Optimized Quantum Steering and Exceptional Points

Parveen Kumar,1 Heinrich-Gregor Zirnstein,2 Kyrylo Snizhko,3, 1 Yuval Gefen,1 and Bernd Rosenow2 1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel 2Institut für Theoretische Physik, Universität Leipzig, Brüderstrasse 16, 04103 Leipzig, Germany 3Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany The state of a quantum system may be steered towards a predesignated target state, employing a sequence of weak blind measurements (where the detector’s readouts are traced out). Here we analyze the steering of a two-level system using the interplay of a system Hamiltonian and weak measurements, and show that any pure or mixed state can be targeted. We show that the optimization of such a steering protocol is underlain by the presence of Liouvillian exceptional points. More speciﬁcally, for high purity target states, optimal steering implies purely relaxational dynamics marked by a second-order exceptional point, while for low purity target states, it implies an oscillatory approach to the target state. The phase transition between these two regimes is characterized by a third-order exceptional point.

Steering of a quantum system towards a pre-designated regimes is characterized by a third-order EP, where all target state can be achieved either by drive-and- three non-zero eigenvalues of the Liouvillian superoper- dissipation schemes [1–11] or through measurement- ator coalesce, and the optimal convergence dynamics of based protocols [12–18]. The former employ a dissipa- the system changes from non-oscillatory to oscillatory, tive environment to relax the quantum system into the reminiscent of a spontaneous breaking of PT -symmetry target state, while in the latter case relaxation (as well [21, 40–44]. We present a general argument that the ap- as back-action on the system) is achieved by measure- pearance of a higher-order exceptional point at the tran- ments. Optimizing the rate of convergence towards the sition is generic. target state is important to render it of practical im- On a more technical level, representing the system portance and minimize external perturbations. Excep- state on or within the Bloch sphere by a vector s, its tional points (EPs), referring to non-Hermitian degen- purity is characterized by P ≡ (1 + s2)/2. We find that eracies where two or more eigenvalues of the evolution the optimal steering dynamics can be characterized by operator coalesce [19–23], play an important role in a three different regimes, summarized in Fig.2. In the low variety of optimization problems [24–27]. Such degen- purity regime with P ≤ 7/8, the convergence rate be- eracies are of particular interest in dynamics with com- comes optimal by choosing the Zeeman field in the sys- plex eigenvalues where unitary dynamics competes with tem Hamiltonian as large as possible, and convergence dissipation or gain, and may be generalized from non- towards the target state is oscillatory. In the medium pu- Hermitian Hamiltonians to Liouvillian dynamics [28]. In rity regime, 7/8 < P ≤ 127/128, optimum convergence is recent years, it has been recognized that operating near achieved when all three non-vanishing eigenvalues of the an EP enables unique functionality such as unidirectional Liouvillian superoperator have equal real parts; the opti- invisibility [29–32] or enhanced sensitivity [33–38]. mal convergence rate is independent of P , while the