line we used the positivity of trum of the master operator in Eq. (1). We now justify pl ≥ 0, l = 1, ..., m, the third line follows from the triangle this assumption. Pm inequality, and the fourth line follows from p˜l = 1 = A higher number of states than m Eq. (14) necessarily Pm l=1 l=1 pl (note that the final result corresponds to the leads to linearly dependent matrices after the projection triangle inequality). We now construct the probability onto the low-lying modes as in Eq. (17). Therefore, the distribution for which the lower bound is saturated. We decomposition of P[ρ(0)] into such states in the space Pm define ∆ = max (−p˜k, 0) and of coefficients in not unique [even with the additional k=1 assumption of the (approximate) positivity of decompo- pl ≡ 0 if p˜l ≤ 0, (C43) sition]. Therefore, when the states in Eq. (14) are well h X i approximated by their projection on the low-lying modes pl ≡ p˜l − min p˜l, ∆ − (˜pk − pk) if p˜l > 0, in Eq. (17), we conclude that there are no more than k0 m states for the decomposition to be uniquely defined [uniqueness here is defined up to corrections in Eq. (14)]. l = 1, ..., m. In Eq. (C43) all the negative coordinates in Furthermore, when some of the states in Eq. (14) dif- (˜p1, ..., p˜m) are replaced by 0 [which saturates the second fer significantly from their projection on the low-lying inequality in Eq. (C42)]. To obtain the sum of probabil- modes, i.e., are not metastable, not all probabilistic mix- ities equal 1, the remaining nonzero positive coordinates tures of the states in Eq. (14) are metastable, and thus of ˜p, which sum up to 1 + ∆, are individually reduced the probabilities, although possibly uniquely defined, do while keeping their positivity [which guarantees satura- not represent degrees of freedom of the MM. Moreover, tion of the first inequality in Eq. (C42)], in Eq. (C43) we although in this case the candidate states can be replaced chose to set the remaining p˜l > 0 either to 0 or reduced 39 by the remaining difference of the probability sum to 1. being the closest state to ρ˜l [cf. Eq. (12)] and kρ˜lk ≤ This gives kρ˜l − ρlk + kρlk ≤ 1 + C+. Therefore, using Eq. (21) we arrive at Eq. (22). k˜p − pk1 = k˜pk1 − 1. (C44)

Note that the choice of optimal p in Eq. (C43) is c. Optimality of test of classicality generally not unique. We now consider how the construction of classical We conclude that the maximal distance of barycentric approximation in Eq. (22) compares to a given set of coordinates to the simplex of probability distributions states in Eq. (14). over all initial states of the system is given by Eq. (21). As a barycentric coordinate p˜ is bounded from below l Bound on C in terms of corrections in Eq. (14). We by the minimal eigenvalue of P˜ in the dual basis, from cl l now bound the corrections to the classicality in Eq. (21) Eq. (C44) we arrive at Eq. (23). by the corrections in Eq. (14). First, we have

m m The average distance Ccl for uniformly distributed pure n o X ˜ ˜ X initial states of the system is bounded by k˜p − pk1 = Tr PlP[ρ(0)] − Pl pkρk (C48) l=1 k=1 m ˜− m m X −Tr(Pl ) C˜ n o Ccl ≡ max k˜pk1 − 1 ≤ 2 (C45) cl X X ρ(0) dim(H) = 1 + Tr PlP[ρ(0)] − Pl pkρk l=1 2 l=1 k=1 m m − ˜ with P˜ being P˜ restricted to its negative eigenval- Ccl X n X o l l ≤ 1 + Tr P P[ρ(0)] − p ρ ues and dim(H) the dimension of the system Hilbert 2 l k k ˜ l=1 k=1 space. By construction we have Ccl ≤ Ccl, and Ccl ≤ ˜ m Ccl X dim(H)Ccl also holds [cf. Eqs. (23) and (24)]. Indeed, = 1 + P[ρ(0)] − plρl ˜− 2 p˜l can be bounded from below by overlap with Pl . As l=1 − p˜l = Tr[P˜lρ(0)], while P ≤ 0, we have max (−p˜l, 0) ≤ l where [˜p(t)] ≡ Tr[P˜ ρ(t)], and P is the POVM deﬁned −Tr(P −ρ(0)), with the right-hand side being linear in l l l l in Eq. (26), l = 1, ...., m , so that P ≥ 0, which we use ρ(0). Averaging ρ(0) over initial states (uniformly dis- l in the inequality, while Pm P = 1, which leads to the tributed pure states or mixed states in Hilbert-Schmidt l=1 l last equality. Furthermore, metric [S8, S9]) gives ρ¯ = 1/dim(H), and thus we arrive at Eq. (C45). m X Furthermore, the average corrections in Eq. (14) are P[ρ(0)] − plρl ≤ kP[ρ(0)] − ρ(t)k bounded as l=1 m m X X + ρ(t) − plρl ρ(t) − plρl . Ccl + C+ + CMM, (C46) l=1 l=1 ≤ C˜+ + C, (C49) [see the derivation of Eq. (22) below]. where C are the maximal corrections in Eq. (14) over the set of metastable states and time t is chosen as in b. Derivation of Eq. (22) in the main text Eq. (13). Therefore,

C˜cl From the triangle inequality we have for the classical Ccl ≤ maxk˜p − pk1 ≤ 1 + C˜+ + C , (C50) approximation of metastable states in the trace norm ρ(0) 2

m m so that [cf. Eq. (21)] X X ρ(t) − plρl ≤ ρ(t) − p˜lρ˜l (C47) mCcl l=1 l=1 ≤ m C˜+ + C , (C51) 1 + mC m m m m cl X X X X + p˜lρ˜l − plρ˜l + plρ˜l − plρl and thus l=1 l=1 l=1 l=1 m m C C˜ + C, (C52) X X cl . + ≤ C + |p˜ − p | kρ˜ k + p kρ˜ − ρ k MM l l l l l l ˜ ˜ l=1 l=1 when m(C+ + C) 1, in which case Ccl 1 follows [cf. Eq. (13) and see Sec.D2]. ≤ CMM + (1 + C+) k˜p − pk1 + C+, where in the second inequality we used Eq. (10), while Bound on the stationarity of states in Eq. (14). We the third inequality follows from kρ˜l − ρlk ≤ C+ for ρl now prove that for states in Eq. (14) we have 40

Bound on corrections for metastable phases replaced by closest states to their projections. Let ρ0 be the closest ˜ l kρl − P(ρl)k . 2 C+ + C , (C53) state to the projection P(ρl) of ρl in Eq. (14), so that 0 kP(ρl) − ρlk ≤ C+ [cf. Eq. (12)]. From the triangle in- when C˜cl 1 or m(C˜+ + C) 1. Therefore, in this case equality we then have [cf. Eq. (C59)] the states in Eq. (14) are metastable. m X 0 X Consider ρl in Eq. (14) as an initial state of the system, ρ(t) − plρl ≤ P[ρ(0)] − plP(ρl) (C60) l = 1, ..., m. We denote ρl(t) the corresponding state at l l=1 time t. By deﬁnition, there exists a probability distribu- m (l) X 0 + pl kP(ρl) − ρlk tion pk , k = 1, ...m such that for all times within the metastability regime l=1 ˜ . CMM + C+ + C + C+. m X (l) Equations (C59) and (C60) actually hold for any number ρl(t) − pk ρk ≤ C, (C54) k=1 of states in Eq. (14) (see the discussion on the correctness of classicality deﬁnition in Sec.C1). We conclude that and therefore from Eq. (C49) corrections in Eq. (14) can increase in the leading order ˜ m by at most CMM + C+ + C+ when the states in Eq. (14) X (l) ˜ are replaced by closest states to their projections. P(ρl) − pk ρk ≤ C+ + C. (C55) k=1 We also have [cf. Eqs. (C49) and (C48)] D. CLASSICAL METASTABLE PHASES m X (l) (l) X (l) ρl − pk ρk ≤ |1 − pl |kρlk + pk kρkk(C56) k=1 k6=l Here, we prove the bounds in Eqs. (28)–(32) of the (l) (l) main text. We begin by discussing the properties of = k˜p − p k1 the dual basis. We then derive relations between dis- ˜ Ccl tances measured by trace norm for density matrices and ≤ 1 + C˜+ + C , 2 L1 norm for barycentric coordinates leading to Eqs. (28) and, thus, we arrive at and (29). We also prove related bounds on scalar products of metastable phases. Finally, we derive Eqs. (31) m m and (32). X (l) X (l) ρl − P[ρl] ≤ ρl − pk ρk + P(ρl) − pk ρk k=1 k=1 C˜ 1. Properties of dual basis in classical metastable ≤ 2 + cl C˜ + C . (C57) 2 + manifolds

Here, we discuss properties of minimum and maximum Bound on corrections for metastable phases replaced by eigenvalues of the dual basis in Eq. (20) and ﬁnd corre- their projections. We have sponding bounds on its norms in terms of Ccl in Eq. (21). ˜ m m We also show that Ccl in Eq. (23) can be understood X X P[ρ(0)] − p P(ρ ) ≤ kPk P[ρ(0)] − p ρ a distance to operators in a certain POVM and discuss l l l l cross-correlations of the elements of the dual basis in l=1 l=1 metastable states. ≤ (1 + C+)(C˜+ + C), (C58) where in the second inequality we used Eq. (C49). There- a. Properties of dual basis fore, from the triangle inequality

X Let p˜max be the maximal eigenvalue of P˜ , ρmax(l) be ρ(t) − plP(ρl) ≤ ρ(t) − P[ρ(0)] l l l the density matrix of the corresponding eigenstate. Let m [˜pmax(l)] =p ˜max(l) = Tr[P˜ ρmax(l)], and pmax(l) be the X k k k max(l) + P[ρ(0)] − plP(ρl) closest probability distribution to ˜p [cf. Eq. (C44)]. l=1 We then have ≤ CMM + (1 + C+)(C˜+ + C) max X max(l) p˜l = 1 − p˜k ˜ . CMM + C+ + C. (C59) k6=l X max(l) Ccl ≤ 1 − max(−p˜ , 0) ≤ 1 + . (D61) k 2 k6=l 41

Furthermore, for the state ρl from which ρ˜l is obtained c. Cross-correlations of dual basis by the projection on the low-lying modes [cf. Eq. (17)] max ˜ we have, by the definition, p˜l ≥ Tr(Plρl) = 1, so that Below we show that the cross-correlation for the dual from Eq. (D61) basis measurement is diagonal in metastable states, m X ˜ ˜ max Ccl |Tr{PkPlP[ρ(0)]} − δk,lp˜l| (D68) |1 − p˜l | ≤ . (D62) 2 k=1 ˜ 3Ccl ˜ Ccl min ˜ . |p˜l| 3C+ + Ccl + + |1 − p˜l| C+ + 3Ccl + , Let p˜l be the minimal eigenvalue of Pl. We obtain that 2 2 min min ˜ p˜l ≤ 0 by considering k 6= l, p˜l ≤ Tr(Plρk) = 0 and where P˜l is the superoperator describing the action of thus [cf. Eq. (21)] measuring P˜l [see Eq. (9) in the main text] and p˜l = Tr[P˜lρ(0)] = Tr{P˜lP[ρ(0)]}. This further leads to min Ccl m − p˜l ≤ (D63) X 2 max |Tr[P˜kP˜l(˜ρn)] − δk,lδn,l| (D69) 1≤n≤m k=1 as well as [cf. Eq. (23)] 3C C max 3C + C˜ + cl , C + 3C˜ + cl , . + cl 2 + cl 2 m X which approximation is responsible for the classical C˜ = (−p˜min). (D64) cl l dynamics of autocorrelations discussed in Sec. VA5 l=1 of the main text. The results here can be seen as a consequence of the approximate disjointness of the As the norm P˜ is by definition equal the maximal l max supports of metastable phases proven in Sec.D3b. or minus the minimal eigenvalue of P˜l, we conclude Derivation of Eq. (D69). For the POVM elements de- 0 ˜ Ccl fined in Eq. (D66), and ρn being the closest state to ρñ Pl ≤ 1 + . (D65) max 2 [cf. Eq. (12)], n = 1, ..., m, we have m X 0 |Tr[PkPl(ρn)] − δk,lδn,l| (D70) k=1 b. Distance to POVM 1 0 0 = Tr[( − Pl)Pl(ρn)] + |Tr[PlPl(ρn)] − δn,l]| 1 0 2 0 Consider operators [cf. Eq. (26)] = Tr[( − Pl)Plρn] + |Tr(Pl ρn) − δn,l| 0 0 where we used Tr[PkPl(ρn)] = Tr[PkPl(ρn)], the positiv- 0 Pm 1 P˜ − p˜min1 ity of Pk and Pl(ρn), l=1 Pl = , and the fact that the l l 1 Pl ≡ ˜ , l = 1, ..., m. (D66) bases of Pl and − Pl are the same. Moreover, 1 + Ccl 2 1 0 0 1 0 Tr[( −Pl)Plρn] ≤ min{Tr(Plρn),Tr[( −Pl)ρn]}, (D71) 2 0 0 Pm 1 Tr[Pl (ρn)] ≤ Tr(Plρn), (D72) We have that Pl ≥ 0 and l=1 Pl = , so that these 1 1 2 0 operators constitute a POVM, i.e., a set of operators for where we used Pl, − Pl ≤ , while |Tr(Pl ρl) − 1| = 2 0 1 0 1 0 which pl ≡ Tr[Plρ(0)], l = 1, ..., m, corresponds to a prob- 1−Tr(Pl ρl) = Tr[( −Pl)ρl]+Tr[( −Pl)Plρl]. Therefore, ability distribution for any initial state ρ(0). m X 0 0 We now estimate the distance of this probability dis- |Tr[PkPl(ρn)]| ≤ 2Tr(Plρn), (D73) tribution to the barycentric coordinates p˜l ≡ Tr[P˜lρ(0)] k=1 [cf. Eq. (20)]. We have m X 0 1 0 |Tr[PkPl(ρl)]| ≤ 3Tr[( −Pl)ρl]. (D74) m ˜ k=1 1 X Ccl min 0 ˜ 0 min k˜p − pk1 = ˜ − p˜l − p˜l (D67) Noting that for n 6= l, Tr(Plρn) = [Tr(Plρn) − p˜ ]/(1 + Ccl 2 l 1 + l=1 2 C˜cl/2) ≤ [(1+Ccl/2)C+ +Ccl/2]/(1+C˜cl/2), while Tr[(1− ˜ Ccl m ! 0 ˜ 0 min ˜ 2 X Pl)ρl] = 1 − [Tr(Plρl) − p˜l ]/(1 + Ccl/2) ≤ 1 − [1 − ≤ ˜ |p˜l| + 1 ˜ ˜ ˜ 1 + Ccl C+Ccl/2]/(1 + Ccl/2) = (Ccl + C+Ccl)/(2 + Ccl), we arrive 2 l=1 at ˜ Ccl m 2 ˜ X 0 2Ccl ≤ ˜ (2 + Ccl) . Ccl, |Tr[P P (ρ )]| ≤ 2C + 2C + C , (D75) 1 + Ccl k l n + ˜ . + cl 2 k=1 2 + Ccl m ˜ X 3Ccl 3 where the first inequality corresponds to the triangle in- |Tr[P P (ρ0)]| C˜ . (D76) k l l . ˜ . 2 cl equality. k=1 2 + Ccl 42 where for the second inequalities in both lines we assumed For L1 norm, we have [cf. Eq. (C48)] C˜cl 1. Finally, we have m (1) (2) X (1) (2) m ˜p − ˜p = |[˜p ]l − [˜p ]l| (D79) 1 X 0 l=1 |Tr[PkPl(ρ )] − Tr[P˜kP˜l(˜ρn)]| (D77) n m k=1 X = Tr[P˜ ρ˜(1)] − Tr[P˜ ρ˜(2)] m l l X 0 ˜ 2 l=1 ≤ |Tr{PkPl[ρn − (1 + Ccl/2) ρñ]}| ˜ ! m k=1 Ccl X = 1 + Tr(P ρ˜(1)] − Tr[P ρ˜(2)] ˜ m l l Ccl X 2 +(−p˜min) 1 + |Tr(P ρ˜ )| l=1 l 2 k n ! k=1 C˜ ≤ 1 + cl min kρ˜(1) − ρ˜(2)k, kρ(1)(0) − ρ(2)(0)k , C˜ C˜ 2 + cl 1 + cl |Tr(P ρ˜ )|, 2 2 l n where Pl is the POVM defined in Eq. (D66). The first inequality in Eq. (28) follows by observing kρ(t) − where we used the fact that the bases of Pl and P˜l are 0 ˜ 2 P[ρ(0)]k ≤ CMM for time t within the metastable regime the same. Noting that |Tr{PkPl[ρn − (1 + Ccl/2) ρñ]}| ≤ 0 2 [cf. Eq. (10)]. Tr{Pk|Pl[ρ −(1+C˜cl/2) ρñ]|} and similarly |Tr(Pkρñ)| ≤ n For the trace norm, we have Tr(Pk|ρñ|), we have m X h i m ρ˜(1) − ρ˜(2) = ρ˜ Tr(P˜ ρ˜(1)) − Tr(P˜ ρ˜(2)) X 0 ˜ ˜ l l l |Tr[PkPl(ρn)] − Tr[PkPl(˜ρn)]| (D78) l=1 k=1 m X 0 ˜ ˜ ≤ (1 + C ) Tr(P˜ ρ˜(1)) − Tr(P˜ ρ˜(2)) ≤ kρn − ρñk + Ccl(1 − Ccl/4)kρñk + l l l=1 C˜cl C˜cl + − p˜min + 1 + kρ˜ k (1) (2) l 2 2 n = (1 + C+) k˜p − ˜p k1, (D80) ˜ ≤ C+ + Ccl(1 + C+) where the inequality corresponds to the triangle inequal- Ccl + C˜cl C˜cl ity and kρ˜kk ≤ 1 + C+ follows from Eq. (12). Therefore, + 1 + (1 + C ) 0(1) 0(2) (1) 2 2 + the distance of the states ρ and ρ closest to ρ˜ (2) C + 3C˜ and ρ˜ , respectively, is bounded as [cf. Eq. (12)] C + cl cl . . + 2 0(1) 0(2) (1) (2) ρ − ρ ≤ (1 + C+) k˜p − ˜p k1 + 2C+. (D81) 0 ˜ 2 0 where we used kPl[ρn −(1+Ccl/2) ρñ]k ≤ kPlkkρn −(1+ C˜ /2)2ρ˜ k ≤ kρ0 − ρ˜ k + C˜ (1 − C˜ /4)kρ˜ k as kP k ≤ 1 Similarly, the distance between states that project on cl n n n cl cl n l ρ˜(1) and ρ˜(2) during metastable regime is bounded by from the complete-positivity of Pl. (1) (2) (1 + C+) k˜p − ˜p k1 + 2CMM [cf. Eq. (10)], which gives the second inequality in Eq. (28). Finally, we note that, since the distance in the trace 2. Trace-norm vs. L1-norm in classical metastable norm is contractive under quantum dynamics [S10], we manifolds have that the following series of inequalities

kρ(1)(0) − ρ(2)(0)k ≥ kρ(1)(t) − ρ(2)(t)k (D82) Here, we discuss how the distances measured in the (1) (2) trace norm for density matrices and by L1 norm for ≥ kρ˜ − ρ˜ k − 2CMM 0(1) 0(2) barycentric coordinates of the MM are related. In partic- ≥ kρ − ρ k − 2CMM − 2C+. ular, we derive Eqs. (28) and (29) of the main text. We also discuss the norms of the long-time generator and the In particular, states that project on ρ˜(1) and ρ˜(2) do not relaxation times. need to be metastable. Therefore, only the distance between their projections [or metastable states, e.g., as in Eqs. (14) or (22)] is bounded in Eq. (D81).

a. Distance between metastable states b. Distance between metastable phases We consider two initial states ρ(1)(0) and ρ(2)(0) projected on low-lying eigenmodes as ρ˜(1) and ρ˜(2) [see From Eqs. (D79) and (D80), we obtain that the dis- Eq. (4)], respectively, that are described by barycentric tance between the projections of the metastable phases coordinates ˜p(1) and ˜p(2). in Eq. (17) is bounded from below as 43

Similarly, as LMM maps any state onto the low-lying modes, analogously to Eq. (D80), we also have 4 ≤ min (kρ˜l − ρ˜kk, kρk − ρlk) k 6= l, (D83) 2 + C˜ m cl X ˜ kLMMk ≡ max ρ˜k Wkl(˜p)l (D89) ρ(0) where k, l = 1, ..., m and ρl is the state that projects k,l=1 on ρ˜l [cf. Eq. (14)]. This follows from k˜pl − ˜pkk1 = 2 ≤ max kρkk maxk˜pk1kW˜ k1 with (˜pl)k ≡ Tr(P˜kρ˜l) = δkl, k, l = 1, ..., m. Note that 1≤k≤m ρ(0) the upper bound from Eq. (D80) is trivial [cf. Eq. (12)]. ˜ 0 ≤ (1 + C+)(1 + Ccl) kWk1, Similarly, for the states ρl closest to the projections in Eq. (17), we have [cf. Eq. (12)] where (˜p)l = Tr[P˜lρ(0)].

4 0 0 − 2C+ ≤ kρk − ρlk , k 6= l. (D84) 2 + C˜cl d. Relaxation time

In analogy to Eq. (D79), by considering the POVM We now show when the relaxation time towards the ˜ min1 max min of two elements: P ≡ (Pl − p˜l )/(˜pl − p˜l ) and stationary state ρss in the trace norm and the relaxation 1 max1 ˜ max min − P = (˜pl − Pl)/(˜pl − p˜l ), we obtain of metastable state within the classical MM are approximately the same. 2 We have max min ≤ min (kρ˜l − ρ˜kk, kρk − ρlk) , k 6= l, (D85) p˜l − p˜l m tW˜ X tW˜ e − P˜ss = max (e )kl − (˜pss)l (D90) so that [cf. Sec.D1] 1 1≤l≤m k=1 ˜ ! 2 Ccl tL ≤ min (kρ˜l − ρ˜kk, kρk − ρlk) , k 6= l, (D86) ≤ 1 + max e (ρl) − ρss 1 + Ccl 2 1≤l≤m 0 ˜ ! while for the states ρl closest to the projections in Ccl tL ≤ 1 + e − Pss . Eq. (17), we have [cf. Eq. (12)] 2

2 0 0 In the second line we used Eq. (D79) by noting that, for − 2C+ ≤ kρk − ρlk , k 6= l, (D87) 1 + C tW˜ tW˜ cl a given l, the vector [(e )1l, ..., (e )ml] corresponds to the barycentric coordinates of etLMM (˜ρ ), which is the which corresponds to the bound in Eq. (29) of the main l projection onto the low-lying modes of the state etL(ρ ) text. l [cf. Eq. (17)]. Therefore, we conclude that the metastable phases are Similarly, we have approximately disjoint with respect to the trace norm tL tL (for the bimodal case m = 2, see also Ref. [S11]). (e − Pss) ≤ (e − Pss)P (D91) tL + (e − Pss)(I − P)

c. Norms of long-time generator with

tL tLMM (e − Pss)P ≤ (1 + Ccl) max ke (˜ρl) − ρssk (D92) We now compare the induced trace norm for LMM in 1≤l≤m Eq. (6) and the induced L1 norm for W˜ in Eq. (34). m X tW˜ Since 1 is 0 eigenvector of L due its trace preserva- ≤ (1 + Ccl)(1 + C+) max (e )kl − (˜pss)l MM 1≤l≤m tion, analogously to Eq. (D79), we have k=1

tW˜ ˜ m = (1 + Ccl)(1 + C+) e − Pss X 1 kW˜ k1 ≡ max |Tr[P˜kLMM(˜ρl)]| (D88) 1≤l≤m and k=1 ˜ ! m tL tL Ccl X (e − Pss)(I − P) = e (I − P) (D93) = 1 + max |Tr[PkLMM(˜ρl)]| 2 1≤l≤m tL tLMM k=1 = e − e P ! n ≤ 2CMM. C˜cl ≤ 1 + max kLMM(˜ρl)k 2 1≤l≤m In Eq. (D92), in the ﬁrst inequality we used P[ρ(0)] = Pm p˜ ρ˜ , where Pm |p˜ | ≤ (1+C ) for any initial state ˜ ! l=1 l l l=1 l cl Ccl ρ(0) [cf. Eq. (21)], and in the second inequality Eq. (D80). ≤ 1 + kLMMk. 2 In Eq. (D93), the inequality holds for times t ≥ t00 and 44

00 min max n such that t/n belongs to the metastable regime, t ≤ where p˜k and p˜k are minimal and maximal eigenval- 0 t/n ≤ t . ues of P˜k, which are introduced to obtain products with When C˜cl 1, from Eqs. (D90)–(D93), the de- positive operators. From the Cauchy-Schwarz inequality cay to the stationary state after the metastable regime we further obtain, [cf. Eqs. (2) and (4)] is equally well captured by the decay √ √ |Tr[ ρ (P˜ − p˜min1) ρ ]| (D100) of the probabilities between metastable phases. There- k k k l q fore, from Eq. (D90), the relaxation time τ˜ with respect min 2 ≤ Tr[(P˜k − p˜ 1) ρl ] to L1-norm is approximately equal the relaxation time τ k q q with respect to the trace norm, max min ˜ min1 ≤ p˜k − p˜k Tr[(Pk − p˜k )ρl ], τ ≈ τ.˜ (D94) ˜ min1 2 max where in the third line we used (Pk − p˜k ) ≤ (˜pk − min ˜ min1 For example, in the case of a perturbation away from a p˜k )(Pk − p˜k ). Analogously, we also have classical first-order phase transition occurring for a finite √ max1 ˜ √ system size, both relaxation times are of the same order |Tr[ ρk (˜pk − Pk) ρl]| (D101) ˜ q in the perturbation. When Ccl 1 does not hold, the ≤ Tr[ρ (˜pmax1 − P˜ )2] relaxation time for the barycentric coordinates of the MM k k k q q is generally longer, τ ≤ τ˜; cf. Eqs. (E190)–(D93). max min max1 ˜ ≤ p˜k − p˜k Tr[ρk (˜pk − Pk)]. Therefore, 3. Orthogonality and disjointness of phases in s ˜ min1 classical metastable manifolds √ √ Tr[(Pk − p˜k )ρl ] Tr( ρk ρl) ≤ (D102) Ccl 1 − 2 a. Bounds on scalar products of metastable phases s Tr[ρ (˜pmax1 − P˜ )] + k k k , Ccl We first derive an upper bound on the scalar product 1 − 2 between square roots of the states ρ that project on ρ˜ l l where we used |p˜max − p˜min| ≥ |p˜max| − |p˜min| ≥ 1 − C /2 [cf. Eqs. (14) and (17)], k k k k cl [cf. Eqs. (D62) and (D63)]. √ √ p For the states ρ that project onto ρ˜ [cf. Eqs. (14) Tr ( ρk ρl) . 2Ccl, k 6= l. (D95) l l and (17)], ρ˜ = P(ρ ), |Tr[(P˜ − p˜min1) ρ ]| = |p˜min| ≤ where k, l = 1, ..., m. For the states ρ0 closest ρ˜ in l l k k l k l l C /2 [cf. Eq. (D63)], while |Tr[ρ (˜pmax1−P˜ )]| = |p˜max− Eq. (17) [cf. Eq. (12)], we instead have cl k k k k 1| ≤ Ccl/2 [cf. Eq. (D62)]. Therefore, from Eq. (D102), q q we arrive at Tr ρ0 ρ0 p2C + 4C , k 6= l. (D96) k l . cl + s √ √ 2Ccl p p Tr( ρk ρl) ≤ , (D103) Since ρ0 and ρ0 are positive and normalized in the Ccl l l 1 − 2 scalar product, the bounds in Eqs. (D95) and (D96) imply that the metastable phases are approximately disjoint which in the leading order gives Eq. (D95). (see also Sec.IV of the main text). For the states closest to the projections ρ˜l 0 Second, we prove the following bounds on the scalar [cf. Eqs. (14)], which we denote ρl, l = 1, ..., m, we have ˜ min1 0 min ˜ 0 product of the metastable phases that Tr[(Pk − p˜k )ρl ] ≤ |p˜k | + kPkkmaxkρl − ρ˜lk ≤ max1 ˜ max ˜ 0 r and Tr[ρk (˜pk − Pk)] ≤ |1 − p˜k | + kPkkmaxkρl − ρ˜lk, Ccl min max |Tr(ρ ρ )| (kρ k +kρ k ), k 6= l, (D97) where |p˜ | ≤ Ccl/2 [cf. Eq. (D63)], |1 − p˜ | ≤ Ccl/2 k l . 2 k max l max k k [cf. Eq. (D62)], kP˜kkmax ≤ 1 + Ccl/2 [cf. Eq. (D65)], and 0 and kρl − ρ˜lk ≤ C+ [cf. Eq. (12)]. Therefore, from r Eq. (D102), we arrive at [cf. Eq. (D103)] 0 0 Ccl 0 0 |Tr(ρkρl)| . +C+(kρkkmax +kρlkmax), k 6= l, (D98) s 2 q q 2C + 2 (2 + C ) C Tr( ρ0 ρ0) ≤ cl cl + , (D104) where k, l = 1, ..., m. The same inequality as in k l Ccl 1 − 2 Eq. (D98) also holds for Tr(˜ρkρ˜l). Note that the scalar product of metastable phases is affected by mixedness of where C+ is a bound from above on the distance kρl −ρ˜lk p 2 p 2 both phases, |Tr(ρkρl)| ≤ Tr(ρk) Tr(ρl ), which may [cf. Eq. (12)]. In the leading order Eq. (D104) gives enhance their approximate orthogonality. Eq. (D96).

