PHYSICAL REVIEW RESEARCH 2, 033347 (2020)

Measurement-induced steering of quantum systems

Sthitadhi Roy ,1,2,* J. T. Chalker ,1,† I. V. Gornyi ,3,4,‡ and Yuval Gefen5,§ 1Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Oxford University, Parks Road, Oxford OX1 3PU, United Kingdom 2Physical and Theoretical Chemistry, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom 3Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany 4A. F. Ioffe Physico-Technical Institute of the Russian Academy of Sciences, 194021 St. Petersburg, Russia 5Department of Condensed Matter Physics, Weizmann Institute of Science, 7610001 Rehovot, Israel

(Received 7 January 2020; revised 6 August 2020; accepted 10 August 2020; published 1 September 2020)

We set out a general protocol for steering the state of a quantum system from an arbitrary initial state toward a chosen target state by coupling it to auxiliary quantum degrees of freedom. The protocol requires multiple repetitions of an elementary step: During each step, the system evolves for a fixed time while coupled to auxiliary degrees of freedom (which we term “detector qubits”) that have been prepared in a specified initial state. The detectors are discarded at the end of the step, or equivalently, their state is determined by a projective measurement with an unbiased average over all outcomes. The steering harnesses backaction of the detector qubits on the system, arising from entanglement generated during the coupled evolution. We establish principles for the design of the system-detector coupling that ensure steering of a desired form. We illustrate our general ideas using both few-body examples (including a pair of spins-1/2 steered to the singlet state) and a many-body example (a spin-1 chain steered to the Affleck-Kennedy-Lieb-Tasaki state). We study the continuous time limit in our approach and discuss similarities to (and differences from) drive-and-dissipation protocols for quantum state engineering. Our protocols are amenable to implementations using present-day technology. Obvious extensions of our analysis include engineering of other many-body phases in one and higher spatial dimensions, adiabatic manipulations of the target states, and the incorporation of active error correction steps.

DOI: 10.1103/PhysRevResearch.2.033347

I. INTRODUCTION rapidly toward zero with increasing system size. With the second approach, the temperature scale required to completely There are many circumstances in which one would like to eliminate thermal excitations from a system also decreases initialize a many-body quantum system in a specified state: toward zero with size. In principle, if the initial state of the Examples of current interest range from quantum information system is precisely known, a further possibility is to act on the processing to studies of nonequilibrium dynamics. Two stan- state with an appropriately chosen perturbation for a precise dard approaches for preparing quantum states are suggested interval of time, so that it evolves into the target state; this, by the laws of quantum mechanics and statistical mechanics. however, requires extreme fine-tuning in a large system. One of these is to make projective measurements of a set of Our aim in the following is to establish a class of protocols observables represented by commuting operators that fully for quantum state preparation that improve on both projective specify the target state. Alternatively, if the target state is measurement and thermal contact with a heat bath. A protocol the ground state of the Hamiltonian for the system, it can of the type we describe will, in an ideal implementation, be reached by putting the system in thermal contact with steer a system from an arbitrary initial state to the target a heat bath that is at a sufficiently low temperature. Both state, with guaranteed success. It does so by coupling the approaches have disadvantages, especially for a system with quantum system of interest to external detector qubits (auxil- a large number of degrees of freedom: In the first approach, iary quantum degrees of freedom) that have been prepared in a general initial state is not an eigenstate of the measurement specified initial states, then evolving the coupled system under operator and hence the measurement outcome is probabilistic. standard unitary quantum dynamics for a fixed time interval, The probability that the target state is reached decreases and finally decoupling and discarding the detector qubits. Multiple repetitions of this process using freshly prepared detector qubits on each occasion, coupled to the system with *[email protected] a suitable interaction Hamiltonian, produce the outcome we †[email protected] ‡ require. In this paper, we illustrate our general approach both [email protected] / §[email protected] for small systems consisting of one or two spin-1 2 degrees of freedom and for a macroscopic many-body system. Published by the American Physical Society under the terms of the This protocol and possible generalizations have the fun- Creative Commons Attribution 4.0 International license. Further damental concepts of quantum entanglement and quantum distribution of this work must maintain attribution to the author(s) measurements as essential ingredients and possess conceptual and the published article’s title, journal citation, and DOI. links to the theory of open quantum systems. First, the effect