con- Here, we propose a family of protocols for steering a vergence dynamics remains oscillatory, similar to the low two-level quantum system towards desired target states. purity regime. The transition to the high purity regime The system’s initial state is assumed unknown to us. In with 127/128 < P ≤ 1 occurs at a third-order EP where our protocol, the quantum system is subject to both a all three non-zero eigenvalues of the Liouvillian superop- Hamiltonian evolution and a measurement-induced evo- erator coalesce. In the high purity regime, the optimal arXiv:2101.07284v3 [quant-ph] 11 Feb 2021 lution, and the combined effect of both can be described convergence dynamics is non-oscillatory since all three by a Liouvillian superoperator. The system’s target state non-zero eigenvalues of the superoperator are real, and is given by the Liouvillian eigenstate having zero eigen- two of them are degenerate, placing the superoperator at value, and is uniquely determined (along with its purity) a conventional second-order EP. by the interplay between the Hamiltonian and the mea- System evolution and the steady state.— Consider a surement protocol. When optimizing our steering [39] two-level quantum system represented by the density ma- protocol in the sense that the target state is reached as trix ρs whose dynamics comprises two contributions: the fast as possible, we find that the optimal steering for high unitary evolution and the measurement evolution. The purity target states is dominated by the measurement- former is governed by the following Hamiltonian acting induced dynamics and described by second-order excep- in the system’s Hilbert space tional points, while optimal protocols for low purity tar- Hs = ω nˆ · σ, nˆ = (cos φ sin θ, sin φ sin θ, cos θ), (1) get states are dominated by the system-Hamiltonian- induced dynamics. The transition between these two where ω is the Zeeman energy of the two levels, θ, φ 2 are spherical coordinates parametrizing the unit vector nˆ, and σ = (σx, σy, σz) is the vector of Pauli matrices. For the measurement evolution, the system needs to couple with the detector, which is chosen to be a two- 0 1 level quantum object prepared in the state ρd = 2 (I+m ˆ · σ), where mˆ is the detector state initialization direction. Before they interact, the joint system-detector state is 0 ρ(t) = ρs(t) ⊗ ρd. At later times, the joint state ρ(t) evolves with the system-detector interaction Hamiltonian Figure 1. (a) Steady state ellipsoid for the detector state s d s d Hs−d = J σ · σ − (m ˆ · σ )(m ˆ · σ ) , (2) initialization direction mˆ = (0, 0, 1)T , see Eq. (5). (b) The steady state ellipsoid can be rotated by rotating the detector where J is the coupling parameter, and σs, σd are the state initialization direction. Hence, any target state (red) on pseudo-spin operators of the system and detector, respec- or inside the Bloch sphere can be reached, as it lies on one or tively. Then the interaction is switched off, and the de- several rotated ellipsoids (e.g. blue, green, yellow). tector state is measured projectively; disentangling the composite system-detector state and generating a mea- 2Ω sin θ(Ω cos θ sin φ − cos φ) surement back-action on the system state ρ (t). In our s sy = 2 2 , (5b) blind measurement protocol [17], the detector readouts 2 + Ω (cos θ + 1) are discarded (i.e., traced out). After each measurement 0 step, the detector state is reset to ρd. We note