Proof of Eqs. (D95) and (D96). For any states ρk and Proof of Eqs. (D97) and (D98). In analogy to ρl, we have Eq. (D99), for any states ρk and ρl, we have √ √ √ √ max min ˜ min1 max min ˜ min1 (˜pk − p˜k )Tr( ρk ρl) = Tr[ ρk (Pk − p˜k ) ρl] (˜pk − p˜k )Tr[ρk ρl] = Tr[ρk (Pk − p˜k ) ρl] (D105) √ max1 ˜ √ max1 ˜ +Tr[ ρk (˜pk − Pk) ρl], (D99) + Tr[ρk (˜pk − Pk) ρl]. 45

From the Cauchy-Schwarz inequality we obtain b. Proof of Eqs. (31)-(33) in the main text

˜ min 1 |Tr[ρk (Pk − p˜k ) ρl]| (D106) We now prove Eqs. (31)-(33). To this aim, we q q ﬁrst derive bounds on the support of general system ≤ Tr[(P˜ − p˜min1)2ρ ] Tr(ρ2 ρ ) k k l k l states. These bounds can also be used to allows show q q approximate disjointness of basins of attractions. We ≤ p˜max − p˜min Tr[(P˜ − p˜min1)ρ ] kρ k , k k k k l k max also discuss the role of the decay subspace. and, similarly, From the discussion in Sec.D1, there exist at least one eigenvalue of P˜l greater or equal 1, and one less or max1 ˜ |Tr[ρk (˜pk − Pk) ρl]| (D107) equal 0. Let Hl be the sum of the eigenspaces of P˜l with q q max 2 2 eigenvalue above or equal ∆, where 0 ≤ ∆ ≤ 1. We will ≤ Tr[ρk (˜p 1 − P˜k) ] Tr(ρk ρ ) k l consider the overlap of system states with Hl. q q ≤ p˜max − p˜min Tr[ρ (˜pmax1 − P˜ )] kρ k k k k k k l max Bounds for general states. For an initial system state ρ(0), we have [cf. Eqs. (D100) and (D101) and see Eq. (D111) below]. Tr[P˜ ρ(0)]≤ Tr[1 ρ(0)]pmax+{1−Tr[1 ρ(0)]}∆(D112) For the states ρl that project onto ρ˜l [cf. Eqs. (14) l Hl l Hl and (17)], ρ˜ = P(ρ ), we have [cf. Eq. (D103)] C l l ≤ Tr[1 ρ(0)] 1+ cl + {1−Tr[1 ρ(0)]} ∆ Hl 2 Hl s Ccl 2 max ˜ Tr(ρk ρl) ≤ (kρkkmax + kρlkmax) ,(D108) where p˜ ≥ 1 + Ccl/2 is the maximal eigenvalue of Pl, Ccl l 1 − 2 and ˜ 1 1 min which in the leading order gives Eq. (D97). Tr[Pl ρ(0)] ≥ Tr[ Hl ρ(0)]∆+{1−Tr[ Hl ρ(0)]} p˜l C For the states closest to the projections ρ˜l 1 1 cl 0 ≥ Tr[ Hl ρ(0)]∆−{1−Tr[ Hl ρ(0)]} ,(D113) [cf. Eqs. (14)], which we denote ρl, l = 1, ..., m, we 2 have [cf. Eq. (D104)] min ˜ where p˜l ≥ −Ccl/2 is the minimal eigenvalue of Pl. s Therefore, from Eq. (D112) we obtain Ccl Ccl 2 + 1+ 2 C+ |Tr(ρkρl)|≤ (kρkkmax +kρlkmax)(D109), ˜ 1 − Ccl Tr[Plρ(0)] − ∆ 2 Tr[1H ρ(0)] ≥ (D114) l Ccl 1 − ∆ + 2 which in the leading order gives Eq. (D98). and from Eq. (D113) Finally,

˜ Ccl ˜ Ccl 0 0 0 0 Tr[Plρ(0)] + 2 Tr[Plρ(0)] + 2 |Tr(ρ ρ )| ≤ |Tr(ρ ρ ) − Tr(˜ρkρ˜l)| + |Tr(˜ρkρ˜l)|, Tr[1H ρ(0)] ≤ ≤ , k l k l l Ccl ∆ ∆ + 2 where [cf. Eq. (D111) below] (D115) as well as [cf. Eq. (23)] 0 0 0 0 |Tr(ρ ρ ) − Tr(˜ρ ρ˜ )| = |Tr[(ρ − ρ˜ )ρ ] + ˜ k l k l k k l ˜ Ccl X 1 − Tr[Plρ(0)] + +Tr[ρ0 (ρ0 − ρ˜ )] + Tr[(ρ0 − ρ˜ )(ρ0 − ρ˜ )]| Tr[1 ρ(0)] < 2 , (D116) k l l k k l l Hk ∆ 0 0 0 0 k6=l ≤ kρk − ρ˜kkkρlkmax + kρl − ρ˜lkkρkkmax 0 0 +kρk − ρ˜kkkρl − ρ˜lk P ˜ ˜ where we used k6=l Tr[Plρ(0)] = 1 − Tr[Pl ρ(0)]. These ≤ C+(kρkkmax + kρlkmax + C+), (D110) bounds are used to argue disjointness of basins of attraction in Sec.IV of the main text. which contributes to Tr(˜ρkρ˜l) in the higher order than Tr(˜ρkρ˜l) [cf. Eq. (D109)]. Proof of Eqs. (31)-(33). For the states ρl that project onto ρ˜l [cf. Eqs. (14) and (17)], ρ˜l = P(ρl), we have A bound on the trace of product of operators. In the ˜ proofs above, we make use of the following results Tr(Plρl) = 1, (D117) Tr(P˜lρk) = 0, k 6= l, (D118) |Tr(XY )| ≤ kXY k ≤ min (kXk kY k, kXkkY k ) . max max where k, = 1, ..., m. Therefore, from Eq. (D114), we (D111) obtain Here, the ﬁrst inequality follows from the deﬁnition of the trace norm, while the second inequality from the Hölder 1 − ∆ Tr(1H ρl) ≥ , (D119) l Ccl inequality for Schatten norms. 1 − ∆ + 2 46 while from Eqs. (D115) and (D116), Considering ∆ = ∆1 in Eq. (D115), we get

C ˜ Ccl 1 cl Tr[Plρ(0)] + 2 Tr( Hk ρl) ≤ , k 6= l, (D120) Tr[1K ρ(0)] ≤ Tr[1H ρ(0)] ≤ (D128) l l Ccl 2∆ ∆1 + ˜ 2 X Ccl Tr(1H ρl) ≤ , (D121) and the same in Eq. (D116), k 2∆ k6=l ˜ 1 − Tr[P˜ ρ(0)] + Ccl X 1 X 1 l 2 which for ∆ = 1/2 in the leading order gives the bounds Tr[ Kk ρ(0)] ≤ Tr[ Hk ρ(0)] ≤ . ∆1 discussed in Sec.IV of the main text. k6=l k6=l For states closest to the projections ρ˜l in Eq. (17), (D129) 0 Therefore, for the state ρ that projects on ρ˜ in which we denote here by ρl, we have l l Eq. (17), l = 1, ..., m, Eq. (D127) gives ˜ 0 Ccl Tr(Plρl) ≥ 1 − 1 + C+, (D122) 1 − ∆2 Ccl 2 Tr(1K ρl) ≤ 1 − , (D130) l Ccl . 1 − ∆2 + 2(1 − ∆2) C 2 |Tr(P˜ ρ0)| ≤ 1 + cl C , k 6= l, (D123) k l 2 + where in the second inequality we assumed 1−∆2 Ccl, while from Eq. (D128) where k, l = 1, ..., m. This follows from Tr(P˜ ρ0) = k l 1 Ccl Ccl ˜ 0 ˜ ˜ Tr( Kk ρl) ≤ ≤ , k 6= l, (D131) |Tr[Pk(ρl − ρ˜l)] + Tr(Pkρ˜l) and Tr(Pkρ˜l) = δkl and 2∆1 + Ccl 2∆1 |Tr[P˜ (ρ − ρ˜ )]| ≤ kP˜ k kρ0 − ρ˜ k ≤ (1 + Ccl )C k l l k max l l 2 + and from Eq. (D129) [cf. Eqs. (12) and (D65)]. Therefore, from Eq. (D114), we obtain ˜ X 1 Ccl Tr( Kk ρl) ≤ . (D132) Ccl 2∆1 0 1 − ∆ − 1 + 2 C+ k6=l Tr(1H ρ ) ≥ , (D124) l l Ccl 1 − ∆ + 2 0 Similarly, for the closest state ρl to ρ˜l we have while Eqs. (D115) and (D116), [cf. Eqs. (D122) and (D123)] Ccl 1 − ∆2 − (1 + )C+ Ccl Ccl 1 0 2 1 + C+ + Tr( Kl ρl) ≤ 1 − (D133) Tr(1 ρ0) ≤ 2 2 , k 6= l,(D125) 1 − ∆ + Ccl Hk l ∆ 2 2

Ccl C˜cl Ccl + 2C+ X 1 + C+ + , Tr(1 ρ0) ≤ 2 2 . (D126) . 2(1 − ∆ ) Hk l ∆ 2 k6=l where in the second inequality we assumed 1−∆2 Ccl, These bounds, in the leading order, correspond to C + (2 + C )C Tr(1 ρ0) ≤ cl cl + (D134) Eqs. (31)-(33) when ∆ = 1/2 is chosen. Kk l 2∆1 + Ccl Finally, for the truncated metastable phases we simply Ccl + 2C+ have |Tr(1 ρ ) − Tr(1 ρ˜ )| ≤ kρ − ρ˜ k ≤ C , which , k 6= l, Hk l Hk l l l + . 2∆ introduces corrections of the same order. 1 and Decay subspace. We now show that the subspaces C˜ + (2 + C )C C˜ + 2C ˜ X 1 0 cl cl + cl + spanned by eigenvectors of Pl with the eigenvalues sep- Tr( Kk ρl) ≤ . . 2∆1 2∆1 arated from 0 and 1 by Ccl (or by Ccl + C+), k6=l l = 1, ..., m, which generally evolve into nontrivial mix- (D135) ture of metastable phases, can be neglected in the sup- We conclude that for ∆1 C˜cl and 1 − ∆2 Ccl (or port of metastable phases, i.e., they belong to the (ap- ∆1 C˜cl +2C+ and 1−∆2 Ccl +2C+), the subspaces Kl, proximate) decay subspace. Therefore, the subspaces Hl l = 1, ..., m can be neglected in the support of metastable 0 0 considered in Sec.IV, that is with ∆ = 1/2, feature phases ρ1, ..., ρm (or ρ1, ..., ρm). states that can be neglected in the support of metastable Importantly, by its deﬁnition, Kl exists outside the phases. This leads to Hl being generally not disjoint. basins of attraction, l = 1, ..., m, as its states evolve into Let Kl be a subspace of the system space spanned by nontrivial mixtures of metastable phases. Indeed, for a ˜ the eigenvectors of Pl with the eigenvalues between ∆1 state ρ(0) supported on Kl, we have ∆1 ≤ p˜l ≤ ∆2. and ∆2, where 0 ≤ ∆1 ≤ ∆2 ≤ 1. Considering ∆ = ∆2 When ∆1 C˜cl and 1−∆2 Ccl, we have that p˜l ≥ ∆1 in Eq. (D114), we obtain is nonnegligible and so is the sum of other barycentric coordinates, as P p˜ = 1−p˜ ≥ 1−∆ . Thus, the ˜ k6=l k l 2 Tr[Plρ(0)] − ∆2 metastable state P[ρ(0)] corresponding to ρ(0) is a mix- Tr[1K ρ(0)] ≤ 1 − Tr[1H ρ(0)] ≤ 1 − . l l Ccl 1 − ∆2 + 2 ture of ρ˜l with at least one other metastable phase ρ˜k, (D127) k 6= l. 47

Pm 0(k) 4. Nonuniqueness of phases in classical metastable where the second inequality follows from l=1 |p˜l | ≤ manifolds 0 1 + Ccl, we have from Eq. (D142)

(l) 0 X (l) 0 Here, we discuss the nonuniqueness of the choice p˜k & 1 − Ccl, p˜k . Ccl + Ccl. (D144) of metastable phases. We prove that considering an n6=k alternate set of m metastable phases leading to the 0 In the trace norm, we have corrections to the classicality Ccl, the distance of the alternative metastable phases to closest phase in the 0 (l) X (l) 0 0 kρ˜l − ρ˜kk ≤ 1 − p˜ kρ˜kk + pñ kρñk (D145) original set is bounded by Ccl + Ccl + min(Ccl, Ccl). k n6=k   We consider two sets of linearly independent 0 0 (l) X (l) metastable phases, ρ˜1, ..., ρ˜m and ρ˜1, ..., ρ˜m with ≤ (1 + C+)  1 − p˜k + pñ  0 the corrections to the classicality Ccl and Ccl, respec- n6=k tively. We can represent the metastable phases using 0 Pm (l) where we used the fact that kρ˜ k ≤ 1 + C from Eq. (12) barycentric coordinates as ρ˜ = p˜ ρ˜k and ρ˜k = n + l k=1 k [see also Eq. (D80)]. Analogously, by considering the Pm p˜0(k)ρ˜0 , so that from the linear independence from n=1 n n decomposition in the barycentric coordinates of ρ0 , ..., 0 Pm Pm (l) 0(k) 0 1 ρ˜ = p˜ pñ ρ˜ we have 0 l k=1 n=1 k n ρm, we arrive at the upper bound by ≤ (1 + C+)(2|1 − m p˜0(k)| + C0 ). Therefore, using Eqs. (D143) and (D144), X (l) 0(k) l cl p˜k pñ = δnl, (D136) we find k=1 kρ˜0 − ρ˜ k C + C0 + min(C , C0 ), (D146) and, analogously, l k . cl cl cl cl m while for the metastable phases chosen as the closest X (l) 0(k) pñ p˜l = δnk. (D137) states to the projections ρ˜l and ρ˜k, the corrections can l=1 increase by 2C+ [cf. Eq. (12)]. Therefore, from the triangle inequality Finally, we observe that k is uniquely defined for each l, i.e., is a function of l. Indeed, this follows directly from m X (l) 0(k) two inequalities Eq. (D144). Furthermore, this function p˜ p˜ ≥ 1 (D138) k l is bijective from the second inequality in Eq. (D143). k=1 and m E. CLASSICAL LONG-TIME DYNAMICS X (l) 0(k) p˜k p˜l ≥ 1. (D139) l=1 Here, first review of properties of classical stochastic From the inequality in Eq. (D138), by noting that dynamics in Sec.E1. We then prove that the long-time Pm (l) 0(k) 0 dynamics is effectively classical as discussed in Sec.V of k=1 |p˜k | ≤ 1 + Ccl [cf. Eq. (21)] and |p˜l | ≤ 1 + Ccl/2 [cf. Eq. (D61)], we observe the main text. First, in Sec.E2, we consider the dynamics of the average system state within a classical MM, and C0 derive bounds on its approximation by classical dynamics p˜(l) p˜0(k) + 1 + C − |p˜(l)| 1 + cl ≥ 1, (D140) k l cl k 2 governed by a classical stochastic generator. Second, in Sec.E3, we prove that classical trajectories of that classi- and thus cal stochastic generator capture the statistics of quantum 0 0 0 0(k) C 2Ccl + C + CclC trajectories for times after the initial relaxation. p˜ ≥ 1 + cl − cl cl . (D141) l 2 (l) 2|p˜k | Analogously, from Eq. (D139) we also have 1. Classical stochastic dynamics C 2C0 + C + C C0 |p˜(l)| ≥ 1 + cl − cl cl cl cl . (D142) k 0(k) Here, we review properties of classical stochastic dy- 2 2|p˜ | l namics and statistics of its trajectories. We also discuss Pm (l) weak symmetries. Since k=1 |p˜k | ≤ 1 + Ccl, from the lower bound in 0(k) Eq. (D138), we also have max1≤k≤m |p˜l | ≥ 1/(1 + Ccl) and we can remove the absolute value as C 1 cl a. Positivity and probability conservation [cf. Eq. (D63)]. Choosing k corresponding to the maximum, that is, Let l = 1, ..., m label m configurations of a classical

0(k) X 0(k) 0 system and p = (p , ..., p )T be a vector of the cor- p˜l & 1 − Ccl, p˜n . Ccl + Ccl, (D143) 1 m n6=l responding probabilities (0 ≤ pl ≤ 1, l = 1, ..., m and 48

Pm k=1 pk = 1). Time-homogeneous dynamics which pre- where R is the pseudoinverse of W. In the second line serves the positivity of the probability vector is generated in Eq. (E154), we used (J − µ)pss = Wpss = 0 as well Pm by a matrix W [cf. Eq. (6)] as l=1(J)lk = (µ)k. Equations (E153) and (E154) follow from the perturbation theory for W with respect to d s p(t) = Wp(t) (E147) (e−s − 1)J. dt Similarly, the cumulants of the statistics of a time- such that integral of system observable which takes value mk in the kth conﬁgurations is encoded by the maximal eigenvalue (W)ll ≤ 0, (W)kl ≥ 0 for k 6= l, (E148) of the biased operator [cf. Eq. (E224)] where k, l = 1, ..., m, while the preservation of the total Ws = W − hm, (E155) probability requires

m where the diagonal matrix of the system observable X (W) = 0. (E149) kl (m) = δ m . (E156) k=1 kl kl l for all l = 1, ..., m. The rate of the average time-integral

m m X X b. Stochastic trajectories mss = (mpss)l = (pss)lml, (E157) l=1 l=1 The matrix W can be considered as a Markovian gen- while the rate of fluctuations of time-integral erator of stochastic trajectories of system configurations. m The waiting time for a transition from the kth configu- 2 X δ = −2 (mRm pss)l, (E158) ration is distributed exponentially with the rate −(W )ll, ss so that the average lifetime is l=1 1 Eqs. (E157) and (E158) follow from the perturbation the- τ = − , (E150) l ory for Wh with respect to hm. (W)ll while the probability that the transition takes place from lth to kth configuration is proportional to (W)kl [equals c. Weak symmetries −(W)kl/(W)ll], k, l = 1, ..., m. The cumulants of the statistics of number of transitions Stochastic dynamics features weak symmetry Π when in a stochastic trajectory [cf. Eq. (44)] is encoded by the maximal eigenvalue of the biased operator [W, Π] = 0, (E159) −s Ws = W + (e − 1)J, (E151) where Π is a permutation matrix between m system configurations. where J = W + µ with The symmetry in Eq. (E159) is equivalent to

(µ)kl = δklµl = −δkl(W)ll (E152) (W)kl = (W)π(k)π(l), (E160) being a diagonal matrix of activities. The (minus) first derivative of the maximal eigenvalue determines the av- where k, l = 1, ..m and π denotes the permutation cor- erage activity responding to Π, that is (Π)kl = δkπ(l), k, l = 1, ..., m. From Eq. (E160), all configurations that belong to the m m X X same cycle, feature the same decay rate, (W)kk = µss = (Jpss)l = (pss)lµl, (E153) (W)πn(k)πn(k), n = 1, 2, .... Furthermore, for two dif- l=1 l=1 ferent cycles of length d1 and d2, with d1,2 being the where pss is the stationary probability of W, i.e., Wpss = greatest common factor of d1 and d2, transitions from the 0, and (pss)l corresponds to the average time spent in first to the second cycle are the same for all elements of the lth configuration, l = 1, .., m. The second derivative a subcycle of length d1/d1,2 and all elements of a subcy- the rate of fluctuations of the number of transitions in cle d2/d1,2 (and analogously for transitions from the sec- stochastic trajectories ond to the first cycle). In particular, the transition rates m m from (or to) an invariant configuration l [π(l) = l] are the 2 X X same for all the elements of a cycle, (W)kl = (W)πn(k)l σss = (Jpss)l − 2 (JRJpss)l (E154) n l=1 l=1 [(W)lk = (W)lπ (l)], n = 1, 2, .... m m From Eq. (E159), the generator W is block diagonal in X X = (µpss)l − 2 (µRµpss)l, an eigenbasis of Π, which can be chosen as plane waves l=1 l=1 over the cycles in the permutation π. Thus, the number 49 of free parameters of W is limited to the sum of squared a. Best classical stochastic approximation of long-time degeneracies of Π eigenvalues, i.e., the plane-wave mo- dynamics generator menta [less 1 from the trace-preservation condition in Eq. (E149)]. Furthermore, the eigenvectors of W can be We now prove that W defined in Eq. (36) is the clos- chosen as eigenvectors of Π, in which case they are linear est classical stochastic generator to W˜ in Eq. (34) with combination of the plane waves with the same eigenvalue. respect to the matrix norm induced by the L1 vector Finally, the symmetric eigenspace of Π is spanned by norm. For any probability-conserving and positive gen- plane waves with 0 momenta that are uniform mixtures erator W, we have of system configurations in each cycle, and thus its di- m mension equals number of cycles in the permutation π. X kW˜ − Wk1 ≡ max (W˜ )kl − (W)kl (E163) 1≤l≤m Moreover, the effective dynamics between those mixtures, k=1 i.e., dynamics of symmetric probabilities, is also stochastic, and given by [cf. Eq. (E147)] while [cf. Eq. (C42)] m h i X ˜ X ˜ d (W)kl − (W)kl = −(W)kl + (W)kl p0(t) = W0 p0(t) (E161) k=1 ˜ dt k:(W)kl<0, k6=l X Pdl−1 + (W˜ ) − (W) where [p0(t)]l = n=0 [p(t)]πn(l) and the index l runs kl kl ˜ over representatives of cycles with dl denoting the corre- k:(W)kl≥0, sponding cycle length, while or k=l X h i ≥ −(W˜ )kl + (W)kl d −1 d −1 k l ˜ 1 X X k:(W)kl<0, (W0)kl = (W)πnk(k) πnl(l). (E162) k6=l dl nk=0 nl=0 h i X ˜ + (W)kl − (W)kl Indeed, p (t) is a probability vector, while W preserves 0 0 k:(W˜ )kl≥0, the positivity and the total probability [cf. Eqs. (E148) or k=l and (E149)]. In particular, the lifetime of a uniform mix- X h i = 2 −(W˜ )kl + (W)kl ture of system configurations in a given cycle is generally k:(W˜ )kl<0, effectively extended, as microscopic transitions to other k6=l elements of the same cycle do not contribute to its ef- X ˜ Pdl−1 ≥ 2 max[−(W)kl, 0], (E164) fective dynamics, (W0)ll = (W)ll + n=1 (W)πn(l) l ≥ k6=l (W)ll [cf. Eq. (E150)]. Analogously, when multiple symmetries are present, the dynamics of probabilities invari- where in the second and last line we used the positiv- ant under all corresponding permutations is also stochas- ity of (W)kl ≥ 0, for k 6= l, the third line follows tic. from the triangle inequality, and the fourth line follows from the probability conservation in both genera- Pm ˜ Pm tors k=1(W)kl = 0 = k=1(W)kl. Finally, for W defined in Eq. (36) the bound in Eq. (E164) is saturated 2. Classical dynamics of average system state [cf. Eq. (C44)] X kW˜ − Wk1 ≡ max 2 max[−(W˜ )kl, 0]. (E165) We now show that the long-time dynamics of the av- 1≤l≤m erage system state is effectively classical. We first derive k6=l the best continuous approximation of the generator of the long-time dynamics in Eq. (34) by classical stochas- b. Derivation of Eq. (37) in the main text tic dynamics between metastable phases in Eq. (36). We then prove an upper bound Eq. (37) on the distance between the two generators. We use this result to show the The vector ˜p of barycentric coordinates between closeness of the effective dynamics to that generated by a metastable phases evolves as classical stochastic generator in Eq. (39) and of the result- t2 ˜p(t) = etW˜ ˜p(0) = ˜p(0) + tW˜ ˜p(0) + W˜ 2 ˜p(0) + ..., ing stationary states in Eq. (41). Similarly, all timescales 2! of the dynamics in the MM can be approximated by the (E166) classical stochastic dynamics. Here, we consider here an [cf. Eqs. (3), (4) and (35)]. As the corresponding state be- approximation of the pseudoinverse of the long-time dy- longs to the MM, at any time t its distance from the sim- namics generator. Finally, we also discuss the approxima- plex of metastable phases is bounded by Ccl in Eq. (21) tion of the long-time dynamics by discrete, rather than m X continuous, positive and trace-preserving dynamics, and k˜p(t)k1 − 1 = 2 max[−p˜l(t), 0] ≤ Ccl. (E167) prove results analogous to Eqs. (37), (39) and (41). l=1 50

Let us consider time t such that which gives t p ˜ Z ˜ tkW˜ k = c C , (E168) tW tW (t−t1)W t1W cl ke − e k1 ≤ dt1 ke k1kW˜ − Wk1ke k1 √ 0 where c is a constant, so that c C 1. In this case, cl ≤ tkW˜ − Wk1(1 + Ccl) the dynamics in Eq. (E166) can be approximated to the p ˜ linear order with corrections . 2 Ccl t kWk1. (E176) 2 In the second inequality we used the fact that W ˜ c ˜p(t) − ˜p(0) − tW ˜p(0) . Ccl. (E169) generates positive trace preserving dynamics and thus 1 2 tW ke k1 = 1, while W˜ transforms any probability vectors In particular, for the system initially in ρl that projects not further away than Ccl from the simplex [cf. Eq. (21)], tW˜ onto ρ˜l in Eq. (17), i.e., p˜k(0) = δkl, from Eq. (E169) we and thus ke k1 ≤ 1 + Ccl. The last inequality follows have from Eq. (37).

X c2 p˜ (t) − 1 − t(W˜ ) + p˜ (t) − t(W˜ ) C . l ll k kl . 2 cl d. Derivation of Eq. (41) in the main text k6=l (E170) Since from the deﬁnition of the absolute value For the stationary states of the open quantum dynamics described within the MM by ˜p and the stationary ss ˜ ˜ state pss of the classical stochastic dynamics W, we have − t(W)kl ≤ −p˜k(t) + p˜k(t) − t(W)kl (E171) tW˜ tW˜ k˜pss − pssk1 ≤ k˜pss − e pssk1 + ke pss − pssk1 and, from the monotonicity of max(x, 0) in x, tW˜ tW˜ tW ≤ ke − P˜ssk1 + ke − e k1

˜ ˜ tW˜ ˜ p ˜ t max[−(W)kl, 0] ≤ max[−p˜k(t), 0] + p˜k(t) − t(W)kl , . ke − Pssk1 + 2 Ccl t kWk1. (E177) (E172) In the first line we used the triangle inequality. In the we arrive at [cf. Eqs. (E167) and (E170)] second line we used that by definition of the projection P˜ss ˜ X X on the stationary probability distribution Psspss = ˜pss, t max[−(W˜ )kl, 0] ≤ max[−p˜k(t), 0] (E173) tW and further exploited e pss = pss and the definition k6=l k6=l of the induced norm. The last inequality follows from

X ˜ Eq. (39). Note that we did not assume uniqueness of p . + p˜k(t) − t(W)kl ss Therefore, when t can be chosen so that ketW˜ −P˜ k 1, k6=l √ ss 1 ˜ 1 + c2 while tkWk1 1/ Ccl, we obtain k˜pss − pssk1 1, but C . . 2 cl this requires For the classical stochastic generator W deﬁned in 1 τkW˜ k1 ≤ τ˜kW˜ k1 √ , (E178) Eq. (36), we have from Eq. (E168) that Ccl

tW˜ ˜ ˜ X ˜ 2 where τ˜ is the relaxation time with respect to ke −Pssk1 tkW−Wk1 ≡ t max 2 max[−(W)kl, 0] . (1+c ) Ccl, 1≤l≤m tL ˜ k6=l rather than ke −Pssk (τ˜ ≈ τ when Ccl 1; cf. Sec.D2). (E174) We note that a related result can be obtained in the so that W approximates well W˜ non-Hermitian perturbation theory [S4], where in the first order ˜ ˜ 2 kW − Wk1 tkW − Wk1 1 + c p ˜ ˜ = . Ccl, ˜pss − pss = R(W − W)˜pss + ..., (E179) kW˜ k1 tkW˜ k1 c where R˜ is the pseudoinverse of W˜ , R˜ W˜ = W˜ R˜ = and for the (optimal) choice c = 1, we obtain Eq. (37). I−P˜ss, with I being the identity matrix (see also Sec.E1). Therefore, the first-order corrections can be bounded as [cf. Eq. (37)] c. Derivation of Eq. (39) in the main text k˜pss − pssk1 ≤ kR˜ k1kW − W˜ k1k˜pssk1 + ...(E180) We have ˜ p . 4tkWk1 Ccl + .... t ˜ Z d ˜ tW tW (t−t1)W t1W For the estimate on kR˜ k 2t, where t ≥ τ˜ is such e − e = dt1 e e (E175) 1 . dt1 tW˜ 0 that ke − P˜ssk1 1, see Eq. (E188) below. Therefore, Z t (t−t1)W t1W˜ when the right-hand-side of Eq. (41) is negligible, so are = dt1 e (W˜ − W)e , 0 the first-order corrections in Eq. (E179). 51

e. Approximation of dynamics resolvent The above inequality holds for any time t. When time t can be chosen as in the case of the discussion of Eq. (41), √ tW˜ Resolvent. We now consider the resolvent R˜ of the that tkW˜ k1 1/ Ccl holds for ke − P˜ssk1 1, the dynamics W˜ at 0, i.e., the pseudoinverse of W˜ , R˜ W˜ = leading corrections in the right-hand side of Eq. (E185) W˜ R˜ = I − P˜ss, where I is the identity matrix (see also are given by the numerator. In this case, the closeness Sec.E1). In Eqs. (E185) and (E186) below, we show it of resolvents occurs when t can be further chosen so that tketW˜ − P˜ k /kR˜ k 1 [note that t2kW˜ k /k kR˜ k can be approximated by the resolvent R of the classical √ ss 1 1 √ 1 1 1 dynamics W in Eq. (36). 1/ Ccl follows from tkW˜ k1 1/ Ccl for t ≥ kR˜ k1, We have ˜ tW˜ while we have kRk1 . 2t for any t such that ke − ∞ ˜ Z ˜ Pssk1 1; see Eq. (E188)]. This, in turn, requires that ˜ t1W ˜ R = − dt1 e − Pss , (E181) the relaxation time 0 s " ˜ ˜ # and thus 1 kWk1kRk1 τ˜kW˜ k1 min √ , √ . (E186) t ∞ Ccl Ccl Z ˜ Z ˜ ˜ t1W ˜ t1W ˜ R + dt1 e − Pss ≤ dt1 e − Pss 0 1 t 1 In fact, this condition on the relaxation time will typically imply existence of time such that tW˜ ˜ ˜ √ ≤ ke − Pssk1kRk1, (E182) tW˜ tkW˜ k1 1/ Ccl and tke − P˜ssk1/kR˜ k1 1, as nτW˜ ˜ τW˜ ˜ n nτke − Pssk1 ≤ nτke − Pssk1 , which decays to 0 tW˜ where in the second inequality we used the fact e P˜ss = as n → ∞ (exponentially at larger n). ˜ ˜ ∞ ˜ ˜ tW ˜ tW ˜ R t1W ˜ Psse = Pss so that (e − Pss) 0 dt1(e − Pss) = ∞ ˜ Discussion of the results in the proximity to a first- R dt (et1W − P˜ ). Equation (E182) holds analogously t 1 ss order phase transition. We note that both Eqs. (40) for the dynamics with W. Furthermore, from Eqs. (39) and (E186) are fulfilled for all perturbations away from and (41) we have a first-order phase transition in a finite-size system, t t Z ˜ Z provided that the degeneracy of stationary states is t1W ˜ t1W dt1 e − Pss − dt1 e − Pss (E183) lifted in the same order. Indeed, in this case kW˜ k1, 0 0 1 1/kR˜ k and 1/τ˜ are of the same order in the per- Z t 1 t1W t1W turbation, while C is of the same or higher order ≤ dt1ke − e k1 + tkP˜ss − Pssk1, cl 0 (see Sec. A2b). Thus, τ˜kW˜ k1 and kR˜ k1kW˜ k1 are p 2 ˜ ˜ tW˜ of zero-order, that is, they remain finite in the limit . 3 Cclt kWk1 + tkPss − e k1. of the perturbation decreasing to 0, while the upper The first inequality follows from the triangle inequality. limits in Eqs. (40) and (E186) diverge in this limit. In the last inequality we used the fact kP˜ss − Pssk1 = In fact, in this case, time t can be chosen so that √ q √ k˜pss − pssk1 and Eq. (E177). Therefore, by applying the τ˜kW˜ k tkW˜ k min[1/ C , kW˜ k kR˜ k / C ] triangle inequality we arrive at 1 1 cl 1 1 cl thus leading to k˜pss − pssk1 1 and kR˜ − Rk1/kR˜ k1 kR˜ − Rk1 (E184) [cf. Eq. (41) and (E185)]. t t Z ˜ Z t1W ˜ t1W ≤ dt1 e − Pss − dt1 e − Pss Bounds on the norm of the resolvent. For the relax-

0 0 1 ation time τ deﬁned as in Eq. (B27), we have that the t t Z ˜ Z ˜ t1W ˜ t1W resolvent S of L fulﬁlls + R + dt1 e − Pss + R + dt1 e − Pss 0 1 0 1 2τ kSk ≤ 2t, (E187) p 2 ˜ ˜ ˜ tW˜ ˜ 1 − keτL − P k . . Ccl 10t + 4tkRk1 kWk1 + 3(t + kRk1)ke − Pssk1 ss

tW˜ p where in the second inequality we consider t ≥ τ for + 2ke − P˜ssk1 + 4 Ccl tkW˜ k1 kR˜ − Rk1. tL which ke − Pssk 1. This follows from S = R ∞ tL tL − 0 dt(e −Pss) [cf. Eq. (E181)], and thus (I−e )S = In the last inequality we used Eqs. (E182) and (E183) t − R dt (et1L − P ). Indeed, we have (1 − keτL − together with ketW − P k ≤ ketW˜ − P˜ k + ketW˜ − 0 1 ss ss 1 ss √1 P k)kSk ≤ (1 − ketL − P k)kSk ≤ kSk − ketLSk ≤ etWk + kP˜ − P k 2(ketW˜ − P˜ k + 2 C tkW˜ k ) ss ss 1 ss ss 1 . ss 1 cl 1 k(I − etL)Sk = kR t(etL − P )k ≤ 2t. Analogously, we and kRk ≤ kR˜ k + kR˜ − Rk . Therefore, 0 ss 1 1 1 have kR˜ − Rk 2˜τ 1 (E185) kR˜ k 2t, (E188) . . τ˜W˜ ˜ . kR˜ k1 1 − 2ke − Pssk1 √ 2 tW˜ Ccl(10t /kR˜ k1 +4t)kW˜ k1 + 3(t/kR˜ k1 +1)ke − P˜ssk1 where the second inequality holds for t ≥ τ˜ such that √ . ˜ tW˜ tW ˜ 1 − 2ke − P˜ssk1 − 4 Ccl t kW˜ k1 ke − Pssk1 1 (cf. Sec.D2). 52