2643-1564/2020/2(3)/033347(20) 033347-1 Published by the American Physical Society ROY, CHALKER, GORNYI, AND GEFEN PHYSICAL REVIEW RESEARCH 2, 033347 (2020) of a coupling between the system and detector qubits is to time-continuum limit maps the dynamics of the density matrix generate entanglement between their states. Second, focusing of the system to a Lindblad equation. For part of the analysis on the behavior of the system after discarding the detector in our work, we exploit this formal connection between our qubits, this entanglement induces steering of its quantum steering protocol and Lindblad dynamics to gain insight into state, of a kind first discussed by Schrödinger in the context of the steady state and the rate of approach to it. the Einstein-Podolsky-Rosen (EPR) paradox [1,2]. A physical The emergence of Lindblad dynamics suggests a compar- setting for the decoupling and discarding of the detectors is ison of our protocol with proposals for preparing nontriv- that of performing strong measurements on these qubits and ial many-body states in open quantum systems, where the then taking an unbiased average over the outcomes. Indeed, dissipative environment is posited to be Markovian and is Schrödinger’s original formulation of steering [3]wasin treated within the framework of Lindblad dynamics [17–26]. terms of a sophisticated experimenter performing suitable The challenge in that context is to find suitable Lindblad measurements on one of the parts of a bipartite system to jump operators that can be cast in terms of physical systems drive the other part to a state, chosen by the experimenter, participating in the dynamical process. Our measurement- with nonzero probability. More recently, the scope of the term induced steering protocol has some technical similarities with quantum steering has been narrowed to the impossibility of this framework, as well as conceptual distinctions from it. local hidden-state models describing the ensemble of states With regard to the former, if the detector qubits we discuss that a system can take upon measurement of another system are viewed as an environment, this environment is Markovian entangled with it [4–6]. We, however, continue to use the term by construction, since the detector qubits are prepared afresh in its broader sense. at the start of every cycle. Hence, the map governing the To set out in more detail the links between our protocol evolution of the density matrix of the system has a rep- for preparing or steering a quantum state and discussions of resentation in terms of Kraus operators, which ultimately quantum measurement, it is useful to recall the distinction leads to a Lindblad equation in the time-continuum limit between strong or projective measurements and the notion of [27,28]. weak quantum measurements [7–13]. The former destroy the An obvious advantage of our protocol is that the jump coherent quantum dynamics of the system. By contrast, under operators in the emergent Lindblad equation are uniquely and the latter, auxiliary degrees of freedom are coupled weakly automatically fixed by the details of the system-detector cou- to the system and projective measurements are performed on pling Hamiltonian, thus facilitating “on-demand” engineering these detector qubits after decoupling them from the system. of Lindbladians. On the other hand, since our protocol is No matter how weak the coupling, a measurement of this fundamentally a microscopic measurement protocol, certain kind unavoidably impacts the system through its backaction, requirements can easily be relaxed to go far beyond what is even after the detector qubits are decoupled from the system. describable within the standard Lindblad framework. We note Instead of viewing the measurement-induced disturbance on earlier works in the context of quantum control, discussing the system’s state as a handicap, it may be considered a re- manipulation of quantum states via repeated interactions with source for quantum control and manipulation. The backaction designed environments [23,29,30]; much of this, however, has of measuring the state of detectors entangled with the system focused on the formal aspects of the theory or on applications can be harnessed to control the system’s evolution and steer restricted to few-body systems. its state toward a desired target state: This constitutes the Another advantage of a measurement-based protocol over central message of our work. In principle, one could make coupling with an environment is that, whereas the latter sim- use of the measurement outcomes to determine whether the ply gets entangled with the system, a detector qubit can be state of the system has been successfully steered in a realiza- read out, yielding information on the system state which could tion of the experiment and discard it if this is not the case. In be further used to accelerate convergence to the desired state. the simplest version of such postselected protocols, the proba- Obvious examples include the use of postselected protocols bility of success then goes down exponentially with the length or using the measurement outcomes to implement active of the outcome sequence or the number of detectors. This error correction and enhance the blind measurement protocols suppression is also likely to be exponentially strong in system [31,32]. size for a many-body system. To avoid such suppression, To summarize, our measurement-based steering protocol our aim is to establish protocols within a setting where the differs fundamentally from and has various advantages over outcomes are averaged over in an unbiased fashion, and hence related protocols for manipulating and stabilizing nontrivial there is no loss of probability. We refer to processes of this quantum states. First, the measurements in our case need not type as blind measurements or nonselective measurements. be weak: The detectors may be strongly coupled to the system. The concept of weak measurements allows for the no- Second, in an appropriate time-continuum limit, the dynamics tion of continuous measurements [14–16]. The latter can be is described by a Lindblad equation, but rather than introduc- viewed as a discrete sequence of weak measurements in the ing a Markovian reservoir or jump operators “by hand,” bath limit of vanishing time interval for each weak measurement. coupling is engineered by the measurement operators, which This results in a sequence of measurement outcomes defining fix the jump operators uniquely. Third, unlike unstructured en- a quantum trajectory. The equation for the density matrix of vironments or baths, the detector qubits can be read out, with the system averaged over the trajectories is governed by a the outcomes exploited to further the efficiency and scope of Lindblad equation. In our case, the steering protocol consists our steering on platforms employing postselected protocols or of a discrete sequence of measurements on detector qubits not using active decision-making protocols based on the readout necessarily coupled weakly to the system. A suitably defined sequence. Finally, we note the nontriviality of applying our