that the 2(1 + Ω2 cos2 θ) system-detector interaction is chosen to be anisotropic, sz = , (5c) 2 + Ω2(cos2 θ + 1) such that only the system’s spin direction orthogonal to the detector state initialization direction is coupled. where Ω = ω/α. The steady state coordinates In our dynamics, the Hamiltonian evolution (cf. Eq. (5)) form an ellipsoid centered at the point m/ˆ 2, (cf. Eq. (1)) and the measurement evolution (cf. Eq. (2)) 2 2 2 in this case sx +sy +2(sz −1/2) = 1/2; conversely, every happen simultaneously. Over a small time step dt, state on this ellipsoid is obtained for at least one choice of the two processes do not interfere with each other (up the Zeeman field (Ω, θ, φ) [Fig.1(a)]. The main features to O(dt)). An infinitesimal time step evolution of the of the steady state ellipsoid are as follows: (i) it remains −iHdt iHdt system is given by ρs(t+dt) = trd[e ρ(t)e ] where fully confined within the Bloch sphere, (ii) its shape re- H = Hs ⊗I+Hs−d. In the continuous time limit dt → 0, mains independent of the protocol parameters, (iii) its 2 and using a scaling of J such that J dt = const ≡ α, we minor axis starts from the center of the Bloch sphere have and ends on its surface, coinciding with the detector state dρ 1 initialization direction mˆ , implying that there exists only s = L[ρ ] = i tr [ρ(t),H] − tr [H, [H, ρ(t)]] dt. (3) dt s d 2 d one pure target state on a given ellipsoid. Each specific choice of protocol parameters (Ω, θ, φ) steers the system Here L is the Liouvillian superoperator acting on the to a unique steady state on this ellipsoid, but the con- system state, and we dropped the terms O[(dt)2] on the verse is not true as a given target state may be stabi- r.h.s. of the above equation. Using Eqs. (1,2,3), we obtain lized by several distinct sets (Ω, θ, φ). Furthermore, a given steady state may belong to several ellipsoids with dρ s = L[ρ ] = i [ρ ,H ]−2α L†Lρ + ρ L†L − 2Lρ L† , different mˆ , and in that case, the minor axis (which is dt s s s s s s (4) determined by mˆ ) of each of these ellipsoids must have where α specifies the measurement strength, and L = a fixed angle with regard to the Bloch vector s of the steady state (cf. Fig.1(b)). |mˆ +ihmˆ −| is the Lindblad jump operator with |mˆ ±i as the eigenstates of the operator mˆ ·σ with eigenvalues ±1. Rotating the detector state initialization direction mˆ , rotates the ellipsoid. Using all possible mˆ , the entire The combined unitary and weak-measurement time Bloch sphere (both surface and interior) can be covered evolution ultimately steer the system towards a steady by the ellipsoids, cf. Fig.1(b). Therefore, any state, irre- (T ) (T ) spective of its purity, can be targeted using our protocol. state determined by the condition dρs /dt = L[ρs ] = (T ) 1 Optimal steering.— Our aim now is to optimize the 0. We parameterize the steady state as ρs = 2 (I+s·σ) protocol such that the target state is reached as fast where s = (s , s , s ) is the steady state Bloch vector. (T ) x y z as possible. While the target state ρ corresponds Assuming a detector state initialization mˆ = (0, 0, 1), s to the eigenvector with zero eigenvalue of the Liouvil- this steady state is given by (T ) (T ) lian, L[ρs ] = λ0ρs with λ0 = 0, the dynamical evo- 2Ω sin θ(Ω cos θ cos φ + sin φ) lution of an arbitrary state is governed by the eigen- sx = , (5a) (j) 2 + Ω2(cos2 θ + 1) vectors ρs with nonzero eigenvalues λj, i.e. ρs(t) = 3