The bound in Eq. (E187) and (E188) also hold when the relaxation times τ and τ 00 are instead defined as the rate of exponential bound on the approach to the sta- Derivation of Eq. (E194). This result follows from the tionary state and the MM, i.e., by Eqs. (B30) and (B31). fact that the dynamics must map any metastable state But in this case we also have [cf. Eq. (E181)] to another state within the MM. tW˜ Z ∞ The closest positive trace preserving dynamics to e t1L Pm tW˜ kSk ≤ dt1 ke − Pssk = τ. (E189) is as follows. Let ∆l ≡ k=1 max[−(e )kl, 0], l = 0 1, ..., m. We then define [cf. Eq. (C43)] Similarly, within the MM [see Eq. (D90)] tW˜ (Tt)kl ≡ 0 if (e )kl ≤ 0, (E198) ˜ tW˜ Ccl − t ke − P˜ssk1 1 + e τ , (E190) tW˜ . 2 and otherwise, when (e )kl > 0, tW˜ which leads to the norm of the resolvent bounded as (Tt)kl ≡ (e )kl (E199) Z ∞ h ˜ ˜ i ˜ C˜cl tW X tW ˜ t1W ˜ − min (e ) , ∆ − (e ) − (T ) , kRk1 ≤ dt1 ke − Pssk1 . 1 + τ. (E191) kl l nl t nl 0 2 n0 Therefore, when C˜cl 1, for example, if m does not scale with Ccl so that mCcl 1 (e.g., m is independent of the k = 1, ..., m. We then have [cf. Eq. (C44)] system size), the norm of resolvent is bounded by the tW˜ tW˜ relaxation time. Furthermore, the condition in Eq. (40) ke − Ttk1 = 2 max ∆l = ke k1 − 1 ≤ Ccl, (E200) 1≤l≤m is implied by the condition in Eq. (E186). Finally, we have lower bounds in terms of the master tW˜ where in the last inequality we used ke k1 ≤ 1 + Ccl, as operator spectrum [cf. Eq. (B36)] the long-time dynamics transforms the MM onto itself. 1 This gives Eq. (E194). kSk ≥ max (E192) Finally, Tt defined in Eqs. (E198) and (E199) is opti- 2≤k≤D2 |λ | k mal, as for any discrete trace preserving dynamics T we and, analogously, have kTk1 = 1 and thus from triangle inequality

˜ ˜ ˜ ˜ 1 − C+ − Ccl ketW − Tk ≥ ketWk − kTk = ketWk − 1. (E201) kRk1 & max . (E193) 1 1 1 1 2≤k≤m |λk| as kPSk ≤ (1 + C+)(1 + Ccl)kR˜ k1 [cf. Eq. (D89)]. Derivation of Eq. (E195). We have

n f. Classical discrete approximation of long-time dynamics ntW˜ n X (k−1)tW˜ tW˜ n−k ke − Tt k1 = e (e − Tt)Tt k=1 1 tW˜ Below, we show how dynamics e at any time t can n be approximated by classical discrete dynamics T [see tW˜ X (k−1)tW˜ n−k t ≤ ke − Ttk1 ke k1kTtk1 Eqs. (E198) and (E199) for the deﬁnition] as k=1

tW˜ ≤ nCcl [1 + Ccl] . nCcl, (E202) ke − Ttk1 ≤ Ccl (E194) where in the second inequality we used Eq. (E194), [cf. Eq. (37)]. At later times nt, where n is an integer, the fact that kTtk1 = 1 from the positivity and the we then have the approximation [cf. Eq. (39)] ktW˜ ktL trace-preservation, and ke k1 ≤ 1 + Ccl as e ntW˜ n transforms the MM onto itself. ke − Tt k1 . nCcl. (E195)

ntW˜ Derivation of Eq. (E197). We have For t chosen long enough, so kP˜ss − e k1 1 for an integer n 1/Ccl, which requires [cf. Eq. (40)] ntW˜ ntW˜ k˜pss − pssk1 ≤ k˜pss − e pssk1 + ke pss − pssk1 t t ˜ ntW˜ ntW˜ n ≥ C , (E196) ≤ kPss − e k1 + ke − Tt k1 τ τ˜ cl ˜ ntW˜ . kPss − e k1 + nCcl. (E203) the stationary state ρss described within the MM by ˜ (˜pss)k = Tr(P˜kρss) is well approximated by the station- In the second line we used that the projection Pss on ˜ ary probability pss of the classical discrete dynamics Tt the stationary probability distribution Psspss = ˜pss, and [cf. Eq. (41)] as further exploited Ttpss = pss and the deﬁnition of the induced norm. The last inequality follows from Eq. (E195). ˜ ntW˜ k˜pss − pssk1 . kPss − e k1 + nCcl. (E197) We did not assume uniqueness of pss. 53

3. Classical statistics of quantum trajectories of Ws. Finally, we note that the corrections can be adjusted to include nonperturbative change of fast modes, −s Here, we first consider generalized statistics of jump by considering perturbation of LQ + (e − 1)QJ Q = number where individual jumps are taken into ac- QLsQ instead of LQ, which leads to replacing SQ by −s + −s count (Sec. E3a). Second, we discuss classical ap- [LQ+(e −1)QJ Q] = SQ+(e −1)SJS +... in the proximations of the statistics for a homodyne mea- above corrections, but this change will only contribute in surement [S12] (Sec. E3b) and for time-integral of a higher order than already considered. system-observables [S12–S17] (Sec. E3c) in the pres- Similarly, one can consider approximating the ence of classical metastability. Third, we derive correc- first and the second derivatives of θ(s), using the −s tions to the approximation of generators of statistics of fact that θ(s) = (e − 1)Tr[J ρss(s)] and thus 0 −s −s 0 jump number [Eq. (49)], time-integral of homodyne cur- θ (s) = −e Tr[J ρss(s)] + (e − 1)Tr[J ρss (s)] 00 −s −s −s rent [Eq. (E212)] and system observables [Eq. (E227)] and θ (s) = e Tr[J ρss(s)] − 2e (e − 0 −s 00 (Sec.E3d). We then prove the relation of the statistics 1)Tr[J ρss (s)] + (e − 1)Tr[J ρss (s)]. In particular, 0 −s −s −s of jumps and system-observables in quantum trajectories θ (s) = −e Tr[J ρss(s)]+(e −1)e Tr[JSsJ ρss(s)]− −s −s to statistics of classical dynamics for finite (Secs. E3e (e − 1)e Tr[SsJ ρss(s)]Tr[J ρss(s)], where Ss is the andE3f) and asymptotic time (Sec.E3g). Finally, we resolvent of Ls at θ(s), while the last term follows discuss multimodal distributions of continuous measure- from the chosen normalization Tr[ρss(s)] = 1. We 0 −s Pm ment results during the metastable regime (Sec.E3h). can then approximate θ (s) = −e l=1[µ˜pss(s)]l + −s −s Pm in in −s For simplicity, we denote here an initial state of the sys- (e − 1)e l=1[(J + µ˜ )Rs(J + µ˜ )pss]l − (e − −s Pm in −s tem as ρ instead of ρ(0) and omit parentheses in the ac- 1)e k,l=1[Rs(J + µ˜ )pss(s)]k[µ˜pss(s)]l − (e − tion of superoperators on system states, e.g., ρ(t) = etLρ −s Pm 1)e l=1[δσ˜2 pss(s)]l/2 + ..., where Rs is the resolvent instead of etL[ρ(0)]. (1) of Ws at θ (s). The last term corresponds to the non-Poissonian contribution from fluctuations in the metastable phases [cf. Eq. (54)], while the rest is the a. Activity in quantum trajectories first-derivative of the classical cumulant generating function for the total activity, i.e., the first derivative of (1) Total activity. In Sec.VB of the main text, we dis- the maximal eigenvalue θ (s) of Ws. cussed the approximation in Eq. (49) of the operator Ls in Eq. (44) by the classical operator Ws in Eq. (47). This Local activity. The joint statistics of the numbers of led to the approximation of the maximal eigenvalue θ(s) individual jumps is encoded by [cf. Eq. (44)] of Ls eigenmode and the corresponding eigenmode ρss(s) X 1 by the maximal eigenvalue and the corresponding eigen- L (ρ) ≡ −i[H, ρ] + e−sjJ ρJ † − {J †J , ρ} , s j j 2 j j mode Ws. Here, we discuss the corrections for those j approximations. (E204) −s We consider LP +(e −1)J as a perturbation of LQ, where (s)j = sj encodes bias values for individual jumps. by means of the degenerate non-Hermitian perturbation In the presence of classical metastability and with in- theory [S4], where Q ≡ I − P is the projection on the ternal activity dominating the long-time dynamics, the fast modes of L [cf. Eq. (4)], with a further (higher order) projection of Ls on the low-lying modes [cf. Eq. (48)] −s approximation of LP + (e − 1)PJP = PLsP by Ws, which is the superoperator corresponding to Ws. The (W˜ s)kl ≡ Tr[P˜kLs(˜ρl)], (E205) degeneracy of zero-eigenspace P of QL is lifted by Ws with the maximal eigenvalue θ(1)(s) [cf. Eq. (51)] cor- k, l = 1, ..., m, can be approximated as [cf. Eq. (62)] (0) Pm responding to the eigenmode ρss (s) ≡ [pss(s)]l ρ˜l l=1 ˜ X [cf. Eq. (50)]. The higher order corrections are given (Ws)kl = W − hsj µ˜j + ... ≡ Whs + ..., (E206) (2) (0) −s 2 j by θ (s) ≡ Tr{L1 (s)[−(e − 1) J SQJ + (Ls − (0) (1) −s (0) W )][ρss (s)]} and ρss (s) ≡ −(e − 1)SQJ [ρss (s)] + −sj s where (hs)j ≡ 1 − e and the individual activities −s 2 (1) + (0) (e − 1) [Ws − θ (s)P] J SQJ [ρss (s)] − [Ws − (1) + (0) † θ (s)P] (Ls − Ws)[ρss (s)], where the first and the µ˜ = Tr J J ρ˜ , (E207) j kl j j l second terms correspond to the corrections from the fast modes [cf. Eq. (48)], while the third term orig- k, l = 1, ..., m. The corrections in Eq. (E206) [and + inate from the approximation in Eq. (49). Here, thus also Eq. (62)] are bounded in the leading or- (0) √ + −sj denotes the pseudoinverse and S = L , while L1 (s) der by 2kW˜ k1 Ccl + maxj |e − 1|[kW˜ k1 + m kH + is the dual eigenmatrix to ρ(0)(s). The normalization P † p ss i j Jj Jj/2kmax 2Ccl + 4C+] [? ]. For the derivation, Tr[ρss(s)] = 1 can be further achieved by additional cor- see Eq. (37) and Sec.E3d. (1) (0) rection −Tr[ρss (s)]ρss (s). Analogously, other m−1 low- Therefore, θ(s) is approximated by the maximal eigen- lying modes of Ls can be approximated by eigenmodes value of Whs in Eq. (E205), while the corresponding den- 54

† sity matrix by dBj (t) the creation operator for the quanta emitted with the action of jump Jj at time t. The statis- m X (0) tics of the homodyne current measured with dX(t) ≡ ρss(s) = [pss(hs)]l ρ˜l + ... ≡ ρss (s) + ..., (E208) P iϕj † −iϕj j[e dBj (t) + e dBj(t)]/2 corresponds to the bi- l=1 ased “tilted” master equation [S12] where pss(hs) is the maximal eigenmode of Whs . 2 To obtain the relevant corrections, we consider r Lr(ρ) ≡ L(ρ) − rX (ρ) + , (E209) the degenerate non-Hermitian perturbation the- 8 P −sj ory with LP + (e − 1)Jj as a perturbation † j where X (ρ) ≡ P (e−iϕj J ρ + eiϕj ρJ )/2. That is, P −sj j j j of LQ, and LP + j(e − 1)PJjP further ap- Θ(r, t) ≡ ln{Tr etLr (ρ)} is the cumulant generating proximated by W , where J (ρ) ≡ J ρJ †, and hs j j j function for the integrated value of the measured ho- W denotes the superoperator corresponding to hs modyne current until time t. The asymptotic statistics W . Thus, the corrections in Eq. (E208) are hs is then determined by θ(r) ≡ lim Θ(r, t)/t, which (1) P (0) t→∞ given by ρss (s) ≡ SQ( j hsj Jj)[ρss (hs)] + [Whs − equals the eigenvalue of Lr with the largest real part. (1) + P P (0) θ (hs)P] ( j hsj Jj)SQ( j hsj Jj)[ρss (hs)] − [Whs − Classical tilted generator for homodyne current. in the θ(1)(h )P]+(L − W )[ρ(0)(h )] and the normaliza- s s hs ss s presence of classical metastability, for the bias parameter tion Tr[ρss(s)] = 1 can be achieved by additional r smaller than the separation to the fast eigenmodes of (1) (0) (1) correction −Tr[ρss (s)]ρss (s). Here, θ (hs) denotes L, the maximal eigenmode of Lr in Eq. (E209) can be ap- the maximal eigenvalue of Whs , which approxi- proximated as a maximal eigenmode of PLrP, which in mates θ(s), with the higher order correction given by the basis of metastable phase corresponds to [cf. Eq. (48)] (2) (0) P P θ (hs) ≡ Tr{L1 (hs)[−( j hsj Jj)SQ( j hsj Jj) + (0) (0) (W˜ r)kl ≡ Tr[P˜kLr(˜ρl)], (E210) (Ls − Whs )][ρss (hs)]}, where L1 (hs) is the dual eigen- (0) matrix to ρss (hs) in Eq. (E208). Analogously, other ˜ ˜ ˜ r2 k, l = 1, ..., m. We have Wr = W − r X + 8 , where m − 1 low-lying modes of Ls can be approximated by (X˜ )kl ≡ Tr[P˜kX (˜ρl)]. X˜ can be approximated by eigenmodes of Wh . s the diagonal matrix of the observable averages in the Metastability and dynamical phase transitions in ac- metastable phases tivity. For the bias ksk1 large enough [but small with R (˜x)kl ≡ δklx˜l, (E211) respect to the gap to fast modes of the dynamics λm − R λ ], the contribution from W in W of Eq. (E206) † m+1 hs where x˜ ≡ P Tr[(e−iϕj J + eiϕj J )˜ρ ]/2, k, l = 1, ..., m. can be neglected. In that case, m low-lying eigenmodes l j j j l Therefore, and eigenvalues of Ls are approximated as metastable phases ρ˜l in Eq. (17). 2 2 ˜ r r In particular, for a homogeneous bias, s = s, Wr = W − r ˜x + + ... ≡ Wr + + ... (E212) j 8 8 and approximation in Eq. (62) we have the correction h SQJ (˜ρ ) − h (M˜ in − µ˜inP)+J SQJ (˜ρ ) + The corrections in Eq. (E212) can be bounded s l s l l √ ˜ in in + ˜ in ˜ (M − µ˜l P) (Ls + hsM )(˜ρl)/hs for the eigenmode in the leading order by 2 CclkWk1 + 2 ˜ P −iϕj p Rl(s) =ρ ˜l + ..., the correction −hsTr[PlJ SQJ (˜ρl)] + |r| m k j e Jjkmax Ccl/2 + C+ [cf. Eq. (49)]. in (W˜ )ll − hs(J˜ − µ˜ )ll to the corresponding eigen- Here, Wr encodes the statistics of time-integral of −s value θl(s) = (e − 1)˜µl + ..., and the cor- the system observable ˜x of Eq. (E211) in classical −s in trajectories. From Eq. (E212), θ(r) is approximated by rection e {−2hsTr[P˜lJ SQJ (˜ρl)] − hs(J˜ − µ˜ )ll − P (W˜ ) (W˜ ) /(˜µin − µ˜in)/h2} to its derivative, the maximal eigenvalue of Wr in Eq. (E212) shifted by k lk kl k l s r2/8. Furthermore, the maximal eigenmode ρ (r) of L −k (s) = −e−sµ˜ + .... Here, M˜ in is the superoperator ss r l l is approximated by the corresponding W˜ eigenmode corresponding to µ˜in. r pss(r) [cf. Eq. (E228)] as

m X (0) b. Homodyne current in quantum trajectories ρss(r) = [pss(r)]l ρ˜l + ... ≡ ρss (r) + .... (E213) l=1

We now discuss the relation of the homodyne mea- (1) surement statistics to the classical dynamics between The corrections in Eq. (E213) are given by ρss (r) ≡ (0) 2 (0) + (0) metastable phases. rSQX [ρss (r)] + r [Wr − θ (r)P] X SQX [ρss (r)] − (0) + (0) [Wr − θ (r)P] (L − rX − Wr)[ρss (r)] in the second or- Statistics of homodyne current. We consider emissions der of the degenerate non-Hermitian perturbation theory of quanta associated with jump occurrence and denote with respect to LP − rX as a perturbation of LQ + r2/8 55 and with a further approximation of LP − rPXP by we denoted by Wr. Here, Wr denotes the superoperator corresponding t R Pm 2 t1W to W , Q ≡ I − P is the projection on the fast modes of dt1 l=1 χ˜ e ˜p r hχ2 (t)i ≡ 0 l (E218) L [cf. Eq. (4)], + stands for a pseudoinverse and S = L+, cl t (1) while θ (r) denotes the maximal eigenvalue of Wr. the rate of the time-integral of fluctuation rates in The normalization Tr[ρss(r)] = 1 can be achieved by metastable phases (χ˜2) ≡ δ {1/2 − 2Tr[X SQX (˜ρ )]} (1) (0) kl kl l considering the additional correction −Tr[ρss (r)]ρss (r). (l) (1) 2 (see Sec. E3f), hXcl i is the average of time-integral The correction θ(r) − θ (r) − r /8, is given by of the homodyne current for lth metastable phase in (2) (0) 2 (0) θ (r) ≡ Tr{L1 (r)[−r X SQX +(L−rX −Wr)][ρss (r)]}, Eq. (E214), i.e., (˜p)k = δkl, k, l = 1, ..., m, and (0) (0) ˜ ˜ ˜ where L1 (r) is the dual eigenmatrix to ρss (r) in Xl ≡ Tr{PlXSQ[ρ(0)]}/Tr[Plρ(0)] is the average con- Eq. (E213). tribution to the integrated homodyne current from before the metastable regime conditioned on the metastable Classical cumulants of homodyne√ current. For times phase that the system evolves into. Therefore, fluc- 00 t ≥ t such that tkW˜ k1 Ccl, the rate of the aver- tuations of the homodyne current stem from classi- age integrated homodyne current can be approximated cal transitions between metastable phases with differ- as [cf. Eq. (52)] ing average homodyne current and fluctuations inside the metastable phases, corrected by the average from before m hX(t)i 1Z t X Tr{X SQ[ρ(0)]} the metastable regime. When time t in Eq. (E216) can = dt ˜xet1W ˜p − +... 1 l be chosen after the final relaxation, the asymptotic rate t t 0 t l=1 of homodyne current fluctuations hXcl(t)i X˜ ≡ + + .... (E214) 2 2 t t 2 hX (t)i−hX(t)i 1 χss ≡ lim = − 2Tr[XSX (ρss)], t→∞ t 2 The first term in Eq. (E214) is the rate of the time- (E219) integral Xcl(t) in classical trajectories of the average ho- is approximated as [cf. Eq. (E212)] modyne current in metastable phases, whose statistics hhX2 (t)i−hX (t)i2 i in encoded by Wr in Eq. (E212) (see Sec. E3f). The 2 cl cl 2 χss = lim + hχcl(t)i (E220) second term represents the constant contribution to the t→∞ t m integrated homodyne current from before the metastable X 2 i = −2 ˜xR˜x + χ˜ pss + ... regime. When time t in Eq. (E214) can be chosen after l the final relaxation, the asymptotic rate is approximated l=1 as [cf. Eq. (53)] (cf. Sec.E1). The corrections to Eqs. (E214) and (E216) are given in Sec.E3e, and to Eq. (E219) in Sec.E3g. hX(t)i hXcl(t)i xss ≡ lim = lim + ... (E215) t→∞ t t→∞ t Classical metastability and dynamical phase transitions m X in homodyne current. When metastable phases differ in = (pss)l x˜l + ..., average homodyne current, for the bias |r| large enough l=1 W can be neglected in Eq. (E212), and Eq. (E213) ap- with corrections bounded in the leading order by proximated as [cf. Sec.VB5 in the main text] max |x˜ |k˜p − p k [note that x = Pm (˜x˜p ) 1≤l≤m l ss ss 1 ss l=1 ss l ρ (r) =ρ ˜ + ..., (E221) and cf. Eqs. (41)]. ss l ˜ For times such that kSQkkX k tk˜xk1 and tkWk1 where l is such that x˜ is maximal (for negative r) q l √ √ or minimal (for positive r) among metastable phases. min[1/ Ccl, kR˜ k1kW˜ k1/ Ccl], the rate of fluctuations The corrections given in the lowest order of the non- of integrated homodyne current is approximated as ˜ [cf. Eq. (58)] Hermitian perturbation theory are rQSX (˜ρl) − r(X − + + x˜lP) X SQX (˜ρl)+(X˜ −x˜lP) [L−r(X −X˜)](˜ρl)/r, where hX2(t)i−hX(t)i2 hX2 (t)i−hX (t)i2 X˜ is the superoperator corresponding to ˜x in Eq. (E211). = cl cl + hχ2 (t)i t t cl Furthermore, m (k) (l) 2 X hXcl (t)i − hXcl (t)i ˜ ˜ r − p˜kp˜l Xk − Xl + ..., (E216) θ(r) = −rx˜ + + ..., (E222) t l 8 k,l=1 r k(r) = −x˜l + + ..., (E223) where the fluctuations of the average homodyne current 4 in classical trajectories with the corrections given in the lowest order of the non- Hermitian perturbation theory by −r2Tr[P˜ X SQX (˜ρ )]+ m l l Z t Z t−t1 ˜ ˜ ˜ 2 X t2W t1W (W)ll − r(X − ˜x)ll and by −2rTr[PlX QSX (˜ρl)] − hX (t)i = −2 dt1 dt2 (˜xe ˜xe ˜p)l, (E217) cl P ˜ ˜ 2 ˜ 0 0 l=1 k6=l(W)lk(W)kl/(˜xk − x˜l)/r − (X − ˜x)ll, respectively. 56

Therefore, the derivative of θ(r) undergoes a sharp Wh in Eq. (E227) encodes the statistics of time- change around r = 0 when metastable phases differ in integral of the system observable ˜m in Eq. (E226) in average homodyne current. This can be interpreted as classical trajectories (cf. Sec.E1). Therefore, for pro- a proximity to a dynamical phase transition [S12]. Fur- nounced enough classical metastability (Ccl, C+ 1), thermore, if the stationary state ρss is different from the θ(h) is approximated by the maximal eigenvalue of Wh in metastable phase with the maximal or the minimal homo- Eq. (E227). Furthermore, the maximal eigenmode ρss(h) dyne current, i.e., features contributions from metastable of Lh in Eq. (E224) is approximated by the maximal W˜ h phases with a different average of the homodyne cur- eigenmode pss(h) [cf. Eq. (50)] rent, the current fluctuations in Eq. (E219) are large m [cf. Eq. (E220)]. X (0) ρss(h) = [pss(h)]l ρ˜l + ... ≡ ρss (h) + .... (E228) l=1 c. Time-integrals of system observables in quantum (1) The corrections in Eq. (E228) are given by ρss (h) ≡ trajectories (0) 2 (1) + (0) hSQM[ρss (h)] + h [Wh − θ (0)P] MSQM[ρss (h)] − (1) + (0) We now discuss the relation of the statistics of time- [Wh − θ (h)P] (Lh − Wh)[ρss (h)] in the degenerate integrated system observables to the classical dynamics non-Hermitian perturbation theory with respect to between metastable phases. LP − hM as a perturbation of LQ, with a further approximation of LP − hPMP by Wh. Here, Wh + denotes the superoperator corresponding to Wh, Statistics of time-integrated system observables. The + statistics of a system observable M time-integrated over stands for the pseudoinverse, S = L and Q ≡ I − P is the projection on the fast modes of L [cf. Eq. (4)]. quantum trajectory corresponds to the biased “tilted” (1) master operator [S13–S17] The maximal eigenvalues of Wh denoted as θ (h) approximates θ(h), with the higher order correction (2) (0) 2 (0) Lh(ρ) ≡ L − hM(ρ), (E224) θ (h) ≡ Tr{L1 (h)[−h MSQM + (Lh − Wh)][ρss (h)]}, (0) (0) where L (h) is the dual eigenmatrix to ρss (h). Finally, where M(ρ) ≡ (Mρ + ρM)/2. Here, Θ(h, t) ≡ 1 the normalization Tr[ρss(h)] = 1 can be achieved by ln{Tr etLh (ρ)} is the generating function for time- considering the additional correction −Tr[ρ(1)(h)]ρ(0)(h). ordered cumulants of the integral of M until time ss ss t. The asymptotic statistics is then determined by Classical cumulants of time-integrated system observ- θ(h) ≡ lim Θ(h, t)/t, which equals the eigenvalue of √ t→∞ ables. For times t ≥ t00 such that tkW˜ k C , the L with the largest real part. 1 cl h rate of the average integrated system observable can be approximated as [cf. Eq. (52)] Classical tilted generator for time-integrated system ob- m servables. In the presence of classical metastability, for hM(t)i 1Z t X Tr{MSQ[ρ(0)]} = dt ˜met1W ˜p − +... the bias parameter h smaller than the separation of the t t 1 l t slow eigenmodes of L to the fast ones, the maximal eigen- 0 l=1 mode can be approximated as a maximal eigenmode of hM (t)i M˜ ≡ cl + + .... (E229) PLhP, which in the basis of metastable phase corre- t t sponds to [cf. Eq. (48)] The first term in Eq. (E229) is the rate of the time- integral M (t) in classical trajectories of the average (W˜ ) = Tr[P˜ L (˜ρ )], (E225) cl h kl k h l observable in metastable phases, whose statistics in encoded by W in Eq. (E227) (see Sec.E3f). The second k, l = 1, ..., m. We have W˜ = W˜ − h M˜ , where h h term represents the constant contribution to the time- (M˜ ) ≡ Tr[P˜ M(˜ρ )] and M˜ can be approximated by kl k l integral from before the metastable regime. When time the diagonal matrix of the observable averages in the t in Eq. (E229) can be chosen after the final relaxation, metastable phases the asymptotic rate is approximated as [cf. Eq. (53)]

( ˜m)kl = δklm˜ l, (E226) hM(t)i hMcl(t)i mss ≡ lim = lim + ... (E230) t→∞ t t→∞ t k, l = 1, ..., m, where m˜ l ≡ Tr(Mρ˜l), and thus m X = (p ) m˜ + ..., ˜ ss l l Wh = W − h ˜m + ... ≡ Wh + .... (E227) l=1 The corrections in Eq. (E227) can be bounded in the with corrections bounded in the leading order by √ max |m˜ |k˜p −p k [note that m = Pm ( ˜m˜p ) leading order by 2 C kW˜ k +|h| m kMk pC /2 + C 1≤l≤m l ss ss 1 ss l=1 ss l cl 1 max cl + and cf. Eqs. (41)]. [see Eq. (37) and Sec.E3d]. For dynamics of classical p ˜ For times such that kSQkkMk tk ˜mk1 and systems, m Ccl/2 + C+ can be replaced by Ccl + m C+ in √ q √ the corrections to Eqs. (E227). tkW˜ k1 min[ Ccl, kR˜ k1kW˜ k1/ Ccl] the rate of 57

+ + time-ordered fluctuations of the integrated system ob- m˜ lP) MSQM(˜ρl)+(M−˜ m˜ lP) [L−h(M−M˜ )](˜ρl)/h, servable is approximated as [cf. Eq. (58)] where M˜ is the superoperator corresponding to ˜m in Eq. (E211). Furthermore, the maximal eigenvalue of Lh hM 2(t)i−hM(t)i2 hM 2 (t)i−hM (t)i2 = cl cl + hδ2 (t)i and its derivative can be approximated by t t cl m (k) (l) θ(h) = −hm˜ l + ..., (E237) X hMcl (t)i − hMcl (t)i − p˜ p˜ M˜ − M˜ + ...,(E231) k(h) = −m˜ l + ..., (E238) k l t k l k,l=1 with the corrections given in the lowest order where the fluctuations of the average system observable of the non-Hermitian perturbation theory by 2 ˜ ˜ ˜ in classical trajectories −h Tr[PlMSQM(˜ρl)] + (W)ll − h(M − ˜m)ll and ˜ P ˜ ˜ by −2hTr[PlMSQM(˜ρl)] − k6=l(W)lk(W)kl/(m ˜ k − m Z t Z t−t1 X m˜ )/h2 − (M˜ − ˜m) , respectively. Therefore, the hM 2 (t)i = −2 dt dt ( ˜met2W ˜met1W ˜p) , l ll cl 1 2 l derivative of θ(h) undergoes a sharp change around 0 0 l=1 (E232) h = 0 when metastable phases differ in observable we denoted by averages, which can be interpreted as a proximity to a dynamical phase transition [S17]. When the stationary t m R P ˜2 t1W state features contributions from metastable phases 2 0dt1 l=1 δ e ˜p hδ (t)i ≡ l (E233) with different observable averages, the fluctuations in cl t Eq. (E234) are large [cf. Eq. (E235)]. the rate of the time-integral of time-ordered fluc- 2 tuation rates in metastable phases (δ˜ )kl ≡ (l) ˜ ˜ ˜ −2δklTr[MSQM(˜ρl)] (see Sec. E3f), hMcl i is the d. Corrections in approximations of Ws, Wh, and Wr average of time-integral of system observable in classical trajectories for lth metastable phase in In this section, we give and prove corrections in Eq. (E229), i.e., (˜p)k = δkl, k, l = 1, ..., m, and approximations of W˜ s, W˜ h, and W˜ r. We also discuss M˜ l ≡ Tr{P˜lMSQ[ρ(0)]}/Tr[P˜lρ(0)] is the average con- how the order of approximation is changed for the tribution to the integrated system observable from before metastability in classical stochastic dynamics. the metastable regime conditioned on the metastable phase that the system evolves into. When time t in Corrections in classical approximation of W˜ s. For W˜ s Eq. (E231) can be chosen after the final relaxation, the in Eq. (48), we have asymptotic rate of time-ordered fluctuations −s W˜ s = W˜ + (e − 1)J˜, (E239) 2 2 2 hM (t)i−hM(t)i where χss ≡ lim = −2Tr[MSM(ρss)], t→∞ t ˜ X ˜ † (E234) (J)kl ≡ Tr(PkJjρ˜l Jj ). (E240) is approximated as [cf. Eq. (E212)] j

2 2 can be related to the jump activity 2 hhMcl(t)i−hMcl(t)i 2 i χss = lim + hδcl(t)i (E235) t→∞ t (µ˜)kl ≡ δkl µ˜l, (E241) m X 2 i as = −2 ˜mR˜m + δ˜ pss + ... l l=1 |(J˜)kl − (W˜ )kl − δkl µ˜k| (E242) i 1 X † p (cf. Sec.E1). The corrections to Eqs. (E229) and (E231) ≤ H − c + Jj Jj 2Ccl + 4C+, 2 max are given in Sec.E3e, and to Eq. (E234) in Sec.E3g. j where c is a real constant chosen to minimize the operator Classical metastability and dynamical phase transitions norm and k, l = 1, ..., m. Equation (E242) illustrates in time-integrated system observables. When metastable that jump operators considered in the activity can lead to phases diﬀer in observable averages, for the bias |h| transitions between metastable phases, but not at rates large enough W can be neglected in Eq. (E227), and higher than the rates in the eﬀective dynamics W˜ (up Eq. (E228) approximated as [cf. Sec.VB5 in the main to corrections to the positivity of metastable phases and text] their projections). Therefore, together with Eq. (37) [and in √ analogously kJ−(W˜ +µ˜−µ˜ )k kW˜ k1 Ccl], we obtain ρss(h) =ρ ˜l + ..., (E236) . in Eq. (49) p where l is such that m˜ l is maximal (for negative h) kW˜ − W k (e−s + 1) C kW˜ k (E243) or minimal (for positive h) among metastable phases. s s 1 . cl 1 −s i X † p The corrections given in the lowest order of the non- +|e − 1| m kH − c1 + J Jjkmax 2Ccl + 4C+. ˜ 2 j Hermitian perturbation theory are hSQM(˜ρl) − h(M − j 58