033347-2 MEASUREMENT-INDUCED STEERING OF QUANTUM … PHYSICAL REVIEW RESEARCH 2, 033347 (2020) measurement-based steering to many-body systems using an Equations (1) and (2) describe the discrete time-evolution extensive number of auxiliary degrees of freedom. map for the density matrix of the system under the steering dynamics. Structure of the paper (iv) The detector qubits are reprepared in their initial states and the above steps are repeated. The rest of the paper is structured as follows. We start with an overview in Sec. II, where we introduce the basic ingredients of the steering protocol and state the main results B. Steering inequalities of the work. Section III presents the details of the steering As we would like the system to get steered toward the | protocol. The guiding principle for designing the protocol target state, denoted by s⊕ (with corresponding density ρ is introduced and derived in Sec. III A,followedbythe matrix s⊕ ), the dynamics induced by Hs-d should satisfy the derivation of the effective Lindblad dynamics in Sec. III B. steering inequality The workings and advantage of the approach is exemplified  |ρ (t + δt )|   |ρ (t )| , ∀ t, (3) via the simple case of a spin-1/2 pair steered toward their s⊕ s s⊕ s⊕ s s⊕ singlet state in Sec. III C. We then turn to a many-body with the inequality ideally becoming an equality only if ρ = ρ quantum system, a spin-1 chain, which we steer to the Affleck- s(t ) s⊕ . Kennedy-Lieb-Tasaki (AKLT) ground state [33,34] in Sec. IV. Note that this inequality is very strong. When combined As a building block of the protocol, the steering of a spin-1 with the condition on equality, they ensure that the system pair is discussed in Sec. IV A, followed by the numerical is steered to the target state irrespective of its initial state. treatment of the spin-1 chain in Sec. IV B. We close with In principle, one could envisage weaker steering inequalities. discussions of possible experimental implementations of the One example is protocol and future directions in Sec. V.  |ρ | = lim s⊕ s(t ) s⊕ 1(4) t→∞ II. OVERVIEW and a still weaker one is  |ρ | >  |ρ | . The central result of this work is a general formalism lim s⊕ s(t ) s⊕ s⊕ s(0) s⊕ (5) t→∞ for measurement-induced steering of a many-body quantum system toward a nontrivial target state, by repeatedly coupling We present in Sec. III A a general strategy for designing and decoupling a set of auxiliary degrees of freedom inter- system-detector couplings so that the strongest of these forms, spersed with unitary dynamics of the composite system. The (3), holds. protocol makes use of the entanglement generated between the system and the auxiliary degrees of freedom to steer the C. Guiding principle for choice of system-detector state. The auxiliary degrees of freedom are simple quantum coupling Hamiltonian systems with small Hilbert-space dimensions such that they With the protocol fixed, it remains to find a guiding are easy to prepare in a desired initial state; in this work, principle for the choice of the Hamiltonian that governs the / we consider a set of decoupled detector qubits (spins-1 2) evolution of the system-detector composite. As a first step, initially polarized along a given direction. consider summing both side of (3) over ρs(t ) representing all − δ possible pure states. If (3) is obeyed, then W ≡ e iHs-d t must A. Steering protocol satisfy ρ ⊗ † ⊗ ρ  ρ ⊗ ρ . The steering protocol can be described as a sequence of Trs,d[W ( d 1s)W (1d s⊕ )] Trs,d[ d s⊕ ] (6) discrete steering events each of which consists of the follow- ing steps: In the special case where the Hilbert spaces of the system H H (i) The detector qubits are prepared in a initial given state, ( s) and the detector ( d) have the same dimension, we see which does not depend on the state of the system. We denote from this that for the inequality to be as strong as possible, W ρ the initial state of the detector qubits as | and the density should be the swap operator between these two spaces and d d ρ should be the image of s⊕ under this swap. More generally, matrix corresponding to this initial state as ρd. (ii) The system is then coupled to the detector qubits and a Hamiltonian which contains direct products of operators (n) H the composite system evolves unitarily under a Hamiltonian Od in the detectors’ subspace d that rotate the detectors from their initial state to an orthogonal subspace and operators for some time. Denoting the density matrix of the system at (n) U in the system’s subspace Hs that rotate the system to the time t as ρs(t ) and the system-detector Hamiltonian as Hs-d, s the state of the system-detector composite after evolution for target state manifold from an orthogonal subspace will steer an interval δt is given by the system toward the target state. Such a Hamiltonian has the form − δ δ ρ + δ = iHs-d t ρ ⊗ ρ iHs-d t . s-d(t t ) e d s(t )e (1) = (n) |  | ⊗ (n) + . ., Hs-d Od d d Us H c (7) (iii) The detector qubits are then decoupled from the n system. Formally, we may trace out the detector degrees of (n) where n labels the detector qubit. Since O connects the state freedom to obtain the density matrix of the system at time d |  |O(n)| = t + δt, d to its orthogonal subspace, it satisfies d d d 0. Additionally, we assert the following properties for the system ρs(t + δt ) = Trdρs-d(t + δt ). (2) operators:

033347-3 ROY, CHALKER, GORNYI, AND GEFEN PHYSICAL REVIEW RESEARCH 2, 033347 (2020)

(n) (n) | = (i) Us annihilates the target state, Us s⊕ 0. detector qubit, there is nothing special about this choice; see (n) (n) (n)† | =| (ii) Us is normalized such that Us Us s⊕ s⊕ . Appendix A. (n) (m) (iii) For different n and m,ifUs and Us share spatial support then U (m)U (n)† = 0. If they do not share a spatial s s E. Many-body states and multiple detectors support, they automatically commute. To ensure that equality in Eq. (3) always implies ρs(t ) = In the context of many-body systems, the subspace orthog- ρ | s⊕ it is necessary that s⊕ is coupled to every state in its onal to the target is exponentially large in the system size. (n)† complement by at least one operator Us . In our study of Hence it may appear that the system-detector coupling Hamil- steering to the AKLT state, we observe guaranteed conver- tonian of the form in Eq. (7) entails an exponentially large { (n)†} gence to the target state despite the set of operators Us number of detectors, or nonlocal system-detector coupling not connecting the AKLT state to all the other states in the Hamiltonians, or both. For implementation of the protocol Hilbert space. However, monotonicity is not ensured and, in to be feasible, we would like to have at most an extensive principle, can depend on the particular measure of distance number of detectors, with couplings that are local. Although from the target state. We discuss this issue in detail in Secs. these constraints appear rather restrictive, they can be satisfied III A and IV B and Appendix D. for versions of our protocol that allow steering to a large class of strongly correlated quantum states which are eigenstates D. Toy example with a spin-1/2 of unfrustrated local projector Hamiltonians. One can steer to such states by locally steering different parts of the system to / As a simple example, consider a single spin-1 2 which we the appropriate eigenstate of the corresponding local projector wish to steer to the fully polarized state along the z direction, part of the Hamiltonian. Since the effective Hilbert dimension | =|↑ | so that s⊕ s . The subspace orthogonal to s⊕ has of a local part of the system is finite, one can steer locally † | =|↓ only a single state: Us s⊕ s . Hence, a single detector with a finite number of detectors, and hence the total number qubit is sufficient to steer the state. Without loss of generality, of detectors required scales linearly with system size, and | =|↑ let us choose it to be initialized as d d . Naturally, not exponentially. Moreover, the couplings are manifestly | =|↓ we also have Od d d .FollowingEq.(7), the system- local. detector coupling Hamiltonian can then be written as The particular example we consider in detail is the Affleck- Kennedy-Lieb-Tasaki (AKLT) state of a one-dimensional Hs-d = J(|↓d↑d|) ⊗ (|↑s↓s|) + H.c. (8) spin-1 chain [33,34]. The AKLT state is a valence bond state = σ −σ + + σ +σ − , J( d s d s ) such that on each bond between two neighboring spins-1, σ ± there is no projection on the total spin-2 sector. Thus, one where s(d) denote the Pauli raising and lowering operators for the system and detector qubits. can steer the spin-1 chain by locally steering each bond out At time t, the state of the system can be generally written of the total spin-2 sector, and the uniqueness of the ground state of the AKLT chain (with periodic boundary conditions) as ρs(t ) = [I2 + s(t ) · σs]/2, where s(t ) is a classical three- component vector. The state of the system spin-1/2 at time guarantees steering to the AKLT state. Steering each bond t + δt can then be obtained by evolving the combined state requires only a finite number of detectors due to the finite + σ z ⊗ ρ / dimension of the total spin-2 subspace for a pair of spins-1 of the system and the detector, (I2 d ) s(t ) 2, with the unitary operator W , and tracing over the detector. This yields and hence the total number of detectors needed to steer a chain the relation for s(t + δt ) is only extensive in system size. Further examples of states which satisfy the above criteria t δ sx (t + δt ) = cos(Jδt )sx (t ) = cos t (Jδt )sx (0), include matrix product states, projected pair entangled states,