(T ) P (j) λj t ρs + j cjρs e where the coefficients cj are deter- L[ρs] = λ ρs and focus on the eigenvalue having small- mined by the system’s initial state ρs(0). Therefore, the est (in magnitude) real part which we maximize over all deviations from the target state decay exponentially in admissible values of Ω. We find that the conditions for time, and the decay rates are determined by the real optimal convergence depend on the degree of purity of the parts of the nonzero eigenvalues of the superoperator L. target state — we identify a low, medium and high pu- The smallest (in magnitude) non-zero real part, i.e. the rity regime, Fig.2(a). In the low purity regime, the con- inverse of the Liouvillian gap, determines the slowest vergence rate becomes bigger the further Ω is increased, convergence rate (Γ), and our aim is to maximize it by implying that steering becomes optimal by choosing Ω as choosing appropriate parameters in the protocol. At first large as possible, Fig.2(b). In the medium purity regime, sight, a straightforward way to speed up the steering pro- we find a critical Ω at which all three nonzero eigenvalues cess would be to increase the measurement strength α, of L have equal real part, and the convergence rate be- as the Liouvillian eigenvalues are directly proportional to comes optimal at this critical Ω, Fig.2(c). The transition α (cf. Eq. (6)). However, α = J 2dt is generically limited from the medium to the high purity regime is marked by by the weak measurement constraint (Jdt)2 1 and by the fact that at this critical Ω, not only the real parts, but the fact that the experimental measurement and read- also the imaginary parts of the three nonzero eigenvalues out time dt cannot be made arbitrarily short. Therefore, of L coincide — the convergence rate becomes optimal at we consider α to be fixed at some maximum strength. a third-order EP where all three nonzero eigenvalues of The protocol parameters that can still be optimized are L coincide, Fig.2(d). In the high purity regime, we find the initialization direction mˆ and the system Hamilto- optimal convergence for a critical Ω where L encounters a nian specified by Ω, θ and φ. These, however, are par- second-order EP, Fig.2(e). The optimal approach to the tially constrained by the choice of a specific target state. target state is oscillatory in the low and medium purity The optimization problem simplifies further, because the regimes, as two of the eigenvalues have nonzero imagi- eigenvalues of the superoperator L remain invariant un- nary parts for all values of Ω, Fig.2(b,c). By contrast, der unitary transformations. Since rotating the detec- the optimal approach to the target state becomes non- tor state initialization direction mˆ corresponds to such a oscillatory (exponential decay) in the high purity regime, unitary transformation [45] on L, the convergence rate because all three eigenvalues of the Liouvillian are real Γ for a given target state becomes independent of mˆ . at the optimal value of Ω, Fig.2(e). Analytical results Therefore, for simplicity, we choose mˆ = (0, 0, 1), for for the transitions between regimes and the optimal con- which the steady state is given by Eq. (5) and the corre- vergence rates are presented in the supplemental mate- sponding steady state ellipsoid is shown in Fig.1(a). For rial [45]. mˆ = (0, 0, 1), we can write the Liouvillian superoperator We highlight that the central feature of our sys- using Eq. (4) as tem is a phase transition at the target state purity P = 127/128 from Hamiltonian-dominated, oscillatory to  0 η η∗ 4  measurement-dominated, non-oscillatory dynamics. This  −η∗ −2(iΩ cos θ + 1) 0 η∗  transition proceeds through a third-order EP where all L = α   ,  −η 0 2(iΩ cos θ − 1) η  three nonzero eigenvalues of the Liouvillian superop- 0 −η −η∗ −4 erator coincide; the eigenvalues are obtained as [45]: (6) λ1 = λ2 = λ3 = −8α/3. While phase transitions have where η = i eiφΩ sin θ. Since L = V L(φ = 0)V † with been associated with second-order EPs, for instance in V = diag(1, e−iφ, eiφ, 1), the eigenvalues and hence the PT -symmetric systems [21, 42, 46], the natural appear- decay rates are independent of φ. Using the ellipsoid ance of a third-order EP seems striking. equation, the purity of a steady state can be expressed as To understand why optimal steering is related to a 2 P = 1 − (1 − sz) /2, also independent of φ (cf. Eq. (5c)). third-order EP, we now present a general principle for Hence, all target states with the same purity on a given optimization. To explain this principle, we first shift our ellipsoid have the same convergence rate. This leaves us perspective and ask: Comparing the target states in dif- with two significant parameters: θ and Ω. We treat Ω ferent regimes, what is the fastest possible convergence as an independent parameter, while θ is determined by rate that can be achieved? From Fig.2(a), we see that Ω and the target state purity P via Eq. (5c). For each the fastest convergence rate is realized in the medium pu- target state purity P , we aim to tune the free parameter rity regime. In fact, in this regime, the convergence rate Ω such that the convergence rate Γ becomes optimal. plateaus at an upper limit, which can be explained as However, we note that, for a given target state, there follows: It turns out that the average of the decay rates exists a lower bound on the allowed values of Ω, which is is independent of the system Hamiltonian, i.e. indepen- an important part of the optimization problem [45]. dent of Ω; we prove this fact below. In our protocol, this The optimal convergence rate and its dependence on fixed average is 8α/3, indicated by a horizontal line in the target state purity are obtained numerically, and are Figs.2(b)–(d). It is a general principle that if one de- shown in Fig.2(a). We solve the eigenvalue equation cay rate is above average, then another one is necessarily 4