We also show for individual jumps that [cf. Eq. (E242)] For a matrix X, we have

˜ ˜ min X Tr(Pk X ρ˜l) = Tr[(Pk − p˜ 1) X ρl] (E252) |Tr(P˜ J ρ˜ J †) − δ Tr(J †J ρ˜ )| − (W˜ ) (E244) k k j l j kl j j l kl min ˜ j +p ˜k Tr(X ρl) + Tr[Pk X (ρl − ρ˜l)], i 1 X † p min ˜ ≤ H − c + Jj Jj 2Ccl + 4C+, where p˜k is the most negative eigenvalue of Pk, and ρl 2 max j is the closest state to ρ˜l [cf. Eq. (12)]. The terms in the second line of (E252) can be bounded as [cf. Eq. (D111)] which bounds the corrections in Eq. (E206). min min |p˜k Tr(X ρl)| ≤ |p˜k |kXkmax, (E253) Corrections in classical approximation of W˜ . For W˜ r r Tr[P˜ X (ρ − ρ˜ )] ≤ kP˜ k kXk kρ − ρ˜ k. (E254) in Eq. (E212), we show that k l l k max max l l The first line in (E252) can be bounded as follows. From ˜ X ˜ −iϕj iϕj † (X)kl ≡ Tr[Pk(e Jjρ˜k + e ρ˜lJj )]/2 (E245) the Cauchy-Schwarz inequality with respect to the oper- j √ ˜ min1 √ ators ρl(Pk − p˜k ) and X ρl we have is approximated by the diagonal matrix ˜x of the observ- ˜ min1 able averages in metastable phases in Eq. (E211) as |Tr[(Pk − p˜k ) X ρl]| (E255) q q ˜ min1 2 † X p ≤ Tr[(Pk − p˜k ) ρl] Tr(X X ρl) |(X˜ ) −(˜x) | ≤ k e−iϕj J k C /2 + C , (E246) kl kl j max cl + q q j max min min ˜ ≤ p˜k − p˜k |p˜k | + Tr(Pk ρl)kXkmax, which together with Eq. (37) gives ˜ min1 2 max where in the second line we used (Pk −p˜k ) ≤ (˜pk − r2 min ˜ min1 ˜ min1 min ˜ p ˜ p˜k )(Pk − p˜k ) and |Tr[(Pk − p˜k ) ρl]| ≤ |p˜k | + kWr − W + r ˜x − k1 . 2 CclkWk1 (E247) 8 |Tr(P˜kρl)|. X −iϕj p +|r|m k e Jjkmax Ccl/2 + C+. For off-diagonal terms (k 6= l), we further obtain j |Tr(P˜kρl)| = |Tr{P˜k(ρl − ρ˜l)| ≤ kP˜kkmaxkρl − ρ˜lk. (E256) Corrections in classical approximation of W˜ h. Simi- Therefore, from (E252) using Eqs. (E253)-(E255) we ob- larly, for W˜ h in Eq. (E225) we show below that tain i (M˜ ) ≡ Tr(P˜ {M, ρ˜ })/2 (E248) ˜ ˜ 1 X † kl k l |(J)kl − (W)kl| ≤ 2 H − c − Jj Jj (E257) 2 max j is approximated by the diagonal matrix ˜m of the observ- q min ˜ able averages in metastable phases in Eq. (E226) as × |p˜k | + kPkkmaxkρl − ρ˜lk q q ˜ p min ˜ max min |(M)kl − ( ˜m)kl| ≤ kMkmax Ccl/2 + C+. (E249) × |p˜k | + kPkkmaxkρl − ρ˜lk + p˜k − p˜k , i Therefore, together with Eq. (37), 1 X † p . H − c − Jj Jj 2Ccl + 4C+, 2 max ˜ p ˜ j kWh − W + h ˜mk1 . 2 CclkWk1 (E250) p min +|h|m kMkmax Ccl/2 + C+, where in the last line we used |p˜k | ≤ Ccl/2 max min max min [cf. Eq. (D63)], |p˜k −p˜k | ≥ |p˜k |−|p˜k | ≥ 1−Ccl/2 which leads to Eq. (E227). [cf. Eq. (D62)] and kP˜kkmax ≤ 1 + Ccl/2 [cf. Eq. (D65)], [cf. Eq. (D95)]. ˜ Derivation of Eq. (E242). We want to compare J Similarly, for diagonal terms [cf. Eq. (E241)] in Eq. (E240) with the effective dynamics in the MM [cf. Eq. (34)] ˜ X 1 ˜ † (J)ll − µ˜l = − Tr[( − Pl)Jjρ˜lJj ], (E258) ˜ X ˜ † j (W)kl = Tr(Pk Jjρ˜lJj ) (E251) j while from Pm P˜ = 1 we also have h i l=1 l ˜ 1 i X † −i Tr Pk H − c − Jj Jj ρ˜l + h.c., 2 ˜ X ˜ X 1 ˜ † j −(W)ll = (W)kl = Tr[( − Pl) Jjρ˜lJj ] (E259) k6=l j where c is an arbitrary real constant. For k 6= l, the h i X i difference of Eqs. (E240) and (E251) corresponds to the −i Tr (1 − P˜ ) H − c1 − J †J ρ˜ + h.c.. l 2 j j l second line of Eq. (E251), which size we now estimate. j 59

Therefore, by replacing P˜l by 1 − P˜l in Eq. (E255) and To bound terms Tr[P˜k X ρ˜l] = Tr[X ρ˜l P˜k], we can noting that again make use of the inequality in Eq. (D111) with the choice Y =ρ ˜lP˜k |Tr[(1 − P˜l)ρl]| = |Tr[(1 − P˜l)(ρl − ρ˜l)| (E260) C |Tr[P˜ X ρ˜ ]| ≤ kXk kP˜ ρ˜ k. (E264) ≤ k1 − P˜ k kρ − ρ˜ k ≤ 1 + cl kρ − ρ˜ k k l max k l l max l l 2 l l Below we show that for commuting ρ˜l and P˜k [e.g., clas- min from |p˜l | ≤ Ccl/2 [cf. Eq. (D63)], we obtain sical stochastic dynamics] we have ˜ ˜ |(J)ll − µ˜l − (W)ll| (E261) X ˜ ˜ kPkρ˜lk . Ccl + C+. (E265) i 1 X † p k6=l . H − c − Jj Jj 2Ccl + 4C+. 2 max j Therefore, we arrive at [cf. Eq. (E261)] Eq. (E242) follows by the triangle inequality. Note that ˜ kXkmax in the bound in Eq. (E242) can be replaced by kJ − W − µ˜k1 (E266) p † 2 i X max1≤l≤m Tr(X Xρ˜l) + kXkmaxC+ [cf. Eq. (E255)]. 2 kH − c1 + J †J k (C˜ + C ) . 2 j j max cl + j Derivation of Eq. (E244). For activities related to individual jumps, we note that and [cf. Eq. (E247)] Tr[(P˜ − p˜min)J ρ J †] = |Tr[(P˜ − p˜min)J ρ J †]|, (E262) ˜ ˜ k k j l j k k j l j kM − ˜mk1 . kMkmax(Ccl + C+). (E267) Tr[(˜pmax − P˜ )J ρ J † = |Tr[(˜pmax − P˜ )J ρ J †|, l l j l j l l j l j Indeed, we have ˜ min max which follows from the positivity of Pk −p˜k and p˜l − X X X kP˜kρ˜lk ≡ |p˜k(n)||ρ˜l(n)| (E268) P˜ and ρ . Therefore, l l k6=l k6=l n X X X ˜ † ˜ = |ρ˜ (n)|[|p˜ (n)| − p˜ (n)] |Tr(Pk Jjρ˜l Jj ) − (µ˜)kl| − [(J)kl − (µ˜j)kl] (E263) l k k j k6=l n min max max min X X . |p˜k |(1−δkl) + |p˜l −1|δkl + (˜pk − p˜k )kρl − ρ˜lk + [|ρ˜l(n)| − ρ˜l(n)] p˜k(n) h X (j) i n k6=l × |µ˜ | + |µ˜l| , l X min j ≤ (1 + C+) (−2˜pk ) + C+ max |1 − p˜l(n)| n k6=l (j) † min where (µ˜ )kl ≡ µ˜ δkl ≡ Tr(J Jjρ˜l)δkl. As |p˜ | ≤ j l j k ≤ 1(1 + C+)C˜cl + (1 + Ccl/2)C+ C˜cl + C+, max max min . Ccl/2, |p˜k − 1| ≤ Ccl/2, (˜pk − p˜k ) ≤ (1 + 2Ccl), and kρl − ρ˜lk ≤ C+, the corrections are of a higher order than where ρ˜l(n) and p˜k(n) are the diagonal entries of ρ˜l and the corrections in Eq. (E242) and thus can be neglected, P˜k, respectively, in their eigenbasis, n = 1, ..., dim(H). P giving rise to Eq. (E244). In the second line we used ρ˜l(n)˜pk(n) = Tr(P˜kρ˜l) = n P δkl. In the first inequality we used n |ρ˜l(n)| ≤ Derivation of Eq. (E246). By considering Eqs. (E252)- P 1 + C+ [cf. Eq. (12)], so that also n |ρ˜l(n)| − (E255) and Eq. (E258) for X = P e−iϕj J /2 and X†, P j j ρ˜l(n) = n |ρ˜l(n)| − 1 ≤ C+, as well as |p˜k(n)| − † min and noting that kXk = kX k we arrive at p˜k(n) = 2 max[−p˜k(n), 0] ≤ −2˜p [cf. Eq. (21)] and max max P k the result. Furthermore, kXkmax in the bound can k6=l p˜k(n) = 1 − p˜l(n) (see Sec.D1). p † 2 be replaced by max1≤l≤m Tr(X Xρ˜l) + kXkmaxC+ [cf. Eq. (E255)]. e. Rates of average and fluctuations in quantum Derivation of Eq. (E249). By considering Eqs. (E252)- trajectories after initial relaxation (E255) and Eq. (E258) for X = M/2 we arrive at the result. Note that kMkmax in the bound in Eq. (E246) can Here, we give corrections to the classical approxi- p † 2 be replaced by max1≤l≤m Tr(M Mρ˜l) + kMkmaxC+ mations of the average and fluctuations rates of jump [cf. Eq. (E255)]. number after the metastable regime given in Eqs. (52) and (54). We also discuss the cases of homodyne Corrections for classical stochastic dynamics. We now measurement and time-integral of system observables show that metastability in dynamics of probability dis- introduced in Secs.E3b andE3c. Derivations rely on tributions rather than density matrices (see Sec. E1) the master operator in Eq. (1) being diagonalizable and leads to linear order of corrections in Eqs. (E242), (E244) can be found at the end of this section. and (E246). 60

Corrections to average rate of jump number. For time is approximated as in Eq. (54) with the corrections 00 t ≥ t , the rate of average jump number is approximated 2 2 2 2 hK (t)i−hK(t)i hKcl(t)i−hKcl(t)i + h∆cl(t)i as in Eq. (52) with the following corrections − (E272) t t m (k) (l) X hKcl (t)i − hKcl (t)i ˜ ˜ − p˜kp˜l Kk − Kl t k,l=1 kSQk h ˜ hK(t)i hKcl(t)i + hKi . kJ k 1 + kSQk 4kJ Pk + 3kJ k − (E269) t t t n kSQk kSQk +4C kµ˜k1 + kJ k kJ kkSQk 2Cn kJ k + kµ˜ − µ˜totk MM t . MM t 1 tot ˜ kµ˜k Z t +tkµ˜ − µ˜ k1 min(kJk1, kJ Pk) 1 t1W˜ t1W + dt1ke − e k 1 t 0 +tkµ˜k kJ˜ − W˜ − µ˜k + kW − W˜ k 1 1 2 1 n kSQk p t t−t 2C kJ k + CclkW˜ k1 1 + tkµ˜k1), 4Z Z 1 . MM t2W˜ t2W 2 t + dt1 dt2ke − e k1 min(kµ˜k1kJ˜k1, kJ Pk ) t 0 0 Z t tot t1W˜ t1W + tkµ˜ − µ˜ k1 + dt1ke − e k1kµ˜k1 2kµ˜k1 0 1 Z t tot t1W˜ t1W 2 where an integer n ≥ 1 is chosen so that t/n belongs to +kµ˜ − µ˜ k1 + dt1ke − e k1kσ˜ k1 t 0 the metastable regime, kJ k = kP J †J k , kµ˜k = j j j max 1 1 Z t tot t1W˜ t1W max1≤l≤m |µ˜l|, and Q ≡ I − P is the projection on the + kµ˜ − µ˜ k1 + dt1ke − e k1kµ˜k1 00 t fast modes of L, so that kSQk . 2t [see Eq. (E294)]. 0 The first correction arises from the approximation of the ×4 min(kK˜ ˜pk1, kJ kkSQk) dynamics of fast modes by their complete decay, while kSQk h kJ k 1 + kSQk4kJ Pk + 3kJ k the second correction stems from replacing the long-time . t dynamics in Eq. (34) by the classical stochastic dynamics n kSQk in Eq. (36) [in the second inequality we used Eq. (37) +4CMM kµ˜k1 + kJ k kJ kkSQk and (39)], so that we require t p + CclkW˜ k1t min(kJ˜k1, kJ Pk) + kµ˜k1 i p X † + 2Ccl +4C+m H + Jj Jj tkµ˜k1 2 max p j tkW˜ k 1/ C . (E270) 1 cl 2p + C kW˜ k t2 min(kµ˜k kJ˜k , kJ Pk2) 3 cl 1 1 1 p + CclkW˜ k1 2 + tkµ˜k1 tkµ˜k1 p 2 When + CclkW˜ k1(1 + tkσ˜ k1) p +2 CclkW˜ k1 2 + tkµ˜k1 min(kK˜ ˜pk1, kJ kkSQk), where 2 tkµ˜k1 kSQkkJ k, (E271) σ˜ kl ≡ δkl[˜µl − 2Tr(J SQJ ρ˜l)], (E273) k, l = 1, ..., m, which can be interpreted as fluctuation rate of metastable phases (see Sec.E3f), and Tr(P˜ JSQρ) the contribution K/t˜ from before the metastable regime (K˜ ) ≡ δ K˜ = δ l (E274) kl kl l kl ˜ can be neglected and the approximation in Eq. (52) holds Tr(Plρ) well [provided that hKcl(t)i/t is of the order of kµ˜k1, encodes the contribution from before the metastable ˜ as kWk1 . kµ˜k1]. In particular, when time t can be regime. We further have kJ Pk . kµ˜k1 + 2kJ kC+ [? chosen after the final relaxation, we obtain the classical ], as well as kJ˜k1 ≤ (1 + C˜cl)kJ Pk and kK˜ ˜pk1 ≤ approximation of the asymptotic average rate in Eq. (53). (1 + C˜cl)kJ kkSQk [cf. Eq. (D79)]. In the second inequality of Eq. (E272) we used Eqs. (39) and (E242). The first and second lines of cor- Corrections to fluctuation rate of jump number. Simi- rections in the first inequality arise from the approxima- larly, for t ≥ t00, the rate of fluctuations in jump number tion of the dynamics of fast modes by the full decay and 61 neglecting any constant contribution to the fluctuations, metastable phases. Finally, the corrections in the third 2 while the rest of corrections originate from replacing the to fifth lines of Eq. (54) are the corrections to [hKcl(t)i− 2 long-time dynamics in Eq. (34) by the classical stochastic hKcl(t)i −hKcl(t)i]/t ≤ tkµ˜k1[min(kJ˜k1, kJ Pk)+kµ˜k1]. approximation in Eq. (36) (and considering the space of They are negligible in comparison with this bound when metastable phases with L1 norm, or the space of density Eq. (E276) is fulfilled together with matrices with trace norm; cf. Sec.D2). In particular, we can identify that the third to fifth lines correspond to the corrections to the non-Poissonian classical fluctu- 2 2 i P † ations of the activity, [hKcl(t)i−hKcl(t)i −hKcl(t)i]/t, the mkH + 2 j Jj Jjkmax 1 sixth line is the correction to the time-integral of fluctua- p . (E278) min(kJ˜k1, kJ Pk) 2Ccl + 4C+ tion rate inside metastable phases, [hKcl(t)i+h∆cl(t)i]/t, while the seventh and eighth lines describe corrections to the contribution from before the metastable regime. When the corrections in Eq. (E272) are negligible in 2 2 It can also be shown that [hK (t)i − hKcl(t)i − comparison with the corresponding (positive) contribu- cl hK (t)i]/t ≤ 2kR˜ k kµ˜k [min(kJ˜k , kJ Pk) + kµ˜k ](1 + tions to the fluctuation rate, we obtain sufficient condi- cl 1 1 1 1 4kR˜ k /t) [cf. derivation of Eq. (54) below], which re- tions on the classical approximation in Eq. (54). As we 1 quires discuss below, the contributions to the fluctuation rate can be bounded in terms of operator norms leading to generalized conditions, which are sufficient for the classical approximation to hold whenever the bounds saturate s ˜ ˜ (for the opposite case, see e.g., Sec.E3f). In particular, kRk1kWk1 tkW˜ k1 √ , (E279) if those conditions are fulfilled at time t after the final re- Ccl laxation, the asymptotic fluctuation rate can be approximated in terms of the classical dynamics [see Eq. (58) and cf. Sec.E3g]. Finally, if there exists a leading contribution, e.g., non-Poissonian classical fluctuations char- which is a stricter condition than Eq. (E276) for ˜ acteristic of a first-order dynamical phase transition (see t ≥ 4kRk1. For those times, no additional condition Sec. VB5 in the main text), the conditions can be re- beyond Eq. (E278) is required, as the correspond- laxed. ing corrections can be shown to be bounded by ˜ ˜ ˜ ˜ The contribution to the time-integral of fluctuation 2kRk1(1 + 4kRk1/t)kµ˜k1kJ − W − µ˜k1. rate inside metastable phases [hKcl(t)i + h∆cl(t)i]/t 2 is bounded by kσ˜ k1, while the contribution to the fluctuation rate from before the metastable regime Corrections to average rate of time-integrated homo- [the second line of Eq. (E272)] is bounded by dyne current. Similarly to Eq. (E269), for the integrated homodyne current we obtain that the corrections 4kµ˜k1 min(kK˜ ˜pk1, kJ kkSQk) (see derivation below). Therefore, the first and the second lines of corrections in Eq. (E214) are bounded by are negligible in comparison with those bounds, when

2 t min(kµ˜k1, kσ˜ k1) kSQkkJ k (E275) hX(t)i hX (t)i + hX˜i [cf. Eq. (E271)]. Furthermore, the corrections to the − cl (E280) time-integral of ﬂuctuation rate inside metastable phases t t and contribution from before the metastable regime in kSQk k˜xk Z t n 1 t1W˜ t1W the last two lines are negligible in comparison with the . 2CMM kX k + dt1ke − e k t t 0 corresponding bounds when kSQk p 2Cn kX k + t C kW˜ k k˜xk , 1 . MM t cl 1 1 tkW˜ k1 √ , (E276) Ccl as well as

kW˜ k1 1 where an integer n ≥ 1 is chosen so that t/n belongs √ , (E277) P −iϕj 2 to the metastable regime, kX k ≤ k e Jjkmax and kσ˜ k1 Ccl j k˜xk1 = max1≤l≤m |x˜l|. [which is the condition for replacing σ˜2 by (σ˜tot)2, while µ˜ can be replaced by µ˜tot in the contribution ˜ from before the metastable regime as kWk1/kµ˜k1 . Corrections to fluctuation rate of time-integrated ho- 2]. Note that Eq. (E277) can be viewed as condi- modyne current. Similarly to Eq. (E272), for the inte- tion on the maximal anti-bunching of fluctuations inside grated homodyne current we obtain that the corrections 62 in Eq. (E216) are bounded by where an integer n ≥ 1 is chosen so that t/n belongs to the metastable regime, kMk ≤ kMkmax and 2 2 2 2 hX (t)i−hX(t)i hXcl(t)i−hXcl(t)i k ˜mk = max |m˜ |. − (E281) 1 1≤l≤m l t t m (k) (l) Corrections to fluctuation rate of time-integrated sys- 2 X hXcl (t)i − hXcl (t)i ˜ ˜ −hχ (t)i p˜kp˜l Xk − Xl tem observables. Analogously to Eq. (E281) for the time- cl t k,l=1 integral of a system observable we obtain that the cor- kSQk2 rections to the classical approximation of time-ordered kX k4kX Pk + 3kX k correlations in Eq. (E231) are bounded by . t 2 2 2 2 n kSQk hM (t)i−hM(t)i hMcl(t)i−hMcl(t)i +4CMM k˜xk1 + kX k kX kkSQk − (E284) t t t +tk˜xk kX˜ − ˜xk m (k) (l) 1 1 2 X hMcl (t)i − hMcl (t)i ˜ ˜ t t−t1 −hδcl(t)i − p˜kp˜l Mk − Ml 4Z Z ˜ t + dt dt ket2W − et2Wk k,l=1 t 1 2 1 0 0 kSQk2 ˜ 2 kMk4kMPk + 3kMk × min(k˜xk1kXk1, kX Pk ) . t Z t t1W˜ t1W 2 n kSQk +2 dt1ke − e k1k˜xk1 +4CMM k ˜mk1 + kMk kMkkSQk 0 t t 1 Z ˜ r t1W t1W 2 Ccl + dt1ke − e k1kχ˜ k1 + + C+mkMkmax tk ˜mk1 t 0 2 t 4 Z ˜ p n h2 i t1W t1W ˜ 2 ˜ 2 2 + dt ke − e k k˜xk kX kkSQk + CclkWk1 t min(k˜xk1kXk1, kX Pk ) + k ˜mk t 1 1 1 3 1 0 o kSQk2 +tkδ˜2k + 2tk ˜mk kMkkSQk , kX k4kX Pk + 3kX k 1 1 . t where kSQk +4Cn k˜xk + kX k kX kkSQk MM 1 t ˜2 δ kl ≡ −2δklTr(MSQMρ˜l). (E285) r C cl X −iϕj ˜ ˜ + + C+m e Jj tk˜xk1 and k ˜mk1 ≤ kMk1 ≤ (1 + Ccl)kMPk [cf. Eq. (D79)]. In 2 max j the inequality we used Eqs. (39) and (E249). 2p ˜ 2 ˜ 2 + CclkWk1t min(k˜xk1kXk1, kX Pk ) Below we drive Eqs. (E269), (E272), (E280), (E281), 3 (E283), and (E284). We start by introducing several p ˜ 2 2 + CclkWk1t k˜xk1 useful facts. p 2 p + CclkW˜ k1tkχ˜ k1 + 2 CclkW˜ k1tk˜xk1kX kkSQk, Exact formulas for cumulants of finite-time distribu- where tion. By noting that

h1 i eΘ(s,t) = Tr[etLs (ρ)] (E286) χ˜2 ≡ δ − 2Tr(X SQX ρ˜ ) , (E282) kl kl 2 l is the moment generating function for the number of jumps K(t) detected up to time t [cf. Eq. (44)], we have and k˜xk1 ≤ kX˜ k1 ≤ (1+C˜cl)kX Pk [cf. Eq. (D79)]. In the ﬁrst inequality, the ﬁrst line of corrections in Eq. (E281) Z t t1L arises from neglecting the dynamics of fast modes, while hK(t)i = dt1Tr(J e ρ) (E287) the rest of corrections originate from replacing the 0 tL long-time dynamics by the classical stochastic dynamics. = tTr(J ρss) + Tr[JS(e − I)ρ], In the second inequality we used Eqs. (39) and (E246). Z t 2 t1L hK (t)i = dt1Tr(J e ρ) (E288) Corrections to average rate of time-integrated sys- 0 Z t Z t−t1 t2L t1L tem observables. Analogously to Eq. (E280), for the + 2 dt1 dt2Tr(J e J e ρ) time-integral of the system observable the corrections in 0 0 tL Eq. (E229) are bounded by = tTr(J ρss) + Tr[JS(e − I)ρ] Z t ˜ t1L hM(t)i hMcl(t)i + hMi + 2Tr(J ρ ) dt (t − t )Tr(J e ρ) − (E283) ss 1 1 t t 0 Z t n kSQk p ˜ + 2 dt Tr{J S[e(t−t1)L − I]J et1Lρ}, . 2CMM kMk + t CclkWk1k ˜mk1, 1 t 0 63

t tLs R (t−t1)Ls t1Ls as ∂se = 0 dt1e ∂sLse = which from Eqs. (39) leads to Eq. (E269) [note that in t −s R (t−t1)Ls t1Ls Eq. (E295) kQρk can be simply replaced by 1]. −e 0 dt1e J e [cf. Eq. (44)]. Similarly, for the integrated homodyne current X(t) we have that Derivation of Eq. (E272). We have [cf. Eq. (E288)]

t t−t1 Θ(r,t) tLr Z Z e = Tr[e (ρ)] (E289) t2L t1L dt1 dt2Tr(J e PJ e Qρ) (E297) is a moment generating function [cf. Eq. (E209)] and thus 0 0 Z t Z t−t2 [cf. Eqs. (E287) and (E288)] t2L t1L = dt2 dt1Tr(J e PJ e Qρ) tL 0 0 hX(t)i = tTr(X ρss) + Tr[XS(e − I)ρ], (E290) Z t t t2L (t−t2)L t Z = dt2Tr{J e PJS[e − I])Qρ} 2 t1L hX (t)i = + 2Tr(X ρss) dt1(t − t1)Tr(X e ρ) (E291) 0 2 0 Z t and +2 dt Tr{X S[e(t−t1)L − I]X et1Lρ}. 1 Z t 0 t2L (t−t2)L dt2Tr{J e PJS[e − I])Qρ} (E298) Analogous formulas hold for time-ordered moments of 0 time-integral of system observables with X replaced by tL +tµssTr(JSQρ) − Tr[JS(e − I)PJSQρ] M and 1/2 removed in Eq. (E291) [cf. Eq. (E224)].

tL 2 +Tr(J e PJS Qρ) Useful facts about norms. We now prove bounds on norms of operators appearing in Eq. (E287) and tL 3 2 tL Eq. (E288). During the metastable regime, t00 ≤ t ≤ t0, ≈ Tr(J e LPJ S Qρ) − Tr(JPJS e Qρ) n 2 from the Taylor series we have [see Eq. (B39)] . (2CMM + CMM)kSQk kJ PkkJ kkQρk, tL tkLMMk ≈ k(e −I)Pk ≤ (1+C+)CMM . CMM. (E292) where in the approximation we used t R dt et2λk e(t−t2)λl = (etλk − etλl )/(λ − λ ) ≈ Note that it follows that Eq. (E292) holds also before the 0 2 k l tλk tλk 2 tλl metastable regime, that is, for t ≤ t00. −e /λl − λke /λl + e /λl for |λk| |λl| [we R For times t ≥ t00, let an integer n ≥ 1 be such that t/n have |λk| −λl for k ≤ m < l; cf. Sec.B2]. In tL belongs to the metastable regime. We have [S6] the inequality we used ke Pk ≤ 1 + C+ together with Eq. (E292) for time kSQk. Analogously to Eq. (E298) tL t L n t L n ke Qk = k(e n −P) Qk ≤ k(e n −P)k kQk (E293) we have n n ≤ CMM(2 + C+) . 2CMM ≤ 2CMM. Z t (t−t1)L t1L dt1Tr{J S[e − I]QJ e Pρ} (E299) Finally, we have that for any time within metastable 0 00 0 regime, t ≤ t ≤ t , tL +tTr(J SQJ ρss) + Tr[J SQJ S(e − I)Pρ] kSQk 2t. (E294) . 2 tL R ∞ tL −Tr(JS QJ e Pρ) Eq. (E294) follows from S = 0 dt(e − Pss) t tL R t1L [cf. Eq. (E181)], and thus (I − e )S = dt1(e − Pss) 0 2 tL 3 tL so that (1−2C )kSQk ≤ (1−ketLQk)kSQk ≤ kSQk− Tr(JS QJ e LPρ) − Tr(JS e QJ Pρ) MM . tL tL R t tL ke SQk ≤ k(I − e )SQk = k 0 e Qk ≤ kQkt ≤ 2t n 2 [cf. Eq. (E187)]. . (2CMM + CMM)kJ kkJ PkkSQk . Finally, from Eq. (E293), we have Derivation of Eq. (E269). For times t ≥ t00, we have

[cf. Eq. (E287)] Z t (t−t1)L t1L dt1Tr{J S[e − I]QJ e Qρ} (E300) tL hK(t)i − tµss − Tr[JS(e − I)Pρ] + Tr(JSQρ) 0 tL n = Tr(JSe Qρ) 2C kJ kkSQkkQρk, (E295) . MM +Tr(J SQJ SQρ) where we used Eq. (E293) with n such that t00 ≤ t/n ≤ t0. Furthermore, we recognize Z t (t−t1)L t1L tL tL = Tr[JSe QJ e Qρ]+Tr(J SQJ Se Qρ) tµss + Tr[JS(e − I)Pρ] (E296) 0 m n p n 2 2 X ˜ tW˜ . 6CMM + 4 CMM kJ k kSQk kQρk, = µ˜ t˜pss + R(e − I)˜p l l=1 where in the last inequality we t t m R (t−t1)L t1L Z used dt1Tr[JSe QJ e Qρ] = X t1W˜ 0 = dt1 µ˜e ˜p , t t l R 2 R (t−t1)L t1L 0 ( + t )dt1Tr[JSe QJ e Qρ], while l=1 0 2 64

Tr[JSe(t−t1)LQJ et1LQρ] can be bounded by We recognize kJ k2kSQkke(t−t1)LQkkQρk or kJ k2kSQkket1LQkkQρk Z t m (used in the ﬁrst and the second integrals, respec- X ˜ tµ + Tr[JS(etL − I)Pρ] = dt J˜et1W ˜p tively). We also assumed that n ≥ 2, so that t/2 ≥ t00 ss 1 l 0 l=1 [cf. Eq. (E293)]. Z t m X ˜ From Eqs. (E295)-(E300) we arrive at [cf. Eqs. (E287) t1W ˜ = dt1 µ˜e ˜p l = hKcl(t)i, (E303) and (E288)] 0 l=1 ˜ where Kcl(t) is the time-integral of the variable with the 2 2 ˜ hK (t)i−hK(t)i (E301) cumulant generating function encoded by Ws in Eq. (48), and, similarly, −tµ − Tr[JS(etL − I)Pρ] + Tr(JSQρ) ss Z t Z t−t1 t t−t t2L t1L Z Z 1 2 dt1 dt2Tr(J e PJ e Pρ) (E304) t2L t1L −2 dt1 dt2Tr(J e PJ e Pρ) 0 0 m 0 0 Z t Z t−t1 X ˜ ˜ tL 2 ˜ t2W˜ t1W + tµss + Tr[JS(e − I)Pρ] − Tr(JSQρ) = 2 dt1 dt2 Je Je ˜p l 0 0 tL l=1 +2tµssTr(JSQρ) + 2Tr[JS(e − I)PJSQρ] m Z t Z t−t1 X ˜ ˜ +2Tr(J etLPJS2Qρ) t2W˜ t1W = 2 dt1 dt2 µ˜e Je ˜p l tL 0 0 l=1 +2tTr(J SQJ ρss) + 2Tr[J SQJ S(e − I)Pρ] = hK˜ 2 (t)i − hK˜ (t)i. 2 tL cl cl +2Tr(JS QJ e Pρ) + 2Tr(J SQJ SQρ) Furthermore, (4Cn + 2C )kJ PkkJ kkSQk21 + kQρk . MM MM tL tµss + Tr[JS(e − I)Pρ] (E305) +12Cn + 8pCn kJ k2kSQk2kQρk MM MM −2tTr(J SQJ ρ ) − 2Tr[J SQJ S(etL − I)Pρ] n ss +2CMMkJ kkSQkkQρk m X ˜ +4Cn kJ khK(t)ikSQkkQρk, 2 ˜ tW MM = σ˜ t˜pss +R e − I ˜p l l=1 m 00 0 Z t where n is such that t ≤ t/n ≤ t . For t kSQk X 2 t W˜ = dt σ˜ e 1 ˜p = hK˜ (t)i + h∆ (t)i [cf. Eq. (E275)], we can further neglect the constant con- 1 l cl cl 0 l=1 tribution, which gives [cf. Eq. (E273)] and 2 2 tL hK (t)i−hK(t)i Tr[JS(e − I)Pρ] 2Tr[JS(etL − I)Pρ]Tr(JSQρ) (E306) − µss − (E302) t t −2Tr[JS(etL − I)PJSQρ] t t−t 2 Z Z 1 t m m t2L t1L Z ˜ − dt1 dt2Tr(J e PJ e Pρ) X t1W X = −2 dt1 µ˜e ˜p K˜ ˜p t 0 0 l k 0 l=1 k=1 tL 2 tµss + Tr[JS(e − I)Pρ] t m + Z ˜ X t1W ˜ t +2 dt1 µ˜e K˜p l Tr(JSQρ) 0 l=1 −2Tr[JS(etL − I)Pρ] t m X ˜ (k) ˜ (l) ˜ ˜ Tr[JS(etL − I)PJSQρ] = p˜kp˜l hKcl (t)i − hKcl (t)i Kk − Kl , +2 t k,l=1 tL Tr[J SQJ S(e − I)Pρ] where K˜ is deﬁned in Eq. (E274) and hK˜ (l)(t)i is +2Tr(J SQJ ρss) + 2 cl t hK˜cl(t)i for the system in the lth metastable phase kSQk kSQk2 during metastable regime, P(ρ) = ρ , k, l = 1, ..., m. kJ kkQρk + 2 kJ kkJ Pk(1 + kQρk) l . t t This, together with Eq. (E296), leads to Eq. (E272) by kSQk2 replacing the long time-dynamics generator W˜ by the + kJ k2kQρk(2 + kQρk) t stochastic dynamics generator W [cf. Eqs. (37), (39) kSQk and (E242)]. +4Cn kµ˜k + kJ k kJ kkSQkkQρk MM 1 t Derivation of Eqs. (E280) and (E283). We replace J by X and M, in Eq. (E295), and µ˜ by ˜x and ˜m in We note that in Eqs. (E295)-(E300) kQρk can be simply Eq. (E296), respectively. replaced by 1. 65

Derivation of Eqs. (E281) and (E284). From time-integral of system-observables (see also Ref. [S20]). Eqs. (E290) and (E291), using Eqs. (E295)-(E300), we Derivations rely on the master operator in Eq. (1) being arrive at [cf. Eq. (E301)] diagonalizable and can be found at the end of this section.