t the Laughlin state of a fractional quantum Hall system [36], + δ = δ = δt δ , sy(t t ) cos(J t )sy(t ) cos (J t )sy(0) the ground state of Kitaev’s toric code [37], and of recent 2 2 interest, fermionic symmetry-protected topologically ordered sz(t + δt ) = 1 − cos (Jδt ) + cos (Jδt )sz(t ) eigenstates of full commuting projector Hamiltonians [38,39]. 2t 2t δ δ = 1 − cos t (Jδt ) + cos t (Jδt )sz(0). (9) An obvious potential concern arises when our protocol is used to steer many-body systems to eigenstates of local The above equation explicitly shows that projector Hamiltonians, because in the cases of interest a lim s(t ) = (0, 0, 1), (10) given quantum degree of freedom appears in more than one t→∞ projector. In particular, in such a scenario it is not guaranteed (n) (m) irrespective of the initial conditions and that the limit is that two system operators Us and Us acting on such a approached exponentially in time. shared degree of freedom, while steering their corresponding This example illustrates how a system-detector coupling parts of the system locally, satisfy the conditions listed at Hamiltonian of the form in Eq. (7) leads to the satisfaction the end of Sec. II C. Steering toward the target space of one of the desired steering inequality, Eq. (3), for generic initial projector may therefore undo, at least partially, the effect states of the system. This guarantees successful steering to the of steering to the target space of a different projector with target state from arbitrary initial states. In turn, the example which the degree of freedom in question is shared. Despite shows how weak measurements can produce control and these complications, we show that our approach enables us to manipulation that projective measurements cannot [35]. select zero-energy eigenstates of local projector Hamiltonians We remark that although in the above example the target as target states and unique stationary states of our dynamics. state of the system qubit is the same as the initial state of the We illustrate this numerically for the AKLT chain, showing