Figure 2. Optimal convergence rate Γ in units of the measurement strength α as a function of target state purity P (a). We identify three regimes: A low purity regime 1/2 ≤ P ≤ 7/8, a medium purity regime 7/8 < P ≤ 127/128, and a high purity regime 127/128 < P ≤ 1. Choosing one target state from each different purity regime (marked by in (a)), we plot the real (solid) and imaginary (dashed) parts of the nonzero eigenvalues λ of the Liouvillian superoperator as a function of Ω in (b), (c) and (e). Here, Ω = ω/α is the ratio of the Zeeman energy ω of the system Hamiltonian, Eq. (1), to the measurement strength α, Eq. (4). Steering becomes optimal when the parameter Ω is tuned such that the topmost real part (solid) becomes as negative as possible (marked by ). At the transition from the medium to the high purity (marked by ? in (a)), the optimal convergence rate occurs at a third-order EP shown in (d). below average; therefore, the optimum convergence rate lying Hilbert space. We now claim that the trace “Tr” is is achieved when all decay rates are equal (to this av- independent of the system Hamiltonian: Since the trace erage). In our protocol, this is realized in the medium is linear and L contains the Hamiltonian in an additive purity regime, whereas in the other regimes, there is no way, it suffices to show that Tr([Hs, ·]) = 0. We can value of Ω for which all three decay rates become equal straightforwardly compute simultaneously. However, we conclude that steering is N X necessarily optimal at the third-order EP, because the Tr([H , ·]) = tr(e† [H , e ]) equality of all nonzero eigenvalues entails the equality of s ij s ij i,j=1 the decay rates. N X In the following, we show that the average of the Li- = (hi|Hs|ii − hj|Hs|ji) = 0. (7) ouvillian’s nonzero eigenvalues is indeed independent of i,j=1 the system Hamiltonian for all superoperators, not just for two-level systems. As argued above, the average This shows that the average convergence rate γ indeed sets an upper limit for the optimal convergence rate. depends only on the jump operators induced by repeated A general Liouvillian superoperator for an open quan- weak measurement. Experimental implementation.—Our protocol can be tum system has the Lindblad form L[ρs] = i[ρs,Hs] + implemented in a variety of experimental platforms. The P L†L ρ + ρ L†L − 2L ρ L† , where H is the sys- j j j s s j j j s j s main ingredient of our protocol, blind measurements sta- tem Hamiltonian and Lj are Lindblad jump operators. bilizing the system at a specific pure state, is particularly The average steering rate γ is the sum of the magnitudes natural for implementation in cold atomic systems such of the real parts of the eigenvalues divided by the total as cold ions [47] or Rydberg atoms [48], but is also sup- 1 PN 2 number of nonzero eigenvalues, γ = N 2−1 j=1 |Re λj|. ported by fluxonium qubits [49–51]. High-fidelity coher- Here N is the dimension of the underlying Hilbert space, ent Hamiltonian manipulation in these systems is now N 2 is the dimension of the linear space of density ma- routinely performed in many laboratories [52–54]. Com- trices, and we have subtracted one for the zero eigen- bining the ingredients and implementing our protocol value corresponding to the steady state. Since the real in these systems appears straightforward. Checking our parts of all the nonzero eigenvalues are negative, we can predictions for the steering optimality and its relation to take the real part of the total sum, and thus express EPs, however, will require a degree of control and sta- the average in terms of the sum over all eigenvalues, bility in the system beyond the standard levels, as the which is equal to the trace of the superoperator, ob- system sensitivity to perturbations in the vicinity of EPs taining γ = −Re (TrL) / N 2 − 1. Here, “Tr” denotes is expected to be enhanced [33, 35, 36, 55]. the trace of superoperators, which can be calculated by Conclusion.— We have proposed a steering proto- PN † TrL = i,j=1 tr(eij L[eij]) where “tr” denotes the ordi- col which uses the interplay of a unitary and a weak- nary trace of operators acting on the underlying Hilbert measurement-induced evolution to steer a two-level quan- space, and the eij = |iihj| are elementary matrices de- tum system towards any desired target state on or within rived from a set of basis states |1i,..., |Ni of the under- the Bloch sphere. The resulting Lindbladian dominates 5 the steering towards high purity states, while the Hamil- Naik, Ziwen Huang, Peter Groszkowski, Eliot Kapit, Jens tonian dynamics dominates the optimal steering towards Koch, and David I. Schuster, “Universal Stabilization of low purity mixed states. 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SUPPLEMENTARY INFORMATION