2 2 t Rates of average and ﬂuctuations of jump number dur- hX (t)i−hX(t)i − (E307) 2 ing metastable regime. For a general initial state ρ, the Z t Z t−t1 rate of average number of jumps within the metastable t2L t1L −2 dt1 dt2Tr(X e PX e Pρ) regime, t00 ≤ t ≤ t0, is approximated as [cf. Eq. (E269)] 0 0 m tL 2 ˜ + txss + Tr[XS(e − I)Pρ] − Tr(XSQρ) hK(t)i X K − p˜lµ˜l + (E309) tL t t +2txssTr(XSQρ) + 2Tr[XS(e − I)PXSQρ] l=1 tL 2 +2Tr(X e PXS Qρ) n1 h C˜cl i kSQko tL . CMM min kJ Pk, 1 + kµ˜k1 + kJ k , +2tTr(X SQX ρss) + 2Tr[X SQX S(e − I)Pρ] 2 2 t ˜ 2 tL where p˜l = Tr(Plρ) [cf. Eq. (20)], +2Tr(XS QX e Pρ) + 2Tr(X SQX SQρ)

X † n 2 µ˜l ≡ Tr Jj Jjρ˜l , (E310) (4C + 2CMM)kX PkkX kkSQk 1 + kQρk . MM j n p n 2 2 + 12CMM + 8 CMM kX k kSQk kQρk and K˜ ≡ −Tr(JSQρ) is the contribution to the num- +4Cn kX khX(t)ikSQkkQρk, MM ber of jumps from before the metastable regime. We 00 0 P † where n is such that t ≤ t/n ≤ t . Thus, [cf. Eq. (E302)] have kµ˜k1 = max1≤l≤m |µ˜l|, kJ k = k jJj Jjkmax, and kJ Pk ≤ kJ k can be replaced by kµ˜k1 + 2C+kJ k [S19]. 2 2 hX (t)i − hX(t)i We can further replace µ˜ by µ˜tot introducing additional (E308) √l l t corrections bounded by CclkW˜ k1. If 2 Z t Z t−t1 t2L t1L 0 − dt1 dt2Tr(X e PX e Pρ) t kµ˜k1 ≥ tkµ˜k1 kSQkkJ k, (E311) t 0 0 tL 2 the contribution K/t˜ from before the metastable regime txss + Tr[XS(e − I)Pρ] + can be neglected [cf. Eq. (E271)]. t For time t in Eq. (E311), the rate of fluctuations of Tr(XSQρ) −2Tr[XS(etL − I)Pρ] jump number is approximated as [cf. Eq. (E272)] t 2 2 Tr[XS(etL − I)PXSQρ] hK (t)i − hK(t)i +2 (E312) t t tL m Tr[X SQX S(e − I)Pρ] Xh1 2 i +2Tr(X SQX ρss) + 2 −t p˜kp˜l(˜µk − µ˜l) − µ˜k(W˜ )kl p˜l t 2 k,l=1 2 kSQk m m . kX k 2kX Pk(1 + kQρk) + kX kkQρk(2 + kQρk) X 2 X ˜ ˜ t − p˜lσ˜ − p˜kp˜l(˜µk − µ˜l)(Kk − Kl) l n kSQk l=1 k,l=1 +4C k˜xk1 + kX k kX kkSQkkQρk MM t kSQk h kJ k 1 + kSQk4kJ Pk + 3kJ k . t Eq. (E284) follows analogously, by replacing X by M kSQk an dropping t/2 and 1/2 in Eqs. (E307) and (E308), re- +4CMM kµ˜k1 + kJ k kJ kkSQk spectively. t 2 2 +t min kLMMkkJ Pk , kW˜ k1kµ˜k1kJ˜k1 +t2 min kL kkJ Pk, kW˜ k kµ˜k kµ˜k f. Rates of average and fluctuations in quantum trajectories MM 1 1 1 i during metastable regime X † p +tkµ˜k1m H + Jj Jj 2Ccl + 4C+ 2 max j We now discuss rates of average and fluctuations ˜ 1 h Ccl 2 i within metastable regime for jump number [Eq. (44)], + CMM min kJ Pk 1 + 2kJ kkSQk , 1 + kσ˜ k1 integrated homodyne current [Eq. (E209)] and time- 2 2 integral of a system observable [Eq. (E224)]. We give h C˜cl i +2C min kJ PkkJ kkSQk, 1 + kµ˜k kK˜ ˜pk , corrections to the classical approximation of the rate of MM 2 1 1 average and fluctuations as given in Eqs. (53), (58) for where jump number, Eqs. (E215) and (E220) for integrated 2 homodyne current, and Eqs. (E230) and (E235) for σ˜l ≡ µ˜l − 2Tr(J SQJ ρ˜l), (E313) 66

[cf. Eq. (E273)] and K˜l ≡ −Tr(P˜lJSQρ)/Tr(P˜lρ), l = [cf. Eq. (E311)], the corrections are negligible, and µ˜l 1, ..., m, is the contribution to the number of jumps in Eq. (E310) is the activity of lth metastable phase (see from before the metastable regime, conditioned on which also Sec.E3h). the metastable phase the state evolves into. We have For the initial state chosen as the closest state to the ˜ ˜ (1 − C+ − Ccl)kLMMk . kWk1 ≤ (1 + Ccl/2)kLMMk projection ρ˜l of lth metastable phase, we also have (cf. Sec.D2). 2 2 m The corrections in Eq. (E312) to be negligible re- hK (t)i−hK(t)i 2 X ˜ − σ˜ − t µ˜k(W)kl (E317) quire conditions (E275), (E276) and Eq. (E278), to- t l k=1 gether with a new condition (1 + C˜ /2)C 1. cl MM kSQk2 Indeed, the first and the second lines of correc- . 2 kJ kkJ Pk tions require Eqs. (E275) and (E276) [cf. Eq. (E272)]. t 2 2 ˜ ˜ The quadratic-in-time corrections in Eq. (E312) orig- +t min kLMMkkJ Pk , kWk1kµ˜k1kJk1 inate from the linear term in the fluctuation rate 2 +t min kLMMkkJ Pk, kW˜ k1kµ˜k1 kµ˜k1 bounded by t min(kJ Pk2, kµ˜k kJ˜k ) + tkµ˜k2. As 1 1 1 i ˜ X † p tkLMMk CMM [Eq. (E292)] and kWk1 ≤ (1 + +tkµ˜k1m H + Jj Jj 2Ccl + 4C+ . 2 max C˜cl/2)kLMMk, the third and fourth lines of corrections j require min(1 + C˜ /2, kJ Pk/kµ˜k ) C min(1 + ˜ cl 1 MM . h 2 Ccl ˜ i ˜ +tC+ min 3kJ Pk , 1 + kµ˜k1 kJk1 + 2kµ˜k1 Ccl/2, 1 + 2C+kJ k/kµ˜k1) CMM 1 [cf. Eq. (E292) and 2 see [S19]] to be negligible in comparison. This con- 1 + C + C dition also guarantees that the seventh line [correc- 2 MM + tions to the fluctuation rate from before the metastable ˜ ˜ h Ccl 2 i regime bounded by 4 min(kJ PkkJ kkSQk, kµ˜k1kK˜pk1)] × min kJ Pk 1 + 2kJ kkSQk), 1 + kσ˜ k1 is negligible. The fifth line of corrections requires 2 Eq. (E278) [cf. Eq. (E272)]. Finally, the correc- +4C+kJ PkkJ kkSQk. tions in the sixth line are negligible when min[1 + The contribution independent of long-time dynamics, 2 C˜cl/2, kJ Pk max(1, kSQkkJ k)/kσ˜ k1] CMM 1. 2 (δσ˜2 )l ≡ σ˜l − (˜µ)l [cf. Eq. (E313)], can thus be viewed tot ˜ We can further replace µ˜l by µ˜l , W by W, and as the non-Poissonian contribution to fluctuations of lth σ˜2 by (˜σtot)2 [cf. Eq. (37)] in Eqs. (E312), introducing 2 l l √ metastable phase (see also Sec.E3h). We note that σ˜l ˜ can be further replaced by (˜σtot)2 introducing additional additional corrections bounded by CclkWk1[t(4kµ˜k1 + √ l ˜ kW˜ k1)+1+4 min(kJ kkSQk, kK˜ ˜pk1)], which to be neg- corrections bounded by CclkWk1 [cf. Eq. (37)], which ligible require Eq. (E277). Furthermore, when the inter- requires Eq. (E279). nal activity dominates transition rates in the long-time The corrections in Eq. (E317) to be negligible in com- 2 dynamics, parison with kσ˜ k1, imply a lower and upper bounds on ˜ ˜ time t, the condition (1 + Ccl/2)CMM 1, as well as the kWk1 kµ˜k1, (E314) following condition on the anti-bunching of fluctuations we can neglect the contribution from the long-time dy- inside metastable phases, namics (non-Poissonian fluctuations contribution to the kJ PkkJ kkSQk 1 classical fluctuation rate); we then obtain Eq. (61). Al- 2 , (E318) ternatively, this contribution can be neglected when kσ˜ k1 C+

˜ Ccl A similar condition in Eq. (E315) guarantees that we can min kJ Pk, 1 + 2 kµ˜k1 1 2 , (E315) neglect the linear-in-time contribution (non-Poissonian kσ˜ k1 CMM fluctuations in the classical dynamics). which condition can be viewed as the lower bound on the We note that, alternatively, for the approximations in anti-bunching of fluctuations inside metastable phases. Eqs. (E316) and (E317) we can consider an initial state ρ such that P(ρ) =ρ ˜l, in which case the corrections are Rates of average and fluctuations of jump number in bounded as in Eqs. (E269) and (E272), respectively. metastable phases. For the initial state ρ chosen as the closest state to the projection ρ˜l of lth metastable phase Rates of average and fluctuations of time-integrated ho- [Eq. (17)], we have modyne current during metastable regime. During the metastable regime, we have for the homodyne current hK(t)i kSQk − µ˜l 2C+kJ k (E316) [cf. Eq. (E280)] t . t m ˜ 1 h C˜ i hX(t)i X X cl − p˜lx˜l + (E319) + CMM + 2C+ min kJ Pk, 1+ kµ˜k1 t t 2 2 l=1 ˜ Therefore, if min(1 + C˜cl/2, kµ˜k1/kJ Pk)(CMM + C+) 1 h Ccl i kSQk . CMM min kX Pk, 1+ k˜xk1 + 2 kX k , 1, and there exists time tkµ˜k1 C+kSQkkJ k 2 2 t 67 where as well as [cf. Eq. (E317)]

X −iϕj iϕj † x˜l ≡ Tr (e Jj + e Jj )˜ρl , (E320) 2 2 j hX (t)i−hX(t)i 2 − σ˜ (E324) t l kSQk2 2 kX kkX Pk . t with k˜xk1 = max1≤l≤m |x˜l| ≤ (1 + C˜cl)kX Pk 2 2 P −iϕj +t min kL kkX Pk , kW˜ k k˜xk kX˜ k [cf. Eq. (D79)] and kX k ≤ k j e Jjkmax, while MM 1 1 1 2 [cf. Eq. (E281)] +t min kLMMkkX Pk, kW˜ k1k˜xk1 k˜xk1 r C X −ϕj cl +tk˜xk1m e Jj + C+ max 2 j 2 2 m ˜ hX (t)i − hX(t)i t X 2 h 2 Ccl i − p˜kp˜l(˜xk − x˜l) (E321) +tC+ min 3kX Pk , 1 + k˜xk1 kX˜ k1 + 2k˜xk1 t 2 2 k,l=1 1 m m + CMM + C+ X 2 X ˜ ˜ 2 − p˜lχ˜ − p˜kp˜l(˜xk − x˜l)(Xk − Xl) l ˜ l=1 k,l=1 h Ccl 2 i × min 2kX PkkX kkSQk, 1 + kχ˜ k1 2 2 kSQk kX k 4kX Pk + 3kX k +4C kX PkkX kkSQk. . t + +4CMMk˜xk1kX kkSQk 2 2 +t min kLMMkkX Pk , kW˜ k1k˜xk1kX˜ k1 2 ˜ +t min kLMMkkX Pk, kWk1k˜xk1 k˜xk1 Therefore, provided that corrections in Eq. (E323) r C and (E324) are negligible, we can conclude that x˜l X −ϕj cl +tk˜xk1m e Jj + C+ in Eq. (E320) and σ˜2 in Eq. (E322) are the rates of max 2 l j average and ﬂuctuations, respectively, of an integrated +2CMMkX PkkX kkSQk homodyne current in lth metastable phase. ˜ 1 h Ccl 2 i + CMM min 2kX PkkX kkSQk, 1 + kχ˜ k1 , 2 2 Rates of average and ﬂuctuations of time-integrated system observable during metastable regime. Analo- gously, for the time-ordered integral of a system observable M we have [cf. Eq. (E283)] where [cf. Eq. (E282)]

m 2 1 hM(t)i M˜ χ˜ ≡ − 2Tr(X SQX ρ˜l) (E322) X l − p˜lm˜ l + (E325) 2 t t l=1 1 h C˜ i kSQk C min kMPk, 1 + cl k ˜mk + 2 kMk , . MM 2 2 1 t and X˜l ≡ −Tr(P˜lXSQρ)/Tr(P˜lρ), l = 1, ..., m.

For the initial state chosen as the closest state to the projection ρ˜l of lth metastable phase, we further have [cf. Eq. (E316)] where

m˜ l ≡ Tr[Mρ˜l], (E326) hX(t)i kSQk − x˜l 2C+kX k (E323) t . t 1 h C˜ i + C + 2C min kX Pk, 1+ cl k˜xk ˜ 2 MM + 2 1 k ˜mk1 = max1≤l≤m |m˜ l| ≤ (1 + Ccl)kMPk [cf. Eq. (D79)] 68 and kMk ≤ kMkmax, while [cf. Eq. (E284)] system-observable in lth metastable phase.

2 2 m hM (t)i − hM(t)i t X 2 Derivation of Eqs. (E309) and (E312). We consider − p˜kp˜l(m ˜ k − m˜ l) (E327) t 2 time within the metastable regime, t00 ≤ t ≤ t0, to in- k,l=1 vestigate the cumulants in Eqs. (E295) and (E302) with m m X ˜2 X ˜ ˜ respect to the metastable state of the system. Therefore, − p˜lδ − p˜kp˜l(m ˜ k − m˜ l)(Mk − Ml) l within the MM, we can expand the long-time dynamics in l=1 k,l=1 the Taylor series, while the contribution from the outside 2 kSQk is bounded as in Eq. (E293). . kMk 4kMPk + 3kMk t We have +4CMMk ˜mk1kMkkSQk hK(t)i Tr(JSQρ) +t2 min kL kkMPk2, kW˜ k k ˜mk kM˜ k − Tr(JPρ) + (E331) MM 1 1 1 t t 2 +t min kLMMkkMPk, kW˜ k1k ˜mk1 k ˜mk1 n1 h C˜cl i kSQk o r . CMM min kJ Pk, 1 + kµ˜k1 + kJ k kQρk , Ccl 2 2 t +tk ˜mk1m kMkmax + C+ 2 from Eq. (E302) and +2CMMkMPkkMkkSQk t t 1 h C˜ i Z Z cl ˜2 dt Tr(J et1LPρ) − t Tr(JPρ) (E332) + CMM min 2kMPkkMkkSQk, 1 + kδ k1 , 1 2 2 0 0 where [cf. Eq. (E285)] Z t . dt1 t1Tr(J LPρ) ˜2 0 δl ≡ −2Tr(MSQMρ˜l). (E328) 2 t ˜ ˜ ˜ ˜ min kLMMkkJ Pk, kWk1kµ˜k1 . and Ml ≡ −Tr(PlMSQρ)/Tr(Plρ), l = 1, ..., m. . 2 For the initial state chosen as the closest state to the tL projection ρ˜l of lth metastable phase, we further have where we used the Taylor series e P = P + tLP + ... [cf. Eq. (E323)] [note that kQρk can be replaced by 1 in derivation of Eq. (E295)]. We also used Eq. (E292) so that

hM(t)i kSQk tkL kkJ Pk C kJ Pk or tkµ˜k kW˜ k ≤ C (1 + − m˜ l 2C+kMk (E329) MM . MM 1 1 MM t . t C˜cl/2)kµ˜k1 via Eq. (D88). If the metastable regime is 1 h C˜ i long enough, for time in Eq. (E311) [cf. Eq. (E294)], + C + 2C min kMPk, 1 + cl k ˜mk 2 MM + 2 1 we can further neglect the contribution from before the metastable regime, and [cf. Eq. (E324)]

hK(t)i 2 2 − Tr(JPρ) (E333) hM (t)i−hM(t)i 2 − δ˜ (E330) t t l 1 h C˜cl i kSQk kSQk2 . CMM min kJ Pk, 1 + kµ˜k1 + kJ k kQρk. 2 kMkkMPk 2 2 t . t 0 +t2 min kL kkMPk2, kW˜ k k ˜mk kM˜ k Similarly, if time t ≤ t can be chosen as Eq. (E275), MM 1 1 1 we have +t2 min kL kkMPk, kW˜ k k ˜mk k ˜mk MM 1 1 1 2 2 hK (t)i−hK(t)i 2 r − tTr(JPJPρ) + tTr(JPρ) (E334) Ccl t +tk ˜mk1m kMkmax + C+ 2 −Tr(JPρ) + 2Tr(J SQJ Pρ) ˜ h Ccl i +tC min 3kMPk2, 1 + k ˜mk kM˜ k + 2k ˜mk + 2 1 1 1 +2Tr(JPJSQρ) − 2Tr(JPρ)Tr(JSQρ)

1 + C + C MM + kSQk 2 2 . kJ k kQρk + kJ kkSQk(2 + 4kQρk + kQρk ) h C˜ i t × min 2kMPkkMkkSQk, 1 + cl kδ˜2k +4C kµ˜k kJ kkSQkkQρk 2 1 MM 1 2 2 ˜ ˜ +4C+kMPkkMkkSQk. +t min kLMMkkJ Pk , kWk1kµ˜k1kJk1 2 ˜ Therefore, provided that corrections in Eq. (E329) +t min kLMMkkJ Pk, kWk1kµ˜k1 kµ˜k1 t and (E330) are negligible, we can conclude that m˜ l in 2 + min kLMMkkJ Pk 1 + 2kJ kkSQk), kW˜ k1kσ˜ k1 Eq. (E326) and δ˜2 in Eq. (E328) are the rates of average 2 l and ﬂuctuations, respectively, of a time-integrated +2t kLMMkkJ Pk, kW˜ k1kµ˜k1 kJ kkSQkkQρk. 69

In Eq. (E334) the ﬁrst two lines of corrections corre- 2C+ and kPρ − ρ˜lk . C+, and thus Eq. (E316) fol- spond to neglecting the dynamics of the fast modes as in lows from Eq. (E331). Similarly, from (E334) we obtain Eq. (E302). The third line corresponds to [cf. Eq. (E304)] Eq. (E317). Derivations of Eqs. (E323)-(E330) are analogous. Z t Z t−t1 t2L t1L 2 2 dt1 dt2Tr(J e PJ e Pρ) − t Tr(JPJPρ) 0 0

Z t Z t−t1 g. Asymptotic rates of fluctuations in quantum trajectories . 2 dt1 dt2 t2Tr(J LPJ Pρ) + t1Tr(J PJ LPρ) 0 0 3 2 ˜ ˜ . t min kLMMkkJ Pk , kWk1kµ˜k1kJk1 . (E335) Here, we give bounds on the corrections in the classical approximations of the asymptotic fluctuation rates in The fourth line is the correction from the square of Eqs. (58), (66), (E235) and (E220). the time-integral of the activity of metastable phases in Eq. (E332) [cf. Eq. (E303)]. The fifth line are correc- Classical approximation of asymptotic rate of jump tions from the time-integral of the fluctuation rate of number fluctuations. The corrections in Eq. (58) origi- metastable phases [cf. Eq. (E305)] and and the sixth line ˜ in ˜ corresponds to corrections to the contribution from be- nate from replacing J by J+µ˜ and R by R in Eq. (57). fore metastable regime [cf. Eq. (E306)] Therefore, the corrections can be bounded as

Z t m t1L t1L X 2 dt1 Tr(J e Pρ)Tr(JSQρ) − Tr(J e PJSQρ) 2 tot tot tot σss − (µ˜ pss − 2µ˜ R˜µ pss + δσ˜2 pss)l (E339) 0 l=1 m −2tTr(JPρ)Tr(JSQρ) − Tr(JPJSQρ) X tot tot tot = µ˜˜pss − 2µ˜R˜ J˜p˜ ss − µ˜ pss − 2µ˜ R˜µ pss l l=1 2 . 2t min kLMMkkJ PkkJ kkSQkkQρk, p ˜ . kµ˜k1k˜pss − pssk1 + CclkWk1 ˜ ˜ kWk1kµ˜k1kK˜pk1 . (E336) h p +2kµ˜k1kR˜ k1 kµ˜k1k˜pss − pssk1 + CclkW˜ k1 We can further estimate the linear-in-time contribution i X † p to the rate of fluctuations as +m H + Jj Jj 2Ccl + 4C+ 2 max j tTr(JPJPρ)−Tr(JPρ)2 (E337) kR˜ − Rk i m 1 +kµ˜k1 , Xh1 2 i ˜ = t p˜ p˜ (˜µ − µ˜ ) − µ˜ (W˜ ) p˜ + ..., kRk1 2 k l k l k kl l k,l=1 ˜ where the corrections are given by where we replaced J = ˜pss by µ˜˜pss in the second line introducing corrections k(J˜ − µ˜)˜pssk = k(J˜ − m W˜ − µ˜)˜p k ≤ kJ˜ − W˜ − µ˜k(1 + C ) mkH + X ˜ ˜ ss cl . t µ˜(J − µ˜ − W)˜p l (E338) P † p i J Jj/2kmax 2Ccl + 4C+ [cf. Eq. (E242)] and further l=1 j j √ tot ˜ i used kµ˜ − µ˜k1 ≤ CclkWk1 [cf. Eqs. (36) and (37)]. X † p . tkµ˜k1 m H + Jj Jj 2Ccl + 4C+, 2 max The first line of corrections in Eq. (E339) is neg- j 2 ligible in comparison with kσ˜ k1, which bounds the and for the inequality we used Eq. (E242). Equa- asymptotic rate of time-integral of fluctuations inside tion (E292) further leads to Eq. (E312). the metastable phases, when Eq. (E277) is fulfilled to- 2 gether with kµ˜k1/kσ˜ k1 1/k˜pss − pssk1 [cf. Eq. (40)]. Derivation of Eqs. (E319), (E321), (E325), and (E327). The rest of corrections are negligible in comparison with ˜ ˜ To obtain Eqs. (E319) and (E321), we use Eqs. (E280) 2kRk1kµ˜k1kJk1, which bounds the non-Poissonian clas- and (E308) and follow the steps in derivation of sical fluctuations of activity between metastable phases, Eqs. (E309) and (E312). The derivations of Eqs. (E325) when Eqs. (E278), (40) and (E186) hold true. Thus, if the and (E327) are analogous. bounds are of the same order as the contributions to the asymptotic fluctuations rate, Eqs. (40), (E186)(E277) Derivation of Eqs. (E316) and (E317). To consider the and (E278) are sufficient conditions for the classical ap- rate of average and fluctuations of the number of jumps proximation in Eq. (58). inside a metastable state, we choose the initial state ρ as When the internal activity dominates transition rate the closest state to the projection of the metastable phase of the-long time dynamics, as assumed in Secs. VB4 ρ˜l in Eq. (17) and time t within metastable regime. In and VB5 of the main text, from Eq. (58) we obtain this case, we have that kQρk = kQ(ρ − ρ˜l)k ≤ kQkC+ . Eq. (66). The additional corrections beyond those in 70

Eq. (E342) are state between metastable phases. See also Sec.VB4 in the main text. Proofs here rely on the master operator m X in Eq. (1) being diagonalizable. µ˜tot−µ˜inp − 2µ˜totR˜µtot−µ˜inR˜µinp ss ss l

l=1 Conditional distribution of continuous measurement. ˜ ˜ . kWk1 1 + 2kµ˜k1kRk1 , (E340) We consider the statistics of jump number conditioned on the ﬁnal system state at time t in a quantum tra- where we used kµ˜tot − µ˜ink ≤ kWk /2 kW˜ k /2 1 1 . 1 jectory evolving into the metastable phase ρ˜l. This ˜ [cf. Eq. (37)] and kRk1 . kRk1 . condition can be described as obtaining outcome l the POVM in Eq. (D66) performed on the system Classical approximation of asymptotic rate of inte- at time t, which takes place with the probability ap- tW grated homodyne current ﬂuctuations. Similarly to proximated by [p(t)]l ≡ (e ˜p)l up to corrections . Eq. (E342), the corrections in Eq. (E220) originate from Pm tW˜ tW ˜ min (vl)k|[(e − e ˜p]k| + Ccl|[p(t)]l|/2 − p˜ ˜ ˜ √k=1 l . replacing X by ˜x [cf. Eq. (E212)] and R by R in ˜ ˜ min 2 CcltkWk1k+Ccl|[p(t)]l|/2 − p˜l [where (vl)k ≡ (δkl − Eq. (E219), so that min ˜ p˜l )/(1 + Ccl/2), k = 1, ..., m, represents Pl in the

m m metastable phase basis]. The conditional average [see 2 X 2 X χss − (pss)l χ˜l − 2 (˜xR˜xpss)l (E341) Eq. (E357)] is approximated by [cf. Eq. (52)] l=1 l=1 m tW m X (e )lkp˜k ˜ X ˜ ˜ hK(t)il = hKcl(t)il + Kk + ..., (E343) = 2 ˜xRX ˜pss − ˜xR˜xpss [p(t)]l l k=1 l=1 h ˜ X −ϕj p where . k˜xk1kRk1 m e Jj 2Ccl + 4C+ max j t R (t−t1)W in t1W 0 dt1[e (J + µ˜ )e ˜p]l kR˜ − Rk1 i hKcl(t)il ≡ (E344) +2k˜xk1 k˜pss − pssk1 + , [p(t)]l kR˜ k1 is the integral of the total activity in the classical trajec- where we used Eq. (E246). tories, conditioned on the system found at time t in lth metastable phase, while the second term encodes the con- Classical approximation of asymptotic rate of inte- tribution from before the metastable regime [cf. Eq. (54)] grated system observable ﬂuctuations. Analogously to reweighed with the probability of observing outcome Eq. (E339), the corrections in Eq. (E235) originate from l at time t. Similarly, the conditional variance [see ˜ ˜ replacing M by ˜m [cf. Eq. (E227)] and R by R in Eq. (E358)] is approximated as [cf. Eq. (54)] Eq. (E234), so that 2 2 2 2 hK (t)il −hK(t)il hKcl(t)il − hKcl(t)il h∆cl(t)il m m = + 2 X ˜2 X t t t δss − (pss)l δl − 2 ( ˜mR˜mpss)l (E342) m tW tW l=1 l=1 1 X (e )lkp˜k (e )lnp˜n + m t [p(t)]l [p(t)]l X k,n=1 = 2 ˜mR˜ M˜ ˜p − ˜mR˜mp ss ss l (k) (n) ˜ ˜ l=1 ×[hKcl (t)il − hKcl (t)il](Kk − Kn) + ..., (E345) ˜ h p . k ˜mk1kRk1 m kMkmax 2Ccl + 4C+ where

kR˜ − Rk1 i 2 +2k ˜mk1 k˜pss − pssk1 + , hKcl(t)il ≡ hKcl(t)il + (E346) kR˜ k1 t t−t R R 1 (t−t1−t2)W in t2W in t1W 2 0dt1 0 dt2 e (J + µ˜ )e (J + µ˜ )e ˜p l where we used Eq. (E249). [p(t)]l is the square of integral of activity in the classical trajec- h. Multimodal distribution of quantum trajectories tories, conditioned on the system found at time t in lth metastable phase [cf. Eq. (E344)], Here, we show that distributions of continuous mea- R t (t−t )W t W 1 ˜ 2 1 surement results are generally multimodal during the 0 dt1 e ∆σ˜ e ˜p l h∆cl(t)il ≡ , (E347) metastable regime. This is a consequence of distinct con- [p(t)]l tinuous measurement statistics in metastable phases of the system. The modes then correspond to measurement with distributions for metastable phases and the probabilities ˜ ˜ are determined by the decomposition of the metastable (∆σ˜2 )kl ≡ −2Tr(PkJ SQJ ρ˜l) (E348) 71 describing the conditional integral of non-Poissonian ﬂuc- regime, which was derived in Sec.E3f, can be understood tuations in metastable phases [cf. Eq. (56)], and as follows. When

t R (t−t1)W in t1W m (k) dt1 e (J + µ˜ )e hK (t)i ≡ 0 lk (E349) X ˜ ˜ min cl l tW p˜l (vl)k|(W˜p)k| + Ccl p˜l /2 + p˜ , (E355) (e )lk l k=1 being the conditional classical average for the system initially in kth metastable phase [cf. Eq. (E344)]. For min where (vl)k ≡ (δkl − p˜ )/(1 + C˜cl/2), k = 1, ..., m, rep- the corrections to Eqs. (E343) and (E345), see the l resents Pl in the metastable phase basis, the conditional derivations below. distribution features the average approximated by the average jump number from the lth metastable phase, µ˜lt, Multimodal distribution during metastable regime. while the fluctuation rate is constant. Therefore, with During the metastable regime, the outcome l is ob- the probability approximated by p˜l, and a long enough served with the probability approximated by p˜l up to metastable regime, t can be chosen so that the activity corrections Pm (v ) |(W˜p˜ ) | + C˜ |p˜ |/2 − p˜min . k=1 l k k cl l l . k(t) ≡ K(t)/t typically takes values close to µ˜l (as the ˜ min CMM + Ccl|p˜l|/2 − p˜l . The rate of the conditional aver- fluctuations decay inversely in t), and thus can be inter- age in Eq. (E343) is approximated by [cf. Eq. (E309)] preted as a single mode in the jump number distribution. When CMM + C˜cl 1, there exists at least one such ˜ ˜ hK(t)il [(W + µ˜)˜p]l Kl probability p˜l. Therefore, up to corrections of the order = + + ... (E350) ˜ t p˜l t 2(CMM + Ccl) + Ccl from replacing by 0 all p˜l that do not [(J + µ˜in)˜p] K˜ fulfill Eq. (E355) [cf. Eq. (C42)], we can understand the = l + l + .... probability distribution of jump number as multimodal, p˜l t with the overall fluctuation rate, Eq. (E312), featuring If time t ≤ t0 can be chosen so that the contribution linear in time contribution due to the different averages from before metastable regime can be further neglected, of the modes. Note that corrections to multimodal dis- the fluctuation rate is approximated by [cf. Eq. (E312)] tribution are always present, even for ideal classical MMs (Ccl, C˜cl = 0), since the long-time dynamics connects the hK2(t)i −hK(t)i2 metastable phases, and we measure pl(t) rather than pl l l (E351) t (CMM > 0). p˜ [(W˜ + µ˜)2˜p] − [(W˜ + µ˜)˜p]2 In Secs.VB4 andVB5 of the main text, we discuss = t l l l p˜2 how measurement of activity of system can be used to l identify the metastable phases a stochastic system state ˜ ˜ n [(W + µ˜+∆σ˜2 )˜p]l X p˜k corresponds to, provided that metastable phases differ + + 2 (W˜ )lk(K˜k − K˜l)+... p˜l p˜l in their internal activity which dominates transition k=1 rates of the long-time dynamics. We note that results in 2 in 2 p˜l[(J + µ˜ ) ˜p]l − [(J + µ˜ )˜p]l analogous to the presented above and in Secs. VB4 = t 2 p˜l andVB5 of the main text hold for integrated homodyne in ˜ n current (cf. Secs. E3b and E3f). Moreover, for the [(J + µ˜ +∆σ˜2 )˜p]l X p˜k + + 2 (J)lk(K˜k − K˜l)+... constant rates of the conditional average and fluctuation p˜l p˜l k=1 rate, we do no longer require the condition in Eq. (E352). For corrections, see the derivation below. When internal activity dominates transition rates of Cumulants of conditional finite-time statistics. Be- the long-time dynamics fore deriving corrections to Eqs. (E343), (E345), (E350), (E351), (E353), and (E353), we introduce cumulant gen- kW˜ k kµ˜k, (E352) erating functions for conditional statistics. Let {Pl}l denote a POVM, that is, Pl is Hermi- P 1 we obtain from Eqs. (E350) and (E351) constant rates tian, Pl ≥ 0 and l Pl = so that pl = Tr(Plρ) is of the conditional average and fluctuation rate (see the the probability of obtaining an outcome l in the corre- derivation below) sponding generalized measurement on the state ρ. The statistics of jump number conditioned on outcome l at hK(t)i l = µ˜in + ..., (E353) time t is encoded by the cumulant generating function t l [cf. Eq. (E286)] 2 2 ˜ hK(t) il −hK(t)il in (∆σ˜2 ˜p)l = µ˜l + + .... (E354) t p˜ tLs l Θl(s, t) ≡ ln[Tr(Ple ρ)], (E356) The result in Eqs. (E353) and (E354) shows that the statistics of jump number K(t) with t in the metastable so that the conditional average of the jump number and 72 its square [cf. Eqs. (E287) and (E287)] to the metastable regime). Furthermore, [cf. Eq. (E312)]

t R t (t−t )L t L Z ˜ ˜ dt Tr[P e 1 J e 1 ρ] T (t−t1)W˜ t1W 0 1 l dt1vl e Je ˜p (E364) hK(t)il = tL , (E357) 0 Tr(Ple ρ) t t Z R (t−t1)L t1L T (t−t1)W in t1W 2 0 dt1Tr[Ple J e ρ] − dt1vl e (J + µ˜ )e ˜p hK (t)il = tL (E358) 0 Tr(Ple ρ) p 2 t t−t C kW˜ k 2t kJ˜k + t kv k R R 1 (t−t1−t2)L t2L t1L . cl 1 1 l max dt1 dt2Tr[Ple J e J e ρ] +2 0 0 . i tL X † p Tr(Ple ρ) +tm H + Jj Jj 2Ccl + 4C+kvlkmax, 2 max j We are now ready to approximate the conditional tL T tW˜ ˜ statistics in terms of classical dynamics. while Tr(Ple JSQρ) = vl e K˜p [cf. Eq. (E274)] and