033347-4 MEASUREMENT-INDUCED STEERING OF QUANTUM … PHYSICAL REVIEW RESEARCH 2, 033347 (2020) steering to the ground state with guaranteed success from We next show that the system is steered toward the target arbitrary initial states. state at every discrete steering event. For brevity, we show here the derivation with a single detector qubit and state the results for multiple detector qubits. Details of the derivation F. Steady states and rate of approach for the latter are presented in Appendix B. Equations (1) and (2) show that the time evolution of the To proceed, we find it most convenient to partition the density matrix of the system is governed by a linear map. composite Hilbert space of the detectors and the system Hd ⊗ On general grounds, the largest eigenvalue magnitude for this Hs into two subspaces, which we denote as Dd ⊗ Hs and map is unity. If the associated eigenvector is unique, this Dd ⊗ Hs, where Dd is the one-dimensional subspace spanned ensures that there exists a unique steady state. The logarithm by |d and Dd ⊕ Dd = Hd. In this convention, the density of the next eigenvalue then encodes the rate of approach in matrix ρd ⊗ ρs(t ) can be represented as time to this steady state. Ideally, we would like the rate to D D d d be finite in the thermodynamic limit so that the timescale }D ρ (t ) 0 d required to steer the many-body state arbitrarily close to the ρ ⊗ ρ (t )= s . (12) d s 00 target state does not diverge with system size. }Dd In the context of the AKLT state, we find that the magni- tude of these eigenvalues is most conveniently studied in the Additionally, the system-detector coupling Hamiltonian, time-continuum limit (δt → 0) of the map in Eq. (2) where Eq. (7), takes the form D D an effective Lindblad equation emerges for the equation of d d ρ }D motion of s with the jump operators being fixed by the 0 U † d system-detectors coupling Hamiltonian H . Unit eigenvalue Hs-d = . (13) s-d U 0 }D for the discrete map corresponds to zero eigenvalue for the d Lindbladian operator. If the corresponding Lindbladian eigen- The unitary time-evolution operator generated by the system- vector is unique, it is the stationary state of the steering detector coupling Hamiltonian is protocol. Successful steering implies that the steady state of ∞ − δ 2k the Lindbladian is the target state. Moreover, the gap in the −iH δt ( i t ) e s-d = spectrum of the Lindbladian between the zero eigenvalue (2k)! k=0 and the rest of the eigenvalues determines the rate of the of − δ approach of the system to the target state. In the case of the (U †U )k i t U †(UU†)k × 2k+1 . (14) AKLT state, analysis of the size dependence of the numer- −iδt † k † k + U (U U ) (UU ) ically obtained gap in the effective Lindbladian’s spectrum 2k 1 indicates that it stays finite in the thermodynamic limit. Following the steering map of Eq. (2), applying the above unitary matrix to the density matrix of Eq. (12), and taking the trace of the detector, one obtains the time-evolved density III. FORMALISM OF QUANTUM STATE STEERING matrix of the system as In this section, we describe the quantum steering protocol (−iδt )2k (iδt )2l in detail. We show explicitly that a system-detector coupling ρ (t + δt ) = [(U †U )kρ (t )(U †U )l ] s (2k)!(2l)! s Hamiltonian of the form given in Eq. (7) leads to dynamics k,l that satisfy the steering inequality, Eq. (3), on rather general (−iδt )2k+1(iδt )2l+1 grounds. Additionally, we make the connection to an effective + (2k + 1)!(2l + 1)! Lindblad equation that describes the time-continuum limit of k,l the steering map, Eq. (2). † k † † l × [U (U U ) ρs(t )U (UU ) ]. (15) | = † | =| A. General derivation of steering inequality Using Us s⊕ 0 and UsUs s⊕ s⊕ , the diagonal ma- from discrete time map trix element of the time-evolved density matrix corresponding to the target state can be obtained from Eq. (15)as Let us discuss in some generality the map governing the  |ρ + δ | discrete time evolution of the system density matrix, Eq. (2), s⊕ s(t t ) s⊕ with the system-detector coupling Hamiltonian Hs-d of Eq. (7). = |ρ | + | ρ †| 2 δ . ρ = ρ =|  | s⊕ s(t ) s⊕ s⊕ Us s(t )Us s⊕ sin ( t ) We will first show that s s⊕ s⊕ s⊕ is indeed a stationary state of the map and then show that the system is Q ρ progressively steered toward the s⊕ after each steering event. (16) That ρ is a stationary state of the map is easily shown by s⊕ Note Q is a diagonal matrix element of a valid density matrix noting that with H from Eq. (7), H (ρ ⊗ ρ ) = 0, which s-d s-d d s⊕ ρ (t ) which ensures that it is non-negative. The change in the naturally implies s diagonal element of the density matrix corresponding to the − δ δ e iHs-d t ρ ⊗ ρ eiHs-d t = ρ ⊗ ρ . (11) target state is proportional to the support of the density matrix d s⊕ d s⊕ | on the subspace orthogonal to s⊕ . Hence, the inequality Upon taking the trace over the detectors, one recovers that in Eq. (3) becomes an equality when ρs(t ) has no remaining − δ δ iHs-d t ρ ⊗ ρ iHs-d t = ρ ρ | Trd[e d s⊕ e ] s⊕ and hence s⊕ is a station- support on the subspace orthogonal to s⊕ —in other words, ary state of the map. ρs(t ) has reached its target form. This concludes the derivation