The main results of the manuscript “Optimized Quantum Steering and Exceptional Points” discuss the state steering of a two-level system towards an arbitrary desired target state and the optimization of the convergence rate. In the manuscript, the convergence rate optimization is linked to the exceptional points. Here we provide supplementary material for the results in the manuscript. In SectionI, we obtain the lower bound on allowed Ω for a ﬁxed target state. In SectionII, we present details on the optimal convergence rates in the three purity regimes.

I. LOWER BOUND ON Ω

In the manuscript, we discuss that there exist a range of Ω to choose from for a given target state, and we vary Ω to maximize the convergence rate. Here we will show that the allowed values of Ω required for a given target state has a lower bound. For a given detector state initialization direction mˆ = (0, 0, 1), the Bloch coordinates of the system’s steady state are given by

2Ω sin θ(Ω cos θ cos φ + sin φ) s = , (8) x 2 + Ω2(cos2 θ + 1) 2Ω sin θ(Ω cos θ sin φ − cos φ) s = , (9) y 2 + Ω2(cos2 θ + 1) 2(1 + Ω2 cos2 θ) s = , (10) z 2 + Ω2(cos2 θ + 1) where Ω = ω/α. We note that the steady state given by Eqs. (8,9,10) lies on an ellipsoid as

12 2(s2 + s2) + 4 s − = 1. (11) x y z 2 Using Eqs. (8) and (9), we get

(s + s tan φ) Ω = x y . (12) cos θ (sx tan φ − sy) Then from Eqs. (10) and (12), we get

2 2 2 2 2 (sx + sy tan φ) − (sx + sy) + 2(1 − sz)(sz + 2(1 − sz) sec φ) Ω = 2 . (13) (sx tan φ − sy) Therefore, for a given steady state, Ω becomes a function of φ only. Assuming Ω ≥ 0, we see that there exist only one extremum with respect to φ (which is a minimum), and the minimum value of Ω is given by

s 2(1 − sz) −1 sx Ωmin = at φ = φc = tan − . (14) sz sy

For a given steady state on the ellipsoid, the Bloch coordinate sz is related to the state purity P as

p sz = 1 − 2(1 − P ). (15) Therefore, from Eq. (14), we get

s 2p2(1 − P ) Ω = . (16) min 1 − p2(1 − P ) This implies that there exist a lower bound on the allowed values of Ω required to steer a system towards a desired target state and Ωmin increases as target state purity decreases. Since Ω = ω/α, this means one needs stronger Zeeman ﬁeld in order to generate a lower purity target state. 9

II. OPTIMAL CONVERGENCE RATES

In this section, we present details on the optimal convergence rates in the three purity regimes.

A. Eigenvalues of the Liouvillian superoperator

In this subsection, we discuss details on how to compute the eigenvalues of the Liouvillian superator. In the main manuscript, Eq. (4) defines the Liouvillian superoperator L of our system. The Liouvillian superoperators for two different detector state initialization directions, say mˆ and nˆ, are related by a unitary transform. Specifically, let U denote the unitary transform that rotates an eigenstate for one direction, |mˆ +i, to an eigenstate for the other direction, |nˆ+i = U|mˆ +i. Then, the Liouvillian superoperators for the two † † directions, denoted Lmˆ and Lnˆ , are related by Lnˆ [ρ] = ULmˆ [U ρU]U . This fact implies that the superoperators (j) (j) (j) † have the same eigenvalues, because if ρ is an eigenvector of Lmˆ , then ρ˜ = Uρ U is an eigenvector of Lnˆ with the same eigenvalue. Therefore, for simplicity, we choose mˆ = (0, 0, 1). For the detector state initialization direction mˆ = (0, 0, 1), the Liouvillian can be expressed as a 4 × 4-matrix acting on the space of density matrices:

 0 i eiφΩ sin θ −i e−iφΩ sin θ 4   i e−iφΩ sin θ −2(iΩ cos θ + 1) 0 −i e−iφΩ sin θ  L = α   . (17)  −i eiφΩ sin θ 0 2(iΩ cos θ − 1) i eiφΩ sin θ  0 −i eiφΩ sin θ i e−iφΩ sin θ −4

The eigenvalues of the superoperator are the zeros of the characteristic polynomial det(λI − L) of L. Using that the steady state corresponds to the eigenvalue λ = 0, we can simplify the characteristic polynomial and ﬁnd that the nonzero eigenvalues satisfy the polynomial equation

C(λ) = λ3 + 8αλ2 + 4α2 5 + Ω2 λ + 8α3 2 + Ω2(1 + cos2 θ) = 0. (18)

Substituting cos2 θ from Eq. (10), and rescaling λ as λ = α Λ, we can write Eq. (18) as

! 1 + Ω2 C(Λ) = Λ3 + 8Λ2 + 4(5 + Ω2)Λ + 16 = 0, (19) 1 + p2(1 − P ) p where we have used sz = 1 − 2(1 − P ). We note that the eigenvalues of L consist of isolated real numbers and pairs of complex conjugate numbers. This fact holds because the coeﬃcients of the characteristic polynomial are real quantities; thus, if λ is an eigenvalue of the Liouvillian superoperator, then its complex conjugate λ∗ is also an eigenvalue. More generally, this fact is a consequence of the fact that the time evolution preserves the Hermiticity of the density matrix, which implies that † † L[ρs] = (L[ρs]) .