T tW˜ ˜ T tW ˜ p ˜ ˜ Derivation of Eq. (E343). For a POVM operator Pl vl e K˜p − vl e K˜p . CcltkWk1kK˜pkkvlkmax. † being a combination of low-lying modes, P (Pl) = Pl, we (E365) have that the probability of observing outcome l Choosing Pl as POVM in Eq. (D66), we arrive at [cf. Eq. (E360)] m tL T tW X tW˜ tW t Tr(Ple ρ)−v e ˜p ≤ (vl)k e −e ˜p ,(E359) Z l k T (t−t1)W in t1W k=1 dt1vl e (J + µ˜ )e ˜p − [p(t)]lhKcl(t)il 0 C˜ where (vl)k = Tr(Plρ˜k). In particular, choosing Pl as cl min [p(t)]lhKcl(t)il + p˜ hKcl(t)i , (E366) POVM in Eq. (D66), we have that it corresponds to a . 2 l classical measurement, 0 ≤ vl ≤ 1, and further where hKcl(t)il is given in Eq. (E344), while C˜ T tW ˜ tW ˜ T tW cl min vl e K − (e K˜p)l (E367) vl e ˜p − [p(t)]l . [p(t)]l − p˜l , (E360) 2 ˜ Ccl tW ˜ min ˜ (e K˜p)l + p˜ K . tW . l where [p(t)]l ≡ (e ˜p)l. Therefore, the corrections 2 [cf. Eq. (39)] The relative corrections to Eq. (E343), that is, the corrections in the leading order are bounded by the tL Tr(Ple ρ) − [p(t)]l (E361) sum of corrections in Eqs. (E363)-(E367) divided by t ˜ Tr(P etLρ)hK(t)i = R dt Tr[P e(t−t1)LJ et1Lρ], plus p ˜ Ccl min l l 0 1 l 2 CcltkWk1kvlkmax + [p(t)]l − p˜ , . 2 l the correction to the probability given in Eq. (E361) tL divided by Tr(Ple ρ). Therefore, if the corrections in where we introduced the max norm kvlkmax ≡ Eqs. (E361–E367) are negligible we obtain the classical max1≤k≤m |(vl)k| ≤ 1, as well as [cf. Eq. (23)] approximation of Eq. (E343).

m Derivation of Eq. (E345). For the square of jump num- X tL p ˜ ˜ Tr(Ple ρ) − [p(t)]l . 2 CcltkWk1 + Ccl. (E362) ber up to time t in Eq. (E358) conditioned on obtaining l=1 lth outcome of a POVM being a linear combination of low-lying modes, at time t ≥ t00, we show below that For the average jump number up to time t in Z t Eq. (E357) conditioned on obtaining lth outcome of a (t−t1)L t1L dt1Tr[Ple J e ρ] POVM being a linear combination of low-lying modes at 0 00 time t ≥ t we have [cf. Eq. (E298)] Z t Z t−t1 (t−t1−t2)L t2L t1L +2 dt1 dt2Tr[Ple J e J e ρ] Z t 0 0 (t−t1)L t1L t t−t1 dt1Tr[Ple J e ρ] Z Z ˜ ˜ ˜ T (t−t1−t2)W˜ t2W˜ t1W 0 −2 dt1 dt2vl e Je Je ˜p t 0 0 Z ˜ ˜ T (t−t1)W˜ t1W tL Z t − dt1vl e Je ˜p + Tr(Ple JSQρ) ˜ ˜ T (t−t1)W˜ t1W ˜ 0 −2 dt1vl e Je K˜p tL 2 tL 0 ≈ Tr(Ple LJ S Qρ) − Tr(PlJSe Qρ) t Z ˜ ˜ n T (t−t1)W ˜ 2 t1W . 2(CMM + CMM)kJ kkSQkkPlkmax,, (E363) − dt1vl e Σ e ˜p 0 where we used Eq. (E292) and Eq. (E293) together . 2(tCMM + kSQk)(1 + kQρk)kSQkkJ PkkJ kkPlkmax with (E294)(n ≥ 1 is an integer such that t/n belongs (1 + 2kSQkkJ k)kSQkkJ kkQρkkPlkmax, (E368) 73

˜ 2 ˜ ˜ where Σ ≡ J + ∆σ˜2 [cf. Eqs. (E273) and (E348)]. In- Similarly, deed, together with Eq. (E363), we have [cf. Eq. (E298)] t Z ˜ ˜ T (t−t1)W˜ t1W ˜ 2 dt1v e Je K˜p (E370) l Z t Z t1 0 (t−t1)L (t1−t2)L t1L t dt1 dt2Tr[Ple JPe JQe ρ] Z T (t−t1)W in t1W ˜ 0 0 −2 dt1vl e (J + µ˜ )e K˜p Z t 0 + dt Tr[P e(t−t1)LJPet1LJSQρ] p 2 1 l 2 CclkW˜ k1 2t kJ˜k1 + t kK˜ ˜pk1kvlkmax 0 . i i Z t X † p ˜ (t−t1)L t1L 2 +2tm H + Jj Jj 2Ccl + 4C+ kK˜pk1kvlkmax. ≈ dt1Tr[Ple JPe LJ S Qρ] 2 max 0 j Z t (t−t1)L t1L and − dt1Tr[Ple JPJSe Qρ] 0 t Z ˜ ˜ dt vTe(t−t1)WΣ˜ 2et1W ˜p (E371) . (tCMM + kSQk)kSQkkJ PkkJ kkQρkkPlkmax, 1 l 0 Z t T (t−t1)W in ˜ t1W where we used Eq. (E292) with respect to time kSQk and − dt1vl e J + µ˜ + ∆σ˜2 e ˜p ˜ ˜ 0 −Tr[P e(t−t1)LJPet1LJSQρ] = vTe(t−t1)WJ˜et1WK˜ ˜p l l p ˜ 2 ˜ 2 [see Eq. (E274)], and, similarly, . CclkWk1 2t kΣ k1 + t kvlkmax i X † p +tm H + Jj Jj 2Ccl + 4C+kvlkmax. Z t Z t−t1 2 max (t−t1−t2)L t2L t1L j dt1 dt2Tr[Ple PJQe JPe ρ]

0 0 Choosing P as POVM in Eq. (D66), we arrive at Z t l (t−t1)L t1L [cf. Eq. (E366)] + dt1Tr[Ple PJ SQJ Pe ρ] 0 Z t Z t−t1 Z t 2 dt dt vTe(t−t1−t2)W(J + µ˜in)et2W(J + µ˜in)et1W ˜p (t−t1)L 2 t1L 1 2 l ≈ dt1Tr[Ple LPJ S QJ Pe ρ] 0 0 0 t 2 Z −[p(t)]l hK (t)il − hKcl(t)il (t−t1)L t1L cl − dt1Tr[PlJSe QJ Pe ρ] 0 ˜ Ccl 2 (tCMM + kSQk)kSQkkJ kkJ PkkPlkmax, [p(t)]l hK (t)il − hKcl(t)il . . 2 cl min 2 + p˜ hK (t)i − hKcl(t)i , (E372) (t−t1)L t1L l cl with −2Tr[Ple PJ SQJ Pe ρ] = ˜ ˜ T (t−t1)W ˜ t1W 2 vl e ∆σ˜2 e ˜p [see Eq. (E348)], while where hKcl(t)il is given in Eq. (E346). Similarly, [cf. Eqs. (E298) and (E300)] Z t T (t−t1)W in t1W ˜ dt1v e (J + µ˜ )e K˜p l Z t Z t−t1 0 (t−t1−t2)L t2L t1L m dt1 dt2Tr[Ple JQe JQe ρ] X (k) tW ˜ 0 0 − hK (t)il(e )lkKkp˜k cl Z t k=1 (t−t1)L t1L ≈ dt1Tr[Ple J SQJ Qe ρ] ˜ m 0 Ccl X (k) tW ˜ hK (t)il(e )lkKkp˜k tL . 2 cl ≈ Tr(Ple J SQJ SQρ) k=1 kSQk2kJ k2kQρkkP k . m . l max min X (k) ˜ + p˜ hK (t)iKkp˜k , (E373) l cl k=1 Furthermore, we observe that [cf. Eq. (E364)] (k) where hKcl (t)il is given in Eq. (E349), as well as Z t Z t−t1 t T (t−t −t )W˜ t W˜ t W˜ Z 2 dt dt v e 1 2 J˜e 2 J˜e 1 ˜p (E369) T (t−t1)W in ˜ t1W 1 2 l dt1vl e J + µ˜ + ∆σ˜2 e ˜p 0 0 0 Z t Z t−t1 T (t−t1−t2)W in t2W in t1W −2 dt1 dt2vl e (J + µ˜ )e (J + µ˜ )e ˜p −[p(t)]l hKcl(t)il + h∆cl(t)il 0 0 p 4 3 2 C˜ 2 CclkW˜ k1kJ˜k1 t kJ˜k1 + t kvlkmax cl . [p(t)]l hKcl(t)il + h∆cl(t)il 3 . 2 i i 2 X † p ˜ +2t m H + Jj Jj 2Ccl + 4C+ kJk1kvlkmax. min 2 max + p˜l hKcl(t)i + h∆cl(t)i , (E374) j 74

Pm where h∆cl(t)il is deﬁned in Eq. (E347). Eqs. (E368)- where µ˜ = l=1 µ˜lp˜l. Noting that [cf. Eq. (E242)] (E374) ˜ ˜ Eqs. (E368)-(E374) describe all corrections to t(J˜p)l − t[(W + µ˜)˜p]l (E381) tL 2 Tr(Ple ρ)hK (t)il. Therefore, we conclude that the i relative corrections to Eq. (E345) are given by the sum of X † p . tm H + Jj Jj 2Ccl + 4C+ 2 max those corrections plus the corrections to the conditional j average in Eq. (E343) multiplied by 2Tr(P etLρ)hK(t)i , l l we arrive at the ﬁrst line of Eq. (E350), and further that all divided by Tr(P etLρ)[hK2(t)i − hK(t)i2], plus the √ l l l ˜ in ˜ correction to the probability in Eq. (E361) divided by kW + µ˜ − (J + µ˜ )k1 ≤ CclkWk1 [cf. Eq. (37)], at the tL second line of Eq. (E350). We also note that the contri- Tr(Ple ρ). When those corrections are negligible, we arrive the classical approximation in Eq. (E345). bution in Eq. (E350) from before the metastable regime can be neglected if t ≤ t0 is long enough, as the resulting ˜ Derivation of Eqs. (E350) and (E351). We now ap- relative correction is bounded by Tr(PlJSQρ)/[˜µlt+(1+ ˜ proximate the conditional statistics in terms of classical Ccl/2)CMM][cf. Eqs. (D88) and (E292)]. dynamics during the metastable regime. Finally, from (E368) for time within the metastable From Eq. (E359) for times within metastable regime, regime we have [cf. Eq. (D79)] t00 ≤ t ≤ t0, we have by the Taylor series expansion Z t (t−t1)L t1L [cf. Eqs. (D88) and (E292)] dt1Tr[Ple J e ρ] (E382) 0 m Z t Z t−t1 X (t−t1−t2)L t2L t1L Tr(P etLρ) − vT˜p t (v ) W˜p˜ , (E375) +2 dt1 dt2Tr[Ple J e J e ρ] l l . l k k 0 0 k=1 2 T˜2 T˜ ˜ T ˜ 2 h C˜ i −t vl J ˜p − 2tvl JK˜p − tvl Σ ˜p C min 1 + cl kv k , kP k , . MM 2 l max l max . 2(tCMM + kSQk)(1 + kQρk)kSQkkJ PkkJ kkPlkmax 2 2 and, choosing Pl as POVM in Eq. (D66) [cf. Eq. (E360)], +2kSQk kJ k kQρkkPlkmax ˜ ˜ 2 h 2 Ccl ˜ 2 i T Ccl min +t CMM min kJ Pk kPlkmax, 1 + kJk1kvlkmax vl ˜p − p˜l . p˜l − p˜l , (E376) 2 2 h +2tCMM min kJ PkkJ k|SQkkQρkkPlkmax, so that C˜cl i tL 1 + kJ˜k kK˜ ˜pk kv k Tr(Ple ρ) − p˜l (E377) 2 1 1 l max h ˜ i ˜ h Ccl Ccl min +tCMM min 2kJ PkkJ k|SQkkPlkmax, CMM min 1 + kvlkmax, kPlkmax + p˜l − p˜ , . 2 2 l C˜ i 1 + cl kΣ˜ 2k kv k . and 2 1 l max m For Pl in Eq. (D66), we further have X tL ˜ Tr(Ple ρ) − p˜l . CMM + Ccl. (E378) 2 T˜2 T˜ ˜ T ˜ 2 l=1 t v J ˜p + 2tv JK˜p + tv Σ ˜p (E383) l l l

Similarly, from Eq. (E363) for time within the 2 ˜2 ˜ ˜ ˜ 2 −t (J ˜p)l − 2t(JK˜p)l − t(Σ ˜p)l metastable regime, t00 ≤ t ≤ t0, we have [cf. Eq. (E331) and (E332)] ˜ Ccl ˜2 ˜ ˜ ˜ 2 . t t(J ˜p)l + 2(JK˜p)l + (Σ ˜p)l Z t 2 (t−t1)L t1L T˜ m dt1Tr[Ple J e ρ] − tvl J˜p + Tr(PlJSQρ) min X ˜ ˜ 0 + p˜ t tµ˜l(J)lk +µ ˜kKk +σ ˜k p˜k . l . 5CMMkJ kkSQkkPlkmax k=1 h C˜ i Note that [cf. Eq. (E242)] +tC min kJ PkkP k , 1+ cl kJ˜k kv k .(E379) MM l max 2 1 l max ˜2 ˜ ˜ ˜ 2 ˜ 2 t t(J ˜p)l + 2(JK˜p)l + (Σ ˜p)l − t[(W + µ˜) ˜p]l (E384) For P in Eq. (D66), we further have [cf. Eqs. (E366) l and (E367)] ˜ ˜ ˜ ˜ −2[(W + µ˜)K˜p]l − [(W + µ˜ + ∆σ˜2 )˜p]l

T˜ ˜ ˜ i tvl J˜p − Tr(PlJSQρ) − t(J˜p)l − Klp˜l (E380) X † p . tm H + Jj Jj 2Ccl + 4C+ 2 max ˜ j Ccl ˜ ˜ min ˜ t(J˜p)l + Klp˜l + p˜ tµ˜ + K , ˜ ˜ . 2 l ×(2tkJk1 + kK˜pk1 + 1). 75

˜ in Replacing W + µ˜ by√J + µ˜ further introduces cor- of metastabilities arising in the proximity to a dissipative rections bounded by CclkW˜ k1(2tkJ˜k1 + kK˜ ˜pk1 + 1) phase transition at a finite-size, see Sec.A2. [cf. Eq. (37)] and we obtain the classical approximation of the conditional square of jump number. Using then already derived Eq. (E350) we arrive at the approxima- 1. Hierarchy of metastabilities tion of the variance, and thus also the fluctuation rate as given in Eq. (E351). Apart from the metastable regime t00 ≤ t ≤ t0 leading R R to the spectrum separation with λm/λm+1 1 as con- Derivation of Eqs. (E353) and (E354). When the sidered in Sec.II of the main text, we assume here that contribution from before the metastable regime in 00 0 there occurs a later metastable regime t2 ≤ t ≤ t2 that Eq. (E350) is negligible, the relative corrections from ne- corresponds to λR /λR 1 where m < m (see also m2 m2+1 2 ˜ 00 glecting the long-time dynamics are bounded by kWk/µ˜l. Ref. [S21]). For times t ≥ t2 , the system state can be Similarly, in the first line of Eq. (E351) we have approximated as

˜ 2 ˜ 2 m2 p˜l[(W + µ˜) ˜p]l − t[(W + µ˜)˜p]l X ρ(t) = eλmtc R + ... (F385) = p [(W˜˜ µ − µ˜W˜ )˜p] + tp˜ (W˜ 2˜p) − t(W˜p˜ )2, k k l l l l l k=1 and thus neglecting the long-time dynamics removes any [cf. Eq. (3) and note that t0 t00 as t0 −1/λR and time-dependence from the fluctuation rate, with the ne- 2 m t00 ≥ −1/λR ; see Sec.B3]. Furthermore, during the glected terms bounded in the leading order by 2(1 + 2 m2+1 second metastable regime t00 ≤ t ≤ t0 , the decay of m ˜ 2 2 2 Ccl/2)CMMkµ˜k1/p˜l [cf. Eqs. (D88) and (E292)]. Further- slower modes is negligible and the system states appear more, the constant fluctuation rate can be further simpli- stationary (cf. Eq. (4) and Ref. [S21]) fied by neglecting the long-time dynamics in the second term, which leads to corrections bounded by kW˜ k /p˜ , m2 1 l X and in the third term, with corrections bounded by ρ(t) = ck Rk + ... = P2[ρ(0)] + ..., (F386) kW˜ k1kK˜ ˜pk1/p˜l. Therefore, we obtain the approxima- k=1 tion in Eq. (E354) whenever the relative corrections 2 with P2 denoting the projection on the second MM. For [2(1 + C˜cl/2)CMMkµ˜k1 + kW˜ k1(1 + kK˜ ˜pk1)]/(Σ ˜p)l are small. the corrections to the stationarity in Eq. (F386) denoted as C2,MM [cf. Eq. (10)] and the corrections to the positiv- (2) ity of projection P2 as C2,+ [cf. Eq. (12)], the errors in n F. CLASSICAL HIERARCHY OF Eq. (F385) are bounded by 2C2,MM, where n is such that 00 0 METASTABILITIES t2 ≤ t/n ≤ t2 (cf. Sec.IIC in the main text).

Here, we discuss the existence of the second metastable regime in the long-time dynamics of the system, which 2. Hierarchy of classical metastable manifolds corresponds to a further separation in the real part of the m low-lying eigenvalues of the master operator [S21]. We We now show that when the first metastability is prove that for the classical metastability of m low-lying classical (see Sec.III in the main text), it follows that the modes, the second metastability is classical as well. This second metastability is classical as well, i.e., metastable is a direct consequence of long-time dynamics in a classi- states during the second metastable regime are mixtures cal MM being well approximated by classical stochastic of m2 metastable phases. For a related discussion, see dynamics (cf. Sec.V of the main text), as metastable Ref. [S21]. states of classical stochastic dynamics are known to be mixtures of as many metastable phases as the number As discussed in Sec.E2, the dynamics etW˜ in the first of low-lying modes [S7, S22]. As a result, we show that MM can be approximated at any time t by the discrete a hierarchy of metastable phases arises: m2 metastable positive and trace-preserving dynamics Tt, with the cor- phases of the second MM are approximately disjoint mix- rections bounded by Ccl [see Eqs. (E194) and (E195)]. 00 0 tures of m metastable phases of the first MM, while any For t ≥ t2 and an integer n such that nt ≤ t2, that is, metastable phase of the first MM that during the sec- both t and nt belonging to the second metastable regime, ond metastable regime evolves into a mixture, rather we have than a single phase, necessarily belongs to the decay sub- ˜ n ˜ ntW˜ ntW˜ n space. Furthermore, we show how the approximation of kP2 − Tt k1 ≤ kP2 − e k1 + ke − Tt k1 (F387) ˜ the long-time dynamics by classical stochastic dynam- . C2,MM + nCcl, ics can be refined to take into account the hierarchy of metastabilities and hold well for times up to and beyond where P˜ 2 denotes the projection P2 on the second MM the relaxation time, e.g., to obtain a classical approxima- in the basis of the metastable phases of the first MM and tion of the stationary state. For a discussion of hierarchy C˜2,MM are corrections to the stationarity of barycentric 76 coordinates during the second metastable regime in L1 k2, l2 = 1, ..., m2. This transformation is close to identity norm. The first inequality follows from the triangle in- m X2 equality, the second from Eqs. (D79) and (E195). We also kC0 − Ik ≡ max |v0T(P˜ − P )p | (F393) 1 k2 2 t l2 ˜ ˜ 1≤l2≤m2 have C2,MM (1+Ccl/2)C2,MM (see Sec.D2), so that for k =1 ˜ ˜ 2 Ccl 1 we can further replace C2,MM by C2,MM [other- ˜0 m2 00 0 C X wise, if needed, a shorter metastable regime t˜ ≤ t ≤ t˜ = 1 + cl max |vT (P˜ − P )p | 2 2 k2 2 t l2 2 1≤l2≤m2 could be considered to decrease the corrections with re- k2=1 spect to L1 norm]. C˜0 From Eq. (F387) we obtain that the discrete classical ≤ 1 + cl kP˜ − P k 2 2 t 1 evolution with the transfer matrix Tt can be approxi- 0 C˜0 mated by the same operator for all n ≤ t2/t such that cl ˜ 0 ≤ 1 + C2,MM + Ccl + CMM , n 1/Ccl, in which case, the corresponding probability 2 distributions are approximately stationary. If this holds where we introduced v ≡ [v0 − min (v0 ) i]/[1 + 00 0 00 k2 k 1≤l≤m k2 l for n = 2 (e.g., for t = t2 when t2 ≥ 2t2 ), it follows that 0 C˜ /2], where (i)l = 1, l = 1, ...., m [cf. Eqs. (26) there exist a separation in the spectrum of Tt (see the cl end of this section) and the probability distributions can and (C48)]. Therefore, be approximated as mixtures of m approximately dis- 0−1 0 2 k˜p2k − 1 ≤ C k˜p k − 1 (F394) joint probability distributions (from Ref. [S7]; note that 1 1 1 (1 + kC0 − Ik ) k˜p0k − 1 there are additional assumptions on Tt). Let Pt be a . 1 1 0 0 projection on the MM of Tt and C be the correspond- C˜ MM ≤ 1 + cl C˜ + 2C + C0 + C0 , ing corrections to the stationarity (in the classical space 2 2,MM cl MM cl of m configurations corresponding to metastable phases) 0 0 ˜0 [cf. Eq. (10)]. From Eq. (F386) we then have where we used k˜p k1 . 1 + Ccl + (1 + Ccl/2)Ccl [it is 1+C0 for barycentric coordinates of projected probability ˜ ˜ cl kP2 − Ptk1 ≤ kP2 − Ttk1 + kTt − Ptk1 (F388) distributions over m metastable phases, cf. Eq. (F389), ˜ 0 . C2,MM + Ccl + CMM. but we project ˜p with k˜pk1 ≤ 1 + Ccl; cf. Eq. (F393)]. Thus, we arrive at We now consider m2 candidate metastable phases 0 for the second MM. Let p1, ... pm2 be the classical C˜ C 1 + cl C˜ + 2C + C0 + C0 (F395) metastable phases for Tt [cf. Eq. (14)], which under 2,cl . 2 2,MM cl MM cl the projection Pt on the slow modes of Tt we denote C + 2C + C0 + C0 , p0 ≡ P p , l = 1, ..., m [cf. Eq. (17)]. Let v0 be the . 2,MM cl MM cl l2 t l2 2 2 k2 dual basis to p0 , i.e., v0Tp0 = δ , k , l = 1, ..., m . ˜ ˜0 l2 k2 l2 k2l2 2 2 2 where the last inequality holds for Ccl 1 and Ccl 1; We introduce the corresponding corrections to the clas- cf. Eqs. (23) and (F390). We conclude that when the sicality [cf. Eqs. (20) and (21)] first MM of an open quantum system is classical, the

m2 second metastable manifold is classical as well. 0 X 0 Ccl ≡ 2 max max −(vk )l, 0 , (F389) 1≤l≤m 2 k =1 Finally, we prove that a separation in the spectrum 2 2 of Tt follows from kTt − Tt k1 1 [which occurs, and its upper bound [cf. Eqs. (23) and (24)] 00 0 e.g., when t can be chosen so that t2 ≤ t ≤ 2t ≤ t2; m 0 2 cf. Eq. (F387)]. First, for lk, being a left eigenvector of 0 0 X 0 0 0 C ≤ C˜ ≡ 2 − min (v ) ≤ m C . (F390) tλk cl cl k2 l 2 cl Tt corresponding to an eigenvalue e , k = 1, ..., m, we 1≤l≤m 0 0 k =1 0T 2 tλk tλk 0T 0T 2 have lk (Tt − Tt )p = e (1 − e )lk p, while |lk (Tt − 2 0 2 We choose m candidate metastable phases for the sec- Tt )p| ≤ lkkmaxkkTt − Tt k1. Therefore, choosing (p)k = 2 0 0 ond MM as the projections of metastable phases in clas- δkl, where l is such that |(lk)l| = klkkmax, we obtain 0R 0R 0 0 tλk tλk tλk tλk 2 sical dynamics Tt [cf. Eq. (17) and see Sec.C2] e (1 − e ) ≤ |e (1 − e )| ≤ kTt − Tt k1. Second, 2 tλ0R 2 m m when kTt −Tt k1 1, it follows that e k . kTt −Tt k1 h X i X tλ0R 2 ˜ or 1−e k kT −T k , as the function x(1−x) is close ρ˜2,l2 ≡ P2 (pl2 )k ρk = P2pl2 k ρ˜k, (F391) . t t 1 k=1 k=1 to 0 only at x = 0, 1. For eigenvalues ordered with a de- tλ0R 0 creasing real part, we thus obtain e k 1 for k > m l2 = 1, ...m2. 2 tλ0R 0 We now estimate the corresponding corrections to the and 1−e k 1 for k ≤ m2. Finally, for Tt approximat- tW˜ 00 00 0 classicality. Let ˜p2 denote the barycentric coordinates ing e where t2 ≤ t ≤ t1 , we have that m2 = m2, as of P2[ρ(0)] in Eq. (F386) in the basis of Eq. (F391) tW˜ e −Tt can be considered as a perturbation of Tt, lead- [cf. Eq. (20)]. We can relate ˜p2 to the barycentric co- ing to the ﬁrst-order correction in the eigenvalue given by 0 0 ordinates ˜p in the basis of metastable phases p of Tt, 0T tW˜ 0 0 l2 l (e −T )r , where r is a right eigenvector of T and 0−1 0 k t k k t as ˜p2 = C ˜p , where 0T tW˜ 0 0 tW˜ 0 |lk (e − Tt)rk| ≤ klkkmaxkTt − e k1krkk1. Indeed, (C0) ≡ v0TP˜ p , (F392) from Eq. (F388) mixtures of approximately disjoint m0 kl k2 2 l2 2 77

(l) Pm (l2) Pm (l2) (l) probability distributions are described by m2 coeﬃcients, max1≤l≤m |p˜2,l | l=1 |p˜l | ≥ l=1 |p˜l ||p˜2,l | we have 0 0 2 2 which requires m2 ≥ m2, but m2 > m2 would correspond to degeneracy of degrees of freedom, we assume does not 1 max p˜(l) ≥ 1 − C , (F399) 2,l2 & cl occur (cf. Sec.C1). 1≤l≤m 1 + Ccl

where we removed absolute value since Ccl, C2,cl 1 3. Hierarchy of classical metastable phases [cf. Eq. (D63)].

