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 opti- mization 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 oscilla- tory 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 = ! n^ · σ; n^ = (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 ! is the Zeeman energy of the two levels, θ, φ 2 are spherical coordinates parametrizing the unit vector n^, 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 ^ · σ), where m^ 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 ^ · σ )(m ^ · σ ) ; (2) initialization direction m^ = (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 = !/α. The steady state coordinates In our dynamics, the Hamiltonian evolution (cf. Eq. (5)) form an ellipsoid centered at the point m=^ 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 m^ , 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 LyLρ + ρ LyL − 2Lρ L† ; different m^ , and in that case, the minor axis (which is dt s s s s s s (4) determined by m^ ) 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)). jm^ +ihm^ −j is the Lindblad jump operator with jm^ ±i as the eigenstates of the operator m^ ·σ with eigenvalues ±1. Rotating the detector state initialization direction m^ , rotates the ellipsoid.