B. Low purity regime

In the low purity regime, we ﬁnd numerically that one nonzero eigenvalue of the Liouvillian L is real, while the other two form a complex conjugate pair. The magnitude of the real eigenvalue increases monotonically as Ω increases. The real eigenvalue always corresponds to the slowest decay rate.

C. Medium purity regime

In this subsection, we discuss the medium purity regime and show that there is a critical value Ω where all three nonzero eigenvalues of the Liouvillian superoperator have equal real parts, and that the convergence rate becomes optimal at this value. 10

We ﬁnd numerically that one nonzero eigenvalue of the Liouvillian L is real, while the other two eigenvalues form a complex conjugate pair. Furthermore, the magnitude of the real eigenvalues increases monotonically, whereas the magnitude of the real parts of the complex eigenvalues decreases monotonically. In contrast to the low purity regime, the three real parts of the eigenvalues do become equal for some critical value of Ω in the medium purity regime. To ﬁnd this critical value analytically, we make the ansatz

Λ1 = a, Λ2 = a + i b, Λ3 = a − i b (20) for the eigenvalues, where a denotes the common real part, and b denotes the imaginary part of the complex conjugate pair. Then, the characteristic polynomial can be factored as

C(Λ) = (Λ − a)(Λ − a − i b)(Λ − a + i b). (21)

Expanding this expression and comparing it with Eq. (19), we can solve for a, b and Ω, obtaining

s s 8 4 8p2(1 − P ) − 1 26p2(1 − P ) − 1 a = − , b = ± , Ω = ± . (22) 3 3 1 − 2p2(1 − P ) 9(1 − 2p2(1 − P ))

Since the convergence rate is defined as the magnitude of the real part of the eigenvalues, we obtain Γ = 8α/3 for the convergence rate at this value of Ω. We have argued in the main text that the equality of the real parts implies that this convergence rate is optimal. The range of the medium purity regime is determined by the constraint that the quantities b and Ω have to be real. We find that this is satisfied precisely for 7/8 < P ≤ 127/128. At P = 127/128, we have b = 0, i. e. all three nonzero eigenvalues becomes purely real and equal to each other, implying that the Liouvillian superoperator features a third-order exceptional point which marks the onset of the high purity regime.

D. High purity regime

In the high purity regime, we ﬁnd numerically that the optimal convergence rate occurs at a critical value of Ω where the Liouvillian superoperator has a second-order exceptional point, that is where two of its eigenvalues coincide. To compute the exceptional points analytically, we follow a procedure similar to the procedure in the previous subsection. Speciﬁcally, we make the ansatz

Λ1 = Λ2 = a, Λ3 = b (23) for the eigenvalues of the Liouvillian L. Since the eigenvalues are either real or have complex conjugate pairs, the equality of two of them implies that a and b are real quantities. The characteristic equation can be factored as

C(Λ) = (Λ − a)2(Λ − b) = 0. (24)

Expanding this expression and comparing with Eq. (19), we obtain two exceptional points:

a = a+, b = b− Ω = Ω+ (25) and

a = a−, b = b+, Ω = Ω−, (26) where 11

q −3 ± 1 − 8p2(1 − P ) a± = , (27) 1 + p2(1 − P ) q 2 −1 − 4p2(1 − P ) ± 1 − 8p2(1 − P )

b± = , (28) 1 + p2(1 − P ) v u 3/2 u p p u16 2(1 − P ) + 20P − 21 ± 1 − 8 2(1 − P ) Ω = u . (29) ± t 2 2 1 + p2(1 − P )

However, we note that for 161/162 < P < 1, the exceptional point at Ω− falls outside the admissible range for Ω computed in SectionI, that is Ω− < Ωmin. Numerically, we ﬁnd that the convergence rate becomes optimal at Ω = Ω+, and is given by |a+|.