Second, to discuss supports and basins of attraction, From Sec.IV of the main text, it follows that supports for each k = 1, ..., m , we consider a subset S consist- and basins of attractions of metastable phases in the sec- 2 2 k2 ing of the labels l for which p˜(l) ≥ ∆, where C /2 ≤ ond MM are approximately disjoint. Here, we discuss 2,k2 2,cl (l) how m2 metastable phases of the second MM originate ∆ ≤ 1. Noting that −C /2 ≤ p˜ ≤ 1 + C /2, we 2,cl 2,k2 2,cl from m metastable phases of the first MM (cf. Ref. [S21]). have We show that for each metastable phase in the second m X (l ) (l) X (l ) MM, there exist at least one metastable phases of the first |p˜ 2 ||p˜ | ≤ ∆ |p˜ 2 | (F400) MM that during the second metastable regime evolves l 2,k2 l l=1 l∈ /S directly into it. Furthermore, metastable phase in the k2 (l) X (l2) second MM are approximately disjoint mixtures of m + max p˜2,k |p˜l |, 1≤l≤m 2 metastable phases of the first MM. Finally, we argue that l∈Sk2 any metastable phase in the first MM during the second P (l2) P (l2) metastable regime that evolves into a nontrivial mixture Using |p˜ | ≤ 1 + Ccl − |p˜ | and l∈ /Sk2 l l∈Sk2 l of metastable phases in the second MM, the second MM (l) max1≤l≤m p˜ ≤ 1 + C2,cl/2, we then obtain is not supported on that phase, i.e., the phase belongs to 2,k2 the decay subspace. X (l ) 1 − ∆(1 + Ccl) |p˜ 2 | ≥ l 1 + C2,cl − ∆ l∈Sl2 2 a. Supports and basins of attraction 2C + C 1 − cl 2,cl , (F401) & 2(1 − ∆) Let us consider an initial system state that evolves into where in the second inequality we assumed 1−∆ C2,cl. l2th metastable phase ρ˜2,l2 of the second MM, and let m (l2) (l) m (l2) (l2) P P Furthermore, l=1 |p˜l |p˜2,k = δk2,l2 − l=1[˜pl − p˜l , l = 1, ..., m, denote their barycentric coordinates 2 with respect to m metastable phases of the first MM. |p˜(l2)|]˜p(l) . As −C /2 ≤ p˜(l) ≤ 1 + C /2 from l 2,k2 2,cl 2,k2 2,cl (l) m (l ) (l ) (l) Furthermore, let p˜ , l2 = 1, ..., m, denote the barycen- P 2 2 2,l2 Eq. (21), we have | [˜p − |p˜ |]˜p | ≤ Ccl(1 + tric coordinates of m metastable phases of the first MM l=1 l l 2,k2 C2,cl/2). Therefore in the basis of m2 metastable phases of the second MM. We have m X (l ) (l) CclC2,cl |p˜ 2 |p˜ ≤ C + , k 6= l . (F402) l 2,k2 cl 2 2 m m2 2 X (l ) X (l) l=1 ρ˜ = p˜ 2 p˜ ρ˜ , (F396) 2,l2 l 2,k2 2,k2 l=1 k2=1 We also have m X (l ) (l) X (l ) and thus |p˜ 2 |p˜ ≥ ∆ |p˜ 2 | (F403) l 2,k2 l

m l=1 l∈Sk2 X (l2) (l) δ = p˜ p˜ . (F397) h (l) i X (l ) k2,l2 l 2,k2 2 + min min 0, p˜2,k |p˜l |, l=1 1≤l≤m 2 l∈ /Sk2

(l) so that from −C2,cl/2 ≤ p˜ ≤ 1 + C2,cl/2 we have We ﬁrst show that for each metastable phase in the 2,k2 second MM, there exist at least one metastable phases of X (l2) 2Ccl + C2,cl + CclC2,cl the ﬁrst MM that during the second metastable regime |p˜l | ≤ (F404) 2∆ + C2,cl l∈S evolves directly into it. From Eq. (F397) and the triangle k2 inequality 2Ccl + C2,cl . , k2 6= l2 m 2∆ X (l ) (l) |p˜ 2 ||p˜ | ≥ 1. (F398) l 2,l2 where we assumed ∆ C2,cl/2. l=1 Eqs. (F401) and (F404) are analogous to Eqs. (31) and(32) in the main text. In particular, Eq. (F401) shows Pm (l2) Since l=1 |p˜l | ≤ 1 + Ccl [cf. Eq. (21)] and that Sl2 captures the support of l2 metastable phases in 78

the second metastable manifold as well as basin of at- For each k2 = 1, ...., m2, let us consider a subset Dk2 traction. Furthermore, due to the classicality of the con- (l) consisting of the labels l for which ∆1 ≤ p˜2,k ≤ ∆2, sidered manifold, the subsets S , k = 1, ..., m , can be 2 k2 2 2 where 0 ≤ ∆ ≤ ∆ ≤ 1. Noting that Pm |p˜(l2)| ≤ chosen disjoint (not only approximately disjoint), e.g., 1 2 l=1 l 1 + C for any ∆, and considering ∆ = ∆ in Eq. (F401) Pm2 (l) cl 2 for the choice ∆ > (1 + C2,cl)/2, since |p˜ | ≤ k2=1 2,k2 we obtain P (l2) 1 + C2,cl. In that case, we have |p˜ | = m l∈∪k26=l2 Sk2 l (l ) X (l2) X (l2) X (l2) P P |p˜ 2 | and |p˜l | ≤ |p˜l | − |p˜l | k26=l2 l∈Sk2 l l∈Dl2 l=1 l∈Sl2 X X (l2) X (l2) |p˜l | ≤ 1 + Ccl − |p˜l | (F405) 2Ccl + C2,cl . Ccl + , (F408) k26=l2 l∈Sk2 l∈Sl2 2(1 − ∆2) 2Ccl + C2,cl where we assumed 1−∆ C . Considering ∆ = ∆ . Ccl + 2 2,cl 1 2(1 − ∆) in Eq. (F404), we get from Eq. (F401). X (l2) 2Ccl + C2,cl |p˜l | . , k2 6= l2. (F409) 2∆1 Finally, we show that, when m(Ccl + C2,cl) 1, a l∈Dk2 metastable phase of the first MM on which a metastable phase of the second MM is supported evolves directly Furthermore, considering ∆1 + ∆2 ≥ 1 + C2,cl, we have into it, or equivalently, a metastable phase in the sec- from the classicality of the second MM that l belonging ond MM cannot be supported on phases in the first MM to the decay subspace Dk2 for any k2 = 1, ..., m does that during the second metastable regime evolve into not belong to the support of any metastable phases of (l2) the second MM captured by Sl2 , l2 = 1, , ..., m provided a different phase. From Eq. (F402) we have |p˜l | . (l) that ∆ > ∆2 is considered. Therefore, when we choose (Ccl + C2,cl)/|p˜ |. Thus, a metastable phase in the sec- 2,k2 1−∆ C +C /2 [which also implies ∆ C +C /2 ond MM cannot be supported on phases in the first MM 2 cl 2,cl 1 cl 2,cl in Eq. (F409)], from Eq. (F401) metastable phases ρ˜l of that during the second metastable regime evolve into a m2 the first MM with l ∈ ∪ Dk contribute negligibly to different phase, i.e. ρ˜ with l such that p˜(l) C +C , k2=1 2 l 2,k2 cl 2,cl the support of the metastable phases in the second MM. where k2 6= l2 [for example, cf. Eq. (F399)]. Similarly, Note that those are the metastable phases that evolve Pm (l2) into nontrivial mixtures of m metastable phases, as for from Eq. (F398) by noting that l=1 |p˜l | ≤ 1+Ccl and 2 (l) (l) (l) l ∈ Dl we have ∆1 ≤ p˜ ≤ ∆2, so that p˜ ≥ ∆1 |p˜2,l | ≤ 1 + C2,cl/2, we obtain 2 2,l2 2,l2 2 is nonnegligible and so is the sum of other barycentric (l ) (l) (l ) C2,cl P (l) (l) 2 2 coordinates, as p˜ = 1−p˜ ≥ 1−∆2. p˜ p˜ + 1+Ccl −|p˜ | 1+ ≥1, (F406) k26=l2 2,k2 2,l2 l 2,l2 l 2 so that C 2C + C + C C 4. Hierarchy of classical long-time dynamics (l) 2,cl cl 2,cl cl 2,cl p˜2,l ≥ 1 + − . (F407) 2 2 2|p˜(l2)| l Here, we discuss how the approximation of the long- Thus, for l such that |p˜(l2)| C + C , we have that time dynamics by classical stochastic dynamics can be l cl 2,cl refined to take into account the hierarchy of metastabil- p˜(l) is approximated by 1, that is, lth phase in the 2,l2 ities. We consider both continuous and discrete approxi- first MM evolves directly into l2th phase in the second mations of the long-time dynamics (cf. Sec.E2). MM. We note, however, that for a highly-dimensional first MM, such probabilities do not need to exist, when mCcl or mC2,cl is of order 1 [in the opposite case, a. Hierarchy of continuous approximations of classical (l2) long-time dynamics max1≤l≤m |p˜l | ≥ 1/m Ccl, C2,cl].

In general the norm kW˜ k1 is dominated by the fastest b. Decay subspace transitions between metastable phases, while the final relaxation time τ˜ describes the longest timescale of Below, we prove that any metastable phase in the the dynamics. As a consequence, in the presence of first MM that evolves into a nontrivial mixture of the second metastable regime, the classical dynamics metastable phases in the second MM, can be neglected W defined in Eq. (36) may not approximate the long in the supports of metastable phases in the second MM. time-dynamics up to and beyond the relaxation time. Therefore, the supports of those metastable phases in Furthermore, the conditions in Eqs. (40) and (E186) the first MM belong to the approximate decay subspace are in general no longer fulfilled, and the stationary of the system space (cf. Sec.D3). state and the resolvent of W˜ in Eq. (34) may no longer approximated well by the stationary state and 79 the resolvent of the classical stochastic dynamics W. Approximating the stationary state. When there ex- p tW˜ 2 Nevertheless, we show below how the approximations ists time t such that t kW˜ 2k1 1/ C2,cl and ke − can be modified to take into account the hierarchy of P˜2,ssk1 1 [here P˜2,ss denotes the projection Pss on the metastable regimes in the system dynamics. stationary state ρss in the basis of m2 metastable phases], which requires Approximating the second MM. Instead of considering the approximation of the long-time dynamics by the dis- ˜ ˜ 1 crete dynamics T we can consider a generally weaker τkW2k1 ≤ τ˜2kW2k1 , (F415) t pC approximation by the classical stochastic generator W 2,cl in Eq. (36). This gives instead of Eq. (F387) tW˜ 2 [τ˜2 is the relaxation time with respect to ke −P˜2,ssk1; ˜ tW ˜ tW˜ tW tW˜ kP2 − e k1 ≤ kP2 − e k1 + ke − e k1 (F410) we have τ˜2 ≈ τ when C˜2,cl 1] the stationary state is captured by the stationary distribution p of W as [cf. Eqs. (39) and (41)]. Therefore, this approximation 2,ss 2 holds well if for time t after the relaxation towards the tW˜ p tW˜ k˜p − p k kP˜ − e 2 k + 2 C tkW˜ k second metastable regime , kP˜ 2 − e k1 C˜2,MM, we 2,ss 2,ss 1 . 2,ss 1 2,cl 2 1 ˜ (F416) also have ketW − etWk 1, as guaranteed, e.g., by 1 [cf. Eqs. (41) and (E177)]. In the basis of m metastable phases of the first MM (or in the trace norm), the dis- 00 ˜ 1 t2 kWk1 √ . (F411) tance between the corresponding vectors (or matrices) Ccl is in the leading order bounded by k˜p2,ss − p2,ssk1 and Furthermore, the second metastability corresponds to the thus by Eq. (F416) (cf. Sec.D2). metastability in the classical dynamics generated by W, with P2 approximated by the projection P on the low- Approximating the dynamics resolvent. We now show lying eigenmodes of W since [Eq. (F388)] that the resolvent of the dynamics can be approximated tW tW in two steps: for the faster modes by the resolvent of W kP˜ 2 − Pk1 ≤ kP˜ 2 − e k1 + ke − Pk1 (F412) and for the slower modes by the resolvent of W2; see tW 0 0 Eq. (F424) below and cf. Sec.E2. and ke − Pk1 ≤ CMM with CMM denoting the corrections to the stationarity of etW. We have [cf. Eqs. (E181) and (E182)]

t2 t2 0 Z ˜ Z ˜ Approximating long-time dynamics for t ≥ t2. After ˜ t1W ˜ ˜ t1W ˜ R + dt1 e − Pss P2 + dt1e I − P2 the second metastable regime, the system dynamics in 0 0 1 Eq. (F386) is generated by P LP [cf. Eq. (6)], which 2 2 t2W˜ ≤ ke − P˜ssk1kR˜ k1. (F417) we denote in the basis of m2 metastable phases by W˜ 2 ˜ [cf. Eq. (34)]. W2 is approximated by classical stochastic The second term in Eq. (F417) can be approximated by dynamics W2 [cf. Eqs. (36), (37) and (39)], so that the classical dynamics after the second metastable regime ˜ as [cf. Eqs. (E183), (F413) and (F416)] tW2 tW2 p ˜ ke − e k1 . 2 C2,cl tkW2k1, (F413) t2 t2 Z ˜ Z where C is in the approximation of the second MM by t1W ˜ ˜ t1W2 ˜ 2,cl dt1 e − Pss P2 − dt1 e − P2,ss P2 the simplex of m2 metastable phases and we consider the 0 0 1 t2 t2 L1-norm in the basis of m2 metastable phases. Instead, Z ˜ Z t1W2 ˜ t1W2 in the basis of m metastable phases of the first MM, we . dt1 e − P2,ss − dt1 e − P2,ss 0 0 1 denote the action of W2 by W˜ 2,1 and we have ˜ 3pC t2kW˜ k + t kP˜ − et2W2 k , (F418) ˜ ˜ . 2,cl 2 2 1 2 2,ss 1 tW tW2,1 ˜ ˜ p ˜ k(e − e )P2k1 . 2kP2k1 C2,cl t kW2k1. p ˜ where in the first line we introduced the projection P2,ss . 2 C2,cl t kW2k1. (F414) on the stationary state of W2, and the second and third where in the second inequality we neglected line refer to operators in the basis of m2 metastable kP˜ 2k1 − 1 ≤ C˜2,MM, i.e., the corrections to the phases of the second MM. For the resolvent R2 for the positivity of the projection P2 on the second MM in dynamics W2, we have [cf. Eq. (F417) and Sec.D2] L1 norm. We have that kP˜ 2k1 − 1 ≤ (1 + C˜cl/2)C2,+, ˜ ˜ Z t2 so that for Ccl 1 we can further replace kP2k1 − 1 t1W2 ˜ R2 + dt1 e − P2,ss P2 (F419) by the corrections C2,+ in the trace norm [cf. Eq. (12) 0 1 ∞ and Sec.D2]. Note that W˜ 2,1 in Eq. (F414) is defined Z t1W2 ˜ on the image of P˜ and in general is not a classical = dt1 e − P2,ss P2 2 t stochastic generator, as it is probability conserving but 2 1 t2W2 only approximately positive. . ke − P2,ssk1kR2k1 t2W2 . ke − P2,ssk1kR2k1. 80

The third term in Eq. (F417) can be approximated by [cf. Eq. (F412)]. Therefore, we arrive at [cf. Eq. (E185)] the classical dynamics W as [cf. Eqs. (E183)] kR˜ − R(I − P) − R k 2 1 (F424) t t ˜ Z ˜ Z kRk1 t1W ˜ t1W ˜ dt1e (I − P2) − dt1e (I − P2) ˜ t2 0 0 t2W ˜ p ˜ 1 . ke − Pssk1 + t2 3 + 8 C2,clkW2k1 Z t kR˜ k1 t1W˜ t1W ≤ dt1ke − e k1kI − P˜ 2k1 t2 t W˜ 0 + + 4 kP˜ − e 2 2 k ˜ 2,ss 1 2p ˜ kRk1 . 2t CclkWk1, (F420) t p +2t + 4 CclkW˜ k1 tW˜ tW˜ kR˜ k1 where we used kI−P˜ 2k1 ≤ kIk1+kP˜ 2−e k1+ke k1 ≤ ˜ ˜ ˜ tW ˜ (t2−t)W ˜ 1 + C2,MM + 1 + Ccl . 2 (for t chosen within the second +ke (I − P2)k1[3 + ke (I − P2)k1] metastable regime), while [cf. Eq. (E182)] t + + 2 kP − P˜ k / ˜ 2 1 Z t2 kRk1 t W˜ dt e 1 (I − P˜ ) (F421) h ˜ 1 2 1 − 4kP˜ − et2W2k + 8pC t kW˜ k t 1 2,ss 1 2,cl 2 2 1 t2−t ˜ Z ˜ p tW˜ i tW ˜ t1W ˜ ˜ ˜ ˜ ≤ ke (I − P2)k1 dt1e (I − P2) −8t CclkWk1 − 2ke (I − P2)k1 − 2kP − P2k1 . 0 1 ˜ h ˜ i The above inequality holds for any choice of times t and ≤ ketW(I − P˜ )k 1+ke(t2−t)W(I − P˜ )k kR˜ k . p 2 1 2 1 1 t2 provided that t ≤ t2. When t2 kW˜ 2k1 1/ C2,cl

t2W˜ 2 and ke − P˜2,ssk1 1 [cf. Eq. (F416)], as well as, For the resolvent R of the classical dynamics W, we fur- √ ˜ t kW˜ k 1/ C and ketW − P˜ k 1 [t ≥ t˜00, ther have 1 cl 2 1 2 cf. Eq. (F411)], the leading corrections in the right- Z t hand side of Eq. (F424) are given by the numerator t1W ˜ ˜ R(I − P) + dt1(e − Pss)[(I − P)+(P − P2)] as ketW(I − P˜ )k 2C˜ [S6]. In this case, we 2 1 . 2,MM 0 1 ˜ ˜ tW can obtain kR − R(I − P) − R2k1/kRk1 1 by fur- ≤ ke (I − P)k1kR(I − P)k1 + tkP − P˜ 2k1. (F422) ther choosing t kR˜ k1 [possible as t can be chosen R within the second metastable regime so that t −1/λk From Eqs. (F417)-(F422) we can approximate R ˜ for k ≥ m2 while (1 − C+ − Ccl)/|λk | . kRk1 from Eq. (E193)], and assuming that we can further choose ˜ kR − R(I − P) − R2k1 (F423) t2W˜ 2 t2 so that t2kP˜2,ss − e k1/kR˜ k1 1 [which implies ˜ t2W ˜ ˜ p 2 ˜ 2p ˜ ˜ . ke − Pssk1kRk1 + 3 C2,clt2kW2k1 t2 C2,clkW2k1/kRk1 1]. Note that the last assump-

t2W˜ 2 tW2 2p tion requires [cf. Eq. (E186)] +t2kP˜2,ss − e k1 +ke − P2,ssk1kR2k1 +2t CclkW˜ k1 ˜ ˜ τkW˜ k ≤ τ˜ kW˜ k (F425) +ketW(I − P˜ )k [1 + ke(t2−t)W(I − P˜ )k ]kR˜ k 2 1 2 2 1 2 1 2 1 1 s ! tW 1 kR˜ k kW˜ k +ke (I − P)k1kR(I − P)k1 + tkP − P˜ 2k1. 1 2 1 min p , p . C2,cl C2,cl t2W2 t2W˜ 2 (2) t2W˜ 2 We have ke − P2,ssk1 ≤ ke − P˜ ss k1 + ke − (2) ˜ et2W2k + kP˜ − P k 2kP˜ − et2W2 k + 1 ss 2,ss 1 . 2,ss 1 b. Hierarchy of discrete approximations of classical p ˜ tW 4 C2,cl t2kW2k1 and, similarly, ke (I − P)k1 ≤ long-time dynamics k(etW − etW˜ )(I − P)k + ketW˜ (I − P˜ )k + ketW˜ (P − √ 1 2 1 ˜ ˜ tW˜ ˜ ˜ P2)k1 . 4t CclkWk1 + ke (I − P2)k1 + kP − P2k1 We now consider approximation of the long time- [cf. Eq. (F410)]. Furthermore, R2 = R2P˜ 2 and thus dynamics by the classical discrete dynamics Tt defined in ˜ ˜ ˜ ˜ ˜ ˜ Eqs. (E198) and (E199). The approximation corrections kR2P2k1 ≤ k(R−R2)P2k1+kRP2k1 . 2kR−R(I−P)− ˜ ˜ ˜ ketW˜ − T k for time t chosen before the final relaxation R2k1 +kR(I−P)k1kP2 −Pk1 +2kRk1 [as kP2k1 . 2 and t 1 2 are dominated by the fastest transitions between m R(I−P)P˜ 2 = R(I−P)(P˜ 2 −P) from (I−P) = (I−P)]. ˜ metastable phases. As a consequence, in the presence of Analogously, kR(I − P)k1 . 2kR − R(I − P) − R2k1 + ˜ ˜ the second metastable regime, for Tt with time before kR2k1kP2 − Pk1 + 2kRk1 [as kI − Pk1 ≤ 1 + kPk1 . 2 00 n the relaxation to the second MM, t < t2 , Tt in general and kR2(I−P)k1 = kR2(P˜ 2 −P)k1 ≤ kR2k1kP˜ 2 −Pk1]. 0 may not approximate the long time-dynamics for nt ≥ t2 Combining these two results we have kR(I − P)k1 . [cf. Eqs. (E194) and (E195)]. We now discuss how the ˜ ˜ ˜ 2(1 + kP2 − Pk1)(kR − R(I − P) − R2k1 + kRk1)/(1 − approximations can be modified to include the hierarchy ˜ ˜ ˜ kP2 − Pk1) . 2(kR − R(I − P) − R2k1 + kRk1) and, of timescales. ˜ ˜ analogously, kR2k1 . 2(kR − R(I − P) − R2k1 + kRk1) 81

0 Approximating the second MM. When time t ≤ t2, pos- order bounded by k˜p2,ss −p2,ssk1 and thus by Eq. (F430); sibly shorter than the second metastable regime, fulﬁlls cf. Sec.D2. We note that this requires t such that [see [cf. Eq. (E196)] Eq. (E196)] t C , (F426) t t t00 cl ≥ C2,cl. (F431) 2 τ τ˜2 the long-time dynamics entL is well approximated by Tn t Here, τ˜2 is the ﬁnal relaxation time with respect to after the relaxation into the second metastable regime ˜ ketW2 − P˜ k ; we have τ˜ ≈ τ when C˜ 1. [cf. Eq. (E195)]. It also follows that the projection of the 2,ss 1 2 2,cl second MM is approximated by the projection Pt on the eigenmodes of Tt with absolute value of eigenvalues close to 1 as [cf. Eq. (F388)] G. CLASSICAL WEAK SYMMETRIES

kP˜ − P k ≤ kP˜ − Tnk + kTn − P k (F427) 2 t 1 2 t 1 t t 1 In this section, we discuss the role of weak symme- ˜ 0 . C2,MM + nCcl + CMM 1, tries in classical metastability. We prove the resulting symmetry properties of classical MMs and the classical 00 ˜ n ˜ while for n ≤ t1 /t we have kP2 − Tt k1 . C2,MM + nCcl long-time dynamics, which are discussed in Sec.VI of the n 0 0 and kTt − Ptk1 ≤ CMM, where CMM denotes the main text. We also provide an example of the classicality n corrections to the stationarity of Tt . test in the presence of a discrete weak symmetry.

0 Approximating long-time dynamics for t ≥ t2. After the second metastable regime, when system states are effectively restricted to the smaller second MM, the sys- 1. Symmetries of low-lying eigenmodes tW˜ 2 tem dynamics e in the basis of m2 metastable phases [cf. Eq. (34)] is approximated by classical discrete dy- We begin by discussing the symmetry of eigenmodes namics T2,t [cf. Eqs. (E194) and (E195)], of the dynamics which follows from a symmetry of the master operator [see Eq. (67)]. These results hold for ntW˜ 2 n ke − T2,tk1 . nC2,cl, (F428) the case of general metastability. Thanks to the weak symmetry in Eq. (67), the eigen- where C2,cl is in the approximation of the second MM by matrices of L can be chosen as eigenmatrices of U. Let Vk the simplex of m2 metastable phases and we consider the be the eigenspace corresponding to an eigenvalue eiϕk of L1-norm in the basis of m2 metastable phases. Similarly, (k) U, and let |ψ i, jk = 1, ..., dim(Vk) be the orthonor- in the basis of m metastable phases of the first MM, jk ˜ mal basis of the eigenspace. The eigenmatrices of U where we denote the action of T2,t by Tt,2 [cf. Eq. (F414)] with the eigenvalue 1 correspond to matrices block diagonal in ⊕ V . In contrast, coherences between V ntW˜ ˜ n ˜ k k k k(e − Tt,2)P2k1 . nC2,cl. (F429) and V , i.e., |ψ(k)ihψ(l)| with j = 1, ..., dim(V ) and l jk jl k k j = 1, ..., dim(V ), correspond to eigenmatrices of U with Note that T˜ is defined on the image of P˜ only. l l t,2 2 an eigenvalue eiφ = ei(ϕk−ϕl). Therefore, eigenmatrices Furthermore, although it is trace preserving, it is of U with an eigenvalue eiφ are in general composed of not positive. We note, however, that since Tt can coherences between all pairs of eigenspaces that differ in approximate the long times dynamics at times nt, where the arguments of their eigenvalues mod 2π by φ. Note n 1/Ccl [cf. Eq. (E195)], for t chosen for long enough that for complex eigenvalues eiφ 6= (eiφ)∗, the eigenma- the dynamics inside the second MM can be also captured n trices are non-Hermitian. by Tt .

Approximating the stationary state. For long enough ntW˜ 2 t such that ke − P˜2,ssk 1 is for an integer n 2. Symmetries of classical metastable manifolds 1/C2,cl, where P˜2,ss denotes the projection Pss on the stationary state ρss in the basis of m2 metastable phases, the Here, we discuss symmetry properties of metastable stationary state is captured by the stationary distribution phases in classical MMs in the presence of a weak sym- p2,ss of T2,t as [cf. Eqs. (E197) and (E203)] metry. We prove that under a weak symmetry the set of metastable phase undergoes an approximate permuta- ˜ ˜ ntW2 k˜p2,ss − p2,ssk1 . kP2,ss − e k1 + nC2,cl. (F430) tion. In particular, we show that under a continuous weak symmetry individual metastable phases are nec- Furthermore, in the basis of m metastable phases of the essarily invariant. We also discuss symmetrization of ﬁrst MM (or in the trace norm), the distance between the set of metastable phases and estimate the resulting the corresponding vectors (or matrices) is in the leading change in the corrections to the classicality in Eq. (21). 82

a. Discrete symmetries of classical metastable manifolds tation of the metastable phases. Let n be a nonzero integer. We have Here, we prove the approximate invariance of the set of n metastable phases under any weak symmetry. It follows n n X j−1 n−j kU − Π k = U (U − Π)Π (G435) that the action of any weak symmetry corresponds to an 1 j=1 approximate permutation of metastable phases. 1 n X ≤ kUjk kU − Πk kΠkn−j First, we argue that the set of metastable phases pro- 1 1 j=1 jections ρ˜1, ..., ρ˜m in Eq. (17) is transformed under a weak symmetry approximately onto itself, by the results ≤ nkU − Πk1(1 + Ccl) of Sec.D4. . 3nCcl, Considering ρ˜ , ..., ρ˜ or U(˜ρ ), ..., U(˜ρ ) as m 1 m 1 m where we used kUjk ≤ 1 + C (as Uj transforms the metastable phases leads to the same corrections in 1 cl MM onto itself), kΠk = 1, and Eq. (G435). Since U n Eq. (21). Let ρ(0) be an initial state corresponding to 1 is also a weak symmetry [cf. Eq. (67)], from Eq. (G435) k˜pk = 1 + C for the choice of ρ˜ , ..., ρ˜ . Then the 1 cl 1 m there exist a permutation with the matrix Π(n) such that barycentric coordinates of U[ρ(0)] in the basis of U(˜ρ1), ..., U(˜ρm) are given by (˜p)l, l = 1, ..., m and thus the n (n) U − Π . 3Ccl, (G436) corresponding corrections to the classicality are at least 1 C . Conversely, considering the transformation under U † cl and thus from the triangle inequality of an initial state corresponding to the corrections to the classicality for U(˜ρ1), ..., U(˜ρm), we obtain a lower bound n (n) n n n (n) Π − Π ≤ kU − Π k1 + U − Π (G437) for Ccl. Thus, the corrections must be the same for both 1 1 choices of the basis. . 3(n + 1)Ccl. From Sec.D4, there exist a bijective function π of Therefore, for n C 1 we can identify the set l = 1, ..., m, i.e., a permutation, such that cl [cf. Eq. (D146)] Π(n) = Πn, (G438)

U(˜ρl) − ρ˜π(l) . 3Ccl, so that [cf. Eq. (G436)] n n as a consequence of the barycentric coordinates of kU − Π k1 . 3Ccl. (G439) U(˜ρ ) = Pm p˜(l)ρ˜ , where p˜(l) = Tr[P˜ U(˜ρ )] ≡ (U) l k=1 k k k k l kl Furthermore, for N divisible by n using Eq. (G439) we [Eq. (70)] obeying [cf. Eq. (D144)] can retrace its proof with U replaced by U n and n replaced by N/n, to arrive at (l) X (l) p˜π(l) & 1 − Ccl, p˜k . 2Ccl. (G432) N N k6=π(l) U − Π 1 . 3Ccl (G440)

Therefore, the action of the symmetry on the set of [provided that (N/n)Ccl 1]. We conclude that in 0 metastable phases is approximated as Eq. (G438) we only require n Ccl 1 for all prime factors n0 of n (and not necessarily n C 1).   cl

(l) X (l) kU − Πk1 = max  1 − p˜ + p˜k  1≤l≤m π(l) k6=π(l) b. No nontrivial continuous symmetries of classical metastable manifolds . 3Ccl, (G433) where we deﬁned As any weak symmetry corresponds to an approximate permutation of metastable phases, we argue that (Π)kl ≡ δkπ(l). (G434) continuous weak symmetries restricted to low-lying modes are necessarily trivial. From Sec.D4 it also follows that Let us now consider a continuous symmetry, i.e., Uφ = −1 −1 iφG U − Π 1 . 3Ccl, e , where

† Pm 0(k) 0(k) −1 G(ρ) ≡ [G, ρ], (G441) as U (˜ρk) = l=1 p˜l ρ˜l, where p˜l = (U )lk (from † −1 U = U ) are the barycentric coordinates of ρ˜k in the with G being a Hermitian operator on the system space. −1 basis of U(˜ρn), n = 1, ..., m, while (Π )kl = (Π)lk. In this case, a weak symmetry [cf. Eq. (67) and see Refs. [S23, S24]] takes place when Second, we argue that a repeated action of the weak symmetry, corresponds to iterations of the same permu- [G, L] = 0. (G442) 83

In the basis of metastable phases [cf. Eq. (70)], Let ρ˜l belong to an approximate cycle of length dl ≤ m, i.e., (G)kl ≡ Tr[P˜kG(˜ρl)]. (G443) dl (Π )kl = δkl, (G451) Let us consider φ such that k = 1, ..., m, where Π is the permutation matrix that p φ kGk1 = Ccl. (G444) approximates U in Eqs. (G435) and (G434). In this case, D 0 D is divisible by dl (as Π = I) and we can deﬁne ρ˜l as From the Taylor expansion we then have D 0 in Eq. (72). As U P = P, we have that ρ˜l is invariant under U dl , U dl (˜ρ0) =ρ ˜0. We further have that φ2 C l l kU − I − iφGk kGk2 = cl , (G445) φ 1 . 1 D 2 2 d −1 d l 0 l X ndl where (I) = δ is the identity matrix. From Eq. (G435) kρ˜ − ρ˜lk ≤ U (˜ρl) − ρ˜l , (G452) kl kl l D we also have n=0 which we now estimate from above. From Eq. (G451) kUφ − Πk1 . 3Ccl, (G446) and Eq. (G440) (for N = ndl) we have where Π is a permutation matrix. If Π 6= I, we from m m Eq. (G445) and the triangle inequality we arrive at X ndl X ndl ndl |(U )kl − (I)kl| = (U − Π )kl (G453) p k=1 k=1 2 − Ccl = kΠ − Ik − φ kGk (G447) 1 1 ndl ndl ≤ U − Π 1 . 3Ccl, ≤ kΠ − I − iφGk1 and thus [cf. Eq. (G432)] ≤ kUφ − I − iφGk1 + kUφ − Πk1 7 m . Ccl (contradiction!) ndl X ndl 2 U (˜ρl) − ρ˜l ≤ |(U )kl − (I)kl|kρ˜kk(G454) k=1 for Ccl 1. The choice Π = I, however, similarly leads ndl ndl to ≤ (1 + Ccl) U − Π 1 p leads to [cf. Eq. (G452)] Ccl = φ kGk1 (G448) ≤ kUφ − I − iφGk + kUφ − Ik (G449) 1 1 0 3(D − dl) 7 kρ˜l − ρ˜lk . Ccl < 3Ccl, (G455) C (contradiction!). D . cl 2 where in the last inequality we used dl ≥ 1. This can only be remedied when Eq. (G444) is not pos- From Eq. (G455) we can replace the approximate cycle sible, which implies kGk1 = 0 and thus the trivial con- ρ˜πn(l), where π is the permutation associated with Π tinuous symmetry of the MM n 0 in Eqs. (G435) and (G434), by U (˜ρl) n = 1, ..., dl − 1 [cf. Eq. (73)]. Indeed, the distance between the new and G = 0, and Uφ = I. (G450) old basis so that all metastable states are invariant under the sym- 0 0 n ρ˜πn(l) − ρ˜πn(l) ≤ kρ˜l − ρ˜lk + U (˜ρl) − ρ˜πn(l) metry UφP = I. . 6 Ccl, (G456) where in the ﬁrst line we used kU n(˜ρ ) − ρ˜0 k = c. Symmetric set of metastable phases l πn(l) n n 0 0 kU (˜ρl) − U (˜ρl)k = kρ˜l − ρ˜lk as U is a unitary transformation, while the second line follows from Eq. (G439) In Sec. G2a, we showed that the set of metastable [cf. Eq. (G455)]. We can then repeat the construction phases is approximately invariant under the weak sym- in Eqs. (72) and (73) for each of the approximate cycle metry [Eq. (67)]. We now assume a nontrivial symmetry between metastable phases under U. of the MM and consider replacing ρ˜ , ..., ρ˜ in Eq. (17) 1 n The metastable phase in Eq. (72) corresponds to pro- by the invariant set of metastable phases. We also jection only on the low-lying eigenmodes invariant under provide corrections to the classicality in Eq. (21) for the U d, where d is the length of the approximate cycle ρ˜ invariant set of metastable phases. l belongs to,

Symmetric set of metastable phases. Let D be the X (l) ρ˜0 = c R (G457) smallest nonzero integer such that U DP = P, and thus l k k D 1≤k≤m: U = I. We assume d Ccl 1 for all the prime factors (eiφk )dl =1 d of D (which follows from m Ccl 1), in which case D D D kΠ − Ik1 ≤ kU − Π k1 . 3Ccl [cf. Eq. (G439)], and [cf. Eq. (17)]. We note, however, that by construction in D 0 thus Π = I. Eq. (72), the closest state to ρ˜l in Eq. (72) is not further 84

2 away than C+ [cf. Eq. (12)], since it is bounded by the with corrections of the order Ccl in the L1 norm. From distance to the state Eq. (G464),

D −1 0 00−1 00−1 d k˜p k = C ˜p ≤ C k˜pk (G465) d l 1 1 1 1 0 l X ndl 00 ρ ≡ U (ρl), (G458) (1 + kC − Ik ) k˜pk l D . 1 1 n=0 . (1 + 6 Ccl) (1 + Ccl) where ρl is the closest state to ρ˜l, and and thus [cf. Eq. (G460)] D −1 dl 0 d Ccl . 7 Ccl, (G466) 0 0 l X ndl ndl kρ˜ − ρ k ≤ U (˜ρl) − U (ρl) (G459) l l D n=0 as well as [cf. Eq. (C45)] = kρ˜l − ρlk ≤ C+. 0 Ccl ≡ k˜p k1 − 1 . Ccl + 6 Ccl. (G467) 0 0 Note that the state ρl in Eq. (G458), in analogy to ρ˜l in Eq. (72), is invariant under U dl . Furthermore, the n 0 3. Symmetries of classical long-time dynamics elements of the cycle U (ρl), n = 0, ..., d − 1 are approx- d imated by U l (ρl) with the same distance kρ˜l − ρlk. Similarly, if ρl is a state that projected on the low- Here, we show that the approximation of long-time dy- lying modes gives ρ˜l, that is, P(ρl) =ρ ˜l [cf. Eq. (17)], namics by classical stochastic dynamics in Eq. (36) fea- 0 0 0 tures a permutation symmetry for the set of metastable from Eq. (68) we have that P(ρl) = ρl, for ρl defined in 0 dl phases chosen invariant under a weak symmetry. This Eq. (G458). Furthermore, ρl is invariant under U and n 0 n 0 corresponds to Eq. (77) in the main text. Furthermore, P[U (ρl)] = U (ρl) holds for all elements of the cycle n = 0, 1, ..., dl − 1. the long-time dynamics restricted to the symmetric mixtures of metastable phases is also approximated by clas- Classicality corrections for symmetric set of metastable sical stochastic dynamics. phases. We now consider how the choice of a symmetric set of metastable phases in Eqs. (72) and (73) affects the corrections to the classicality in Eq. (21). We are a. Derivation of Eq. (77) in the main text interested in bounding For the set of metastable phases is chosen invariant 0 0 Ccl ≡ max k˜p k1 − 1, (G460) ρ(0) under a weak symmetry [cf. Eq. (67)], the corresponding stochastic classical dynamics W0 in Eq. (36) that approx- 0 0 ˜0 ˜0 ˜ 0 where (˜p )l =p ˜l ≡ Tr[Pl ρ(0)] with an operator Pl being imates W [cf. Eq. (37)] fulfills [cf. Eq. (75) and (76)] an element of the invariant dual basis, i.e., Tr(P˜0 ρ˜0) = k l [W˜ 0, Π] = 0, (G468) δkl, k, l = 1, ..., m. The transformation from the basis in Eq. (17) to the invariant basis defined in Eqs. (72) which is equivalent to [cf. Eq. (E160)] and (73) is given by [cf. Eq. (18)] ˜ 0 ˜ 0 (W )kl = (W )π(k)π(l), (G469) (C00) ≡ Tr(P˜ ρ˜0) = (C0−1C) , (G461) kl k l kl where π is the permutation corresponding to Π and k, l = and the dual basis [cf. Eq. (20)] 1, ..., m. We now prove that this condition holds for the closest classical stochastic generator W0 as well, as given m X by Eq. (77). P˜0 ≡ C00−1 P˜ . (G462) l lk k First, we have k=1 0 0 (W )kl ≡ max[(W˜ )kl, 0] (G470) 00 PD/dl ndl Note that (C )kl = (dl/D) n=1 (U )kl for l as in ˜ 0 0 00 n 00 = max[(W )π(k)π(l), 0] ≡ (W )π(k)π(l). Eq. (72), while (C )kπn(l) = (U C )kl, n = 1, ..., dl − 1 from Eq. (73). Therefore, from Eq. (G451) and Second, Pm 00 Eq. (G440) we obtain k=1 |(C )kl −δkl| . 3Ccl [for N = 0 0 X 0 Pm 00 (W )ll ≡ (W˜ )ll + min[(W˜ )kl, 0] (G471) ndl], while from Eq. (G439) we have k=1 |(C )kπn(l) − Pm n 00 Pm n 1≤k≤m: δkπn(l)| ≤ k=1 |[U (C − I)]kl| + k=1 |(U )kl − k6=l (Πn) | 3C (kUnk + 1) ≤ 6C , where we used kl . cl 1 cl ˜ 0 X ˜ 0 n n n n = (W )π(l)π(l) + min[(W )π(k)π(l), 0] kU k1 ≤ kU −Π k1 +kΠk1 . 1+3Ccl from Eq. (G439) and kΠk = 1. We conclude 1≤k≤m: 1 k06=l kC00 − Ik 6 C , (G463) ˜ 0 X ˜ 0 1 . cl = (W )π(l)π(l) + min[(W )k0π(l), 0] 0 and thus we can approximate 1≤k ≤m: k06=π(l) 00−1 00 0 C = I − (C − I) + ..., (G464) ≡ (W )π(l)π(l), 85 where we introduced k0 ≡ π(k). This ends the proof is governed by the generator [cf. Eq. (E160)]. dk−1 dl−1 0 1 X X 0 (W˜ ) = (W˜ ) n n , (G475) 0 kl d π k(k) π l(l) l n =0 n =0 b. Classical dynamics of symmetric degrees of freedom k l which is trace preserving as a consequence of the trace Here, we consider the long-time dynamics W˜ 0 in the preservation of W˜ . invariant set of metastable phases [Eq. (34)] restricted We now prove that the generator in Eq. (G475) is to the symmetric subspace of the symmetry Π, that is approximately positive up to corrections bounded by uniform mixtures of metastable phases in each cycle of p 0 2 Ccl. As a consequence of the symmetry of the classical the permutation π corresponding to Π (cf. Sec.VI in stochastic generator W0 closest to W˜ 0 [cf. Eq. (77)], when the main text). We show that that dynamics is trace- restricted to the symmetric subspace of Π, which we preserving and approximately positive with corrections 0 p 0 0 denote W0, it remains trace-preserving and completely- bounded by 2 Ccl, where Ccl denote the corrections positive (cf. Sec.E1c). From the triangle inequality, we to the classicality in Eq. (21) for the invariant set of then have metastable phases [cf. Eq. (37)]. ˜ 0 0 kW0 − W 0k1 (G476) Classical symmetric degrees of freedom. Symmetric dk−1 dl−1 mixtures of metastable phases, or the symmetric part of 1 X X X ˜ 0 ˜ ≡ max (W )πnk(k) πnl(l) − (W)πnk(k) πnl(l) states within the MM, can be described by (cf. Sec.E1c) l dl k nk=0 nl=0

dl−1 m dl−1 dl−1 h i 1 X X 0 0 X 0 X ≤ max (W˜ ) nl − (W˜ ) nl n ˜ n kπ (l) kπ (l) (˜p0)l = (˜p )π (l) = Tr Pπ (l) ρ(0) , (G472) l dl nl=0 k=1 nl=0 nl=0 m q X ˜ 0 ˜ ˜ 0 0 0 ≤ max (W )kl − (W)kl ≡ kW − W k1 2 Ccl. where the index l runs over representatives of cycles with 1≤l≤m . k=1 dl denoting the corresponding cycle length, and ˜p are barycentric coordinates of the system state [cf. Eqs. (19) [cf. Eq. (37)]. We note, however, that W0 is gen- and (80b)], so that the number of parameters equals 0 erally not the closest classical stochastic generator the number of cycles in the permutation π, minus 1 to W˜ 0 [cf. Eq. (36)]. The distance to the closest for P (˜p ) = 1. Since Eq. (G473) corresponds to 0 l 0 l classical stochastic generator is instead bounded by an invertible linear transformation of the coefficients 2pmax k˜p0 k − 1 ≤ 2pC0 [cf. Eq. (G473)], where c = Tr[L ρ(0)] for the symmetric low-lying eigenmodes ρ(0) 0 1 cl k k max k˜p0 k −1 bounds the distance of ˜p to the simplex U(L ) = L , k = 2, ..., m [cf. Eq. (82)], it is an equivalent ρ(0) 0 1 0 k k formed by the uniform mixtures of metastable phases for representation of the symmetric degrees of freedom. each cycle (in particular, for only two cycles, W˜ 0 is a Operationally, ˜p describes the barycentric coordi- 0 0 classical stochastic generator; cf. Refs. [S5, S11]). nates of symmetric states in the MM decomposed between uniform mixtures of metastable phases in each cycle, and, from the triangle inequality, we have 4. Example of classicality test with weak symmetry dl−1 0 X X 0 0 0 k˜p0k1 ≡ (˜p )πn(l) ≤ k˜p k1 ≤ 1 + Ccl (G473) Here, we provide a simple example, why, even in the l nl=0 case of the symmetry fully determining the eigenmodes of the dynamics [cf. Eq. (80)], the classicality test is needed. [cf. Eq. (G460)]. Therefore, the corrections to the classicality for symmetric states are not larger than 0 We consider a finite system with m disjoint station- Ccl (in particular, in the case of only two cycles, those ary states ρ1, ...ρm and the corresponding projections corrections will be 0, as in the bimodal case with m = 2; Pm 1 P1,...,Pm [Tr(Pkρl) = δkl, l=1 Pl = ]. We assume cf. Refs. [S5, S11]). Among others, it then follows that that the stationary states are connected by a weak sym- the uniform mixtures of metastable phases in each cycle metry as U(ρl) = ρl+1, l = 1, ..., m (with periodic and their basins of attraction are approximately disjoint boundary conditions on the label, m + 1 ≡ 1). We (cf. Sec.IV in the main text). iϕ have that eigenmatrices of the symmetry fulfill e k Rk = m −i2π jk l −iϕ m i2π jk l (1/m) P (e m ) ρ and e k L = P (e m ) P Trace-preserving and approximately positive dynamics. l=1 l k l=1 l [cf. Eq. (80) and the normalization kL k = 1], where The dynamics of the symmetric degrees of freedom, k max eiϕk is an arbitrary global phase, k = 1, ..., m, and i2π jk d e m is the corresponding symmetry U eigenvalue, jk ∈ ˜p (t) = W˜ 0 ˜p (t), (G474) dt 0 0 0 {0, ..., m − 1}. As the invariant set of candidate phases, 86 let us choose we assume Hermitian Lk, by replacing non-Hermitian † R I m conjugate pairs of eigenmatrices Lk, Lk by Lk and Lk in 1 − p X max min ρ˜l = p ρl + ρk, (G477) Eq. (83), and choose the normalization ck − ck = 1, m max min k=1 where ck and ck are extreme eigenvalues of Lk (thus, kLkkmax ≤ 1). l = 1, ..., m, where 0 ≤ p ≤ 1. We then have [cf. Eq. (82)] From Eq. (H480), there exists at least one metastable max 1 1 1 − p phase ρ˜l with l = lk chosen so that the kth coefficient ˜ max Pl = Pl − 1 (G478) (lk ) (l) max p m p is closer than Ccl, i.e., δk ≤ Ccl, where δk ≡ ck − (l) max ck . Furthermore, the metastable state P(ρk ) for the so that [cf. Eq. (21)] max initial state ρk chosen as the maximal Lk eigenstate, m − 1 1 − p approximates the closest metastable ρ˜lmax as C = 2 (G479) k cl m p max 2Ccl ρ˜ max − P ρ + C , (H481) lk k . (lmax) cl (achieved for any of ρl, l = 1, ..., m), which diverges to ∞ ∆ k as p → 0 and the basis in Eq. (G477) stops being linearly k independent. We note, however, that when expressed in max (l) where ∆k ≡ minl6=lmax δ is the distance in kth coeffi- √the basis of ρ˜l, Rk is still proportional (with the factor k k m/p for k ≥ 2) to Eq. (80a), as so is L to Eq. (80b) cient to the next closest phase (see the derivation below). k √ For ∆max C , the corrections in Eq. (H481) are negligi- when expressed in the basis of P˜ (with the factor p/ m k cl l ble. Otherwise P(ρmax) is a mixture of metastable phases for k ≥ 2). k with δ(l) of the order of C . The discussion is analogous In this example, the proportionality factor between Rk k cl min (l) (l) min and Eq. (80a) [or Lk and Eq. (80b)] indicates that the for the minimal Lk eigenvalue ck with δk ≡ ck −ck . choice of the basis in Eq. (G477) is not optimal. In the The corrections in Eq. (H481) also hold for the L1 presence of classical metastability, however, rather than distance in the barycentric coordinates of the metastable for a classical phase transition as considered in this ex- phases. Thus, the corrections to the classicality in the ample, the proportionality factor for correctly identified basis chosen from the extreme eigenstate of dynamics phases does not equal 1, as ρ˜l in Eq. (17) are only approx- modes are bounded by 2Ccl/(min1≤l≤m ∆l) + Ccl, where imately disjoint (see Sec.IV in the main text), as well as ∆l is the minimal distance to the next metastable P˜l are not bounded by 1 or orthogonal (see Sec.D1). phase in the coefficients for which ρ˜l was extreme, max max min We would expect, however, that the proportionality fac- ∆ = min(mink: lmax=l ∆ , min min ∆ ) (also l k k k: lk =l k tor can be related to the corrections in Eq. (21), and including rotated left eigenbases) (cf. the corrections should be close to 1 for classical metastable phases. to the classicality for the symmetric set of metastable phases in Sec.G2c).

H. QUANTITATIVE ANALYSIS OF Derivation of Eq. (H481). The metastable phase ρ˜ max lk ALGORITHM IN SEC.VIIA OF THE MAIN (l) TEXT exists, as δk ≥ 0 and, thus, from Eq. (H480) we have Pm (l) l=1 plδk ≤ Ccl. Furthermore, the metastable state max Here, we give quantitative analysis of the effectiveness P(ρk ) is approximated as a mixture of metastable (l) P of the numerical approach introduced in Sec.VIIA of the phases with δk of the order of Ccl, since (l) pl = l:δk <∆ main text, in terms of corrections to the classicality Ccl P P (l) 1 − (l) pl ≥ 1 − l plδk /∆ ≥ 1 − Ccl/∆ is close to in Eq. (21) and the assumption of a nonnegligible volume l:δk ≥∆ 1 for ∆ Ccl. Therefore, when there exists only a single of the MM in the space of coefficients. max (lk ) metastable phase ρ˜lmax with ck in the proximity to max k max c , we have p˜ max ≥ 1 − C /∆ and P(ρ ) can be re- k lk cl k 1. Extreme eigenstates of dynamics eigenmodes for placed by that metastable state as given in Eq. (H481), metastable phases P as kρ˜l − P[ρ(0)]k ≤ (|1 − p˜l| + l06=l |p˜l0 |)(1 + C+) and P l06=l |p˜l0 | ≤ 1 + Ccl − |p˜l| ≤ |1 − p˜l| + Ccl [cf. Eq. (21)], In the coefficient space, the MM is well approximated where p˜ = Tr[P˜ ρ(0)]. by the simplex S with vertices given by metastable phases l l coefficients (see Fig.2 in the main text) m 2. Rotations of eigenmodes to expose metastable X (l) min ck − plc ≤ kLkkmax mink˜p − pk1 ≤ Ccl, phases p k p l=1 (H480) We now explain how rotations of the basis of eigen- (l) where ck = Tr[Lkρ(0)] [cf. Eq. (4)] and ck = Tr(Lkρ˜l), modes allow to solve problem of degeneracy in the coef- while (˜p)l = Tr[P˜lρ(0)] [cf. Eq. (20)], k, l = 1, .., m. Here, ficients, as well as they ensure that a given metastable 87 phases corresponds to extreme value of one of the co- 3. Maximal simplex in coefficient space as simplex efficients when the volume of the simplex of metastable of metastable phases phases in the coefficient space is nonnegligible. 0 The required lack of degeneracy (up to order Ccl), be- We show below that the volume of a simplex S with tween metastable phases with the maximal (minimal) kth m vertices inside the MM is bounded by [cf. Eq. (H482)] coefficient in Eq. (H480), corresponds to kth axis in the 0 √ space of coefficients being normal to the supporting hy- Vol(S ) . Vol(S) + Ccl msm−1, (H483) perplane at lth vertex of the simplex in the coefficient space where l = lmax (l = lmin). In convex geometry it k k where S is the simplex of m metastable phases and Ccl is known that for any vertex in a simplex, there exists denotes the corresponding corrections to√ metastability a supporting hyperplane, that is, there exist a rotation in Eq. (21). Therefore, when Vol(S) Ccl m − 1sm−1, such that the rotated kth axis is perpendicular to a sup- the simplex of metastable phases is approximately the porting hyperplane. In our case, the lack of degeneracy maximal simplex within the MM. We note that this is a up to order Ccl, additionally requires the distance from stronger condition than in Eq. (H482). (l) max that hyperplane of other vertices δk Ccl for l 6= lk min (l 6= lk ). As we argue below, this condition can be Derivation of Eq. (H483). First, we note that the dis- translated into a nonnegligible volume of the metastable tance in the space of coefficients of any point within the phases simplex S in the space of coefficients, which is MM√ to the simplex S of metastable phases is bounded guaranteed by by m − 1Ccl in L2 norm. Second, consider lth vertex in a simplex S0 with m vertices. From the derivation of 0 0 0 sm−1 Eq. (H482), we have that Vol(S ) = kδc¯lk2Vol(Sl)/(m − Vol(S) = |det(C)|/m! Ccl , (H482) 0 0 1), where Sl is the simplex obtained from S after remov- m − 1 0 ing lthe vertex andkδc¯lk2 is the length of the component of lth vertex orthogonal to that simplex. Since by re- where sm−1 is the maximal volume of a simplex with m− placing lthe vertex by the closest point in the simplex S 1 vertices inside the (m−1)-dimensional unit hypercube. of metastable phases with respect to L2 norm,√ the or- The condition in Eq. (H482) also guarantees that the thogonal component can decrease at most by m − 1Ccl, separation between any two metastable phases in the we obtain√ that the volume can decrease at most by Cclsm−1/ m − 1. Repeating the procedure with respect space of coefficients equipped with L2 norm is Ccl, and thus the metastable phases are distinguishable in to remaining m − 1 vertices, we arrive at a simplex of the space of coefficients [cf. Eq. (H480)]. m vertices inside S at the√ cost of the√ volume decrease at most by . Cclsm−1m/ m − 1 ≤ mCclsm−1. Noting that volume of any simplex inside S is less than Vol(S), Derivation of Eq. (H482). Let δc¯l be the component of we arrive at Eq. (H483). (l) (1) the shifted coefficient vector (c¯l)k ≡ ck − ck (l 6= 1) orthogonal to the (m − 2) dimensional subspace spanned 0 by other vertices, c¯l0 with l 6= 1, l. The distance of 4. Hierarchy of metastable manifolds other vertices to the supporting hyperplane at c¯l normal to δc¯l is equal to its L2 norm kδc¯lk2. Furthermore, the simplex volume Vol(S) = kδc¯lk2Vol(Sl)/(m − 1), In the presence of hierarchy of metastabilities with a where Sl is the simplex of all vertices√ but lth one. further separation at m2 < m in the spectrum of the Indeed, we have that Vol(S) = det CT C/(m − 1)!, master operator, any metastable state during the second ¯ where (C)k−1,l−1 = (¯cl)k with k, l = 2, ..., m [? ], and metastable regime is approximated as a mixture of m2 p T Vol(Sl) = det (C¯ l) C¯ l/(m − 2)!, where C¯ l is obtained metastable phases up to the corrections to the classicality from C¯ by removing lth column (we assumed l 6= 1). C2,cl in the second MM (see Sec.F). In the space of the

Therefore, up to rotations, in order for metastable phases coefficients (c2, ..., cm2 ), we have an analogous bound to to correspond to extreme eigenstates of dynamics eigen- Eq. (H480), modes, we require a nonnegligible volume of the MM in the coefficient space. Finally, we note that due to the m2 X (2,l2) chosen normalization of Lk, the MM is enclosed by a min ck − p2,l2 ck ≤ kLkkmax mink˜p2 − p2k1 p2 p2 (m − 1)-dimensional unit [cf. Eq. (H482)]. l2=1 We also note that the distance between two vertices, ≤ C2,cl, (H484) kth and lth, is bounded from below by δc¯l in the coor- (2,l2) dinate system shifted to the kth vertex (earlier k = 1). where k ≤ m2, ck denotes the coefficients for m2 Therefore, it the distance between any two vertices with metastable phases of the second MM, ˜p2 is the vector of respect to L2 norm in the space of coefficients is Ccl barycentric coordinates in their basis, and (p2)l2 = p2,l2 when Eq. (H482) is fulfilled. is a vector of a probability distribution. 88

5. Weak symmetries where ϕk is the arbitrary phase chosen to obtain Hermi- R inkφk tian Lk in Eq. (83), and we used that e = 1. Sim- R(min) Here, we discuss degeneracy of coefficients in classical ilarly, the minimum coefficient ck can be approx- MMs in the presence of a weak symmetry U (see Sec.VI imated by considering minimal values of the cosine in I in the main text) and argue how this degeneracy can be Eq. (H486), while the extreme values of the coefficient ck I addressed by the metastable construction in Sec.VIIA for Lk can be approximated by considering the sine in- of the main text. Furthermore, we explain how the con- stead of the cosine. Therefore, degeneracy in the extreme R I struction can be further refined, to exploit the structure coefficients ck or ck can arise in two ways: of the MM arising due to the weak symmetry. 0(l) A. the coefficients ck together with choice of ϕk lead to degeneracy of extreme values attained by differ- a. Symmetric eigenstates of dynamics eigenmodes ent plane waves,

For a weak symmetry U in Eq. (67), let an eigenmode B. some of the cycles contributing to Lk are of length longer than n , so that they have d /n -fold degen- L be chosen such that U †(L ) = eiφk L . Let n > 0 k l k k k k k eracy in the amplitude of the corresponding plane then denote the minimal integer such that einkφk = 1, †n wave [cf. Eq. (80b)]. i.e., U k (Lk) = Lk (we consider a discrete symmetry, without loss of generality, as relevant for the classical Case B degeneracy is a direct consequence of from the MM). We have that the Hermitian and anti-Hermitian presence of the weak symmetry, as discussed in the ﬁrst iϕ part of e k Lk in Eq. (83) commute with the symmetry paragraph, and is always present, e.g., in symmetric R nk I nk applied nk times, [Lk ,U ] = 0 = [Lk,U ]. Therefore, eigenmodes. Nevertheless, both degeneracy cases can be R I eigenstates of Lk and Lk can be chosen as eigenstates of remedied; see below. U nk . Such states and their projections on the MM, are symmetric under U nk , so that under U they form cycles with length that divides nk. c. Reﬁned metastable phase construction

b. Degeneracy of coefficients We now explain how to remedy degeneracy of coefficients arising in the presence of a weak symmetry in the approach of Sec.VIIA in the main text. Although We now discuss the degeneracy of coefficients in clas- any degeneracy can be generally resolved by random sical MMs for a weak symmetry U. Let an eigenmode Lk rotations as argued in Sec.H2, here, we explain how the † iφk be chosen as a symmetry eigenmatrix, U (Lk) = e Lk structure of the MM imposed by the weak symmetry inkφk and nk > 0 be the minimal integer such that e = 1. in the coefficient space can be exploited to simplify the For a state ρ(0), the coefficient ck ≡ Tr[Lkρ(0)] = approach; see also the discussion in Sec.VIIA of the nk Tr{LkU [ρ(0)]} is the same for all states generated from main text. ρ(0) under U nk . In particular, as a weak symmetry acts on an invariant set of metastable phases in a classical MM Case A degeneracy. In general, cycles of various as their permutation π all metastable phases connected lengths contribute to L in Eq. (H485). If we consider nk k under π have the same coefficient ck. L such that there does not exists L with n divisible Let us now focus on the extreme values of coefficients. k l l by nk, however, Lk is supported on only cycles with the From Eqs. (81) and (82), an eigenmode Lk is a linear length n (as there are no longer cycles with the length combination of plane waves over cycles with the length k divisible by nk). In particular, the number of cycles of divisible by nk, length nk equals the degeneracy of the symmetry eigen- inφk X 0(l) 0 value e among low-lying eigenmodes. Therefore, the L = c L j , (H485) k k π kl (l) only coefficient degeneracy that can be present belongs l: nk|dl to Case A discussed in Sec.H5b. We now explain how it can be remedied by an appropriate choice of the phase where l indexes cycles, d is their length, and j ≡ l kl ϕ in LR of Eq. (83). φ d /(2π) mod d [cf. Eq. (80b)]. Since the maximal co- k k k l l First, let n = 1, 2, ..., n −1 be such that it is the closest efficient of LR can be approximated up to corrections k k to 1 among einφk and with the positive imaginary part. C by considering states inside the simplex of metastable cl For any ϕ in Eq. (83), the difference of the maximal and phases [cf. Eqs. (H480)], its value corresponds to the max- k the next in value coefficient for the metastable phases in √ imum value among plane waves weighted by the coeffi- 0(l) 2 0(l) lth cycle is less than 2 2|ck | sin(δφk) , where δφk = cients ck , (nφk mod 2π)/2, but it also can be 0 in the worst case R(max) (1) (2) c = (H486) scenario. By considering both ϕk = 0 and ϕk = δφk k √ in Eq. (83), however, the bigger among the differences 0(l) 0(l) √ max max 2 ck cos nφk + ϕk + arg ck + ..., 0(l) l 0≤n≤nk−1 is no less than 2 2|ck | sin(δφk/2) sin(δφk); this choice 89

R iφk0 iφk can be effectively facilitated by considering both Lk and teger; without repetitions, i.e., without e 6= e with I Lk, whenever dl is not divisible by 4. nk0 = nk). Otherwise, Ll is supported on cycles that 0(l) 0(l0) have not been considered yet. In that case, by choos- Second, let |c | = maxl0 |c | be the maxi- k k ing Ll with nl such that the only Lk with nk divisible mal weight among the plane waves in Eq. (H485). by n have already been considered, we obtain that all The minimal difference between maximal coefficients l new cycles on which Ll is also supported are exactly of for metastable phases in different cycles is at least the length n (and their number is the difference between 0(l) 0(l0) (1) l iφ max[|ck | cos(δφk/2)−maxl06=l |ck |, 0] either for ϕk or the degeneracy of e l and the sum of degeneracies of rel- (2) iφk ϕk . If needed, this bound can be improved by consider- evant e ). By considering rotations of all eigenmodes iφ (n) n with the symmetry eigenvalue e l , a metastable phase in ing additionally ϕk = δφk/2 , n = 1, .., N in Eq. (83), N+1 each cycle with the length nl can be found from extreme in which case cos(δφk/2) is replaced by cos(δφk/2 ). eigenstates of rotated eigenmodes, as discussed in Case Therefore, there is only a single metastable phase R A above. We note that equal mixtures of already found corresponding to the maximal coefficient of L when n /n k phases in cycles of length nk connected by π k l will also 0(l) the maximal weight |ck | is nondegenerate up to be found here (due to Case B degeneracy; cf. Sec.H5a), Ccl [also when multiplied by cos(δφk/2)] and when but those candidate states will be discarded by choos- sin(δφk/2) sin(δφk) Ccl. The former condition can be ing the candidates yielding the maximal volume simplex. achieved for any plane-wave [any l in Eq. (H485)] by a ro- Finally, other elements of the cycles with the length nl tation of all Lk with the same symmetry eigenvalue pro- can be recovered by applying the symmetry nl times, 0(l) 0 vided that the simplex of the coefficients |ck | for those while other modes supported on those cycles, i.e., Ll iφ 0 iφ eigenmodes has a nonnegligible volume (cf. Sec.H2). with e l 6= e l but with nl0 = nl, do not need to be Finally, for each cycle, the metastable phase chosen considered. R If not all choices of the eigenmodes are exhausted from an extreme eigenstate of (rotated) Lk can be used to recover other elements of the cycle by applying in the way discussed above, i.e., there exist Lj with nj < nl, such that nj divides nl, we again repeat the the symmetry U nk times. Therefore, there is no need I procedure described in the above paragraph, but with to consider Lk or other modes supported on those respect to all Ll and with nl that are divisible by nj. cycles, i.e., Lk0 with a different symmetry eigenvalue iφ 0 iφk In particular, Lj may be discarded if is supported on e k 6= e but with nk0 = nk. Furthermore, this choice iφj corresponds to the symmetric set of metastable phases already considered cycles, i.e., degeneracy of e is equal iφl in Eq. (72) (cf. Sec.H5a). to the sum of degeneracies of all relevant e . Example. In Fig.5 of the main text, we have one cy- Case B degeneracy. After finding the extreme states cle of length 2 corresponding to the metastable phases ρ˜1 of Lk as described in the above paragraph, we still need and ρ˜ and two cycles of length 1 corresponding to invari- iφ 3 to consider Ll with the symmetry eigenvalue e l and nl ant metastable phases ρ˜2 and ρ˜4. Here, we would first such that there exists Lk with nk divisible by nl. In that consider the eigenmode L3 corresponding to the symme- case Ll features Case B degeneracy discussed in Sec.H5b try eigenvalue −1 (nk = 2), which would give as candi- with respect to metastable phases connected by πnk/nl in dates approximately ρ˜1 and ρ˜3 (twice; due to cycles of the the already found cycles [cf. Eq. (H485)]. symmetry). We would then be left with two symmetric We are interested in whether the metastable states on eigenmodes L2 and L4 (we do not need to consider trivial which Ll is supported via plane waves have been already L1 = 1), which would give pairs of symmetric candidates found by considering Lk. This is the case when the de- approximately as ρ˜2, (˜ρ1 +ρ ˜3)/2, and ρ˜2, ρ˜4. By clus- iφ generacy of e l among the low-lying eigenmodes equals tering those candidates would be reduced to: ρ˜1, ρ˜2, ρ˜3, the number of already considered cycles with their length (˜ρ1 +ρ ˜3)/2, and ρ˜4. Finally, by considering the maximal divisible by nl (which is equal the sum of degeneracies of volume simplex, we would obtain approximately the four iφ e k for already considered Lk such that nk/nl is an in- metastable phases ρ˜1, ρ˜2, ρ˜3 and ρ˜4.

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