033347-5 ROY, CHALKER, GORNYI, AND GEFEN PHYSICAL REVIEW RESEARCH 2, 033347 (2020) of the steering inequality, Eq. (3), which in turn shows that the for the density matrix of the form system is steered toward the target state progressively at each ∂t ρs = L[ρs] steering event. In the case of multiple detectors, the equation for the time- 1 =−i[H ,ρ ] + L ρ L† − {L†L ,ρ } , (20) evolved density matrix has the form as in Eq. (16)butQ is s s i s i 2 i i s given by (see Appendix B for derivation) i where H is the intrinsic Hamiltonian of the system and s the L sarethequantum jump operators. They are fixed in Q =  |U (n)ρ (t )U (n)†| , (17) i s⊕ s s s s⊕ our derivation by the form of the system-detector coupling n Hamiltonian. As in Sec. III A, we sketch the derivation with a single which is again a sum of diagonal elements of a density matrix detector and state the result for multiple detectors. To proceed and hence is non-negative. Also, it is the total support of the with the derivation, we again partition the composite Hilbert | density matrix on the subspace orthogonal to s⊕ spanned space and represent ρ ⊗ ρ (t )asin(12). The general system- { (n)† | } | d s by the set of states Us s⊕ . Provided s⊕ is connected (n)† detector coupling Hamiltonian can be represented by a matrix to every state in its complement by an operator Us , then of the form = ρ = ρ Q 0 implies s(t ) s⊕ . D D For a many-body system, if a sufficiently large number d d }D of operators is employed that the target state is connected to V U † d H = . (21) all others by the operators U (n)†, monotonic convergence to s-d UV s }Dd the target state is straightforwardly guaranteed. The required number of operators may, however, be exponentially large in Expanding the map in Eq. (2) to second order in δt one obtains system size. By contrast, we are interested in steering a many- ρ (t + δt ) − ρ (t ) 1 s s = i V,ρ t − V, V,ρ t δt body system using only an extensive number of operators, δ [ s( )] [ [ s( )]] each coupled locally to the system. It can then happen that t 2 = ρ 1 Q 0forsome s(t ) at a given step, but that further steps + Uρ (t )U † − {U †U,ρ (t )} δt. successfully steer the system to the target. In Sec. IV B,we s 2 s show how in a such a setting with extensive, local steering (22) operators a spin-1 chain is steered uniquely to the AKLT √ state. In this case, we also examine whether or not steering Taking the limit δt → 0 with U˜ = U δt while requiring is monotonic. We show that this may depend on the choices ||V || ∼ O(1) and ||U˜ || ∼ O(1) [40], one obtains a time- of initial state and of the measure of the distance between the continuum equation of motion time-evolving state and the target state; see Appendix D for ∂ ρ =− ,ρ + ˜ ρ ˜ † − 1 { ˜ † ˜ ,ρ } . t s(t ) i[V s(t )] U s(t )U 2 U U s(t ) details. We will quantify the distance of ρs(t ) from the target ρ (23) state, s⊕ , via the Frobenius and trace norms, denoted as DF and D1 respectively. The are defined as Comparing with Eq. (20), it is trivial to see that the equation ρ of motion for s is indeed of the Lindblad form with the jump operator given by U˜ . The general Hamiltonian in Eq. (21) D (t ) = Tr[ρ (t ) − ρ ]2, (18a) F s s⊕ reduces to the specific one in Eq. (7) with the choice of operators V = V = 0. Hence, the effective Lindblad equation = ρ − ρ 2 / . D1(t ) Tr[ ( s(t ) s⊕ ) ] 2 (18b) which describes the dynamics of the steering protocol in the time-continuum limit consists only of the jump operators. In Additionally, since the AKLT ground state is an unique the case of multiple detectors, the Lindblad equation obtained zero-energy ground state of the AKLT Hamiltonian (with has multiple jump operators periodic boundary conditions), another good measure for the † 1 † distance is the energy of the state measured with respect to the ∂t ρs =−i[V,ρs(t )] + U˜nρs(t )U˜ − {U˜ U˜n,ρs(t )} . n 2 n AKLT Hamiltonian n (24)

EAKLT(t ) = Tr[HAKLTρs(t )]. (19) A few comments regarding the Lindblad equation are in order which will eventually prove useful in the analysis of the concrete examples we discuss. Note that the Lindblad ρ D B. Effective Lindblad dynamics equation is linear in s. Hence, if the Hs -dimensional density matrix ρs(t ) is unravelled as a supervector, ρs(t ), of dimension We now show that the time-continuum limit of the map in 2 DH , the time evolution is simply generated by the superoper- Eq. (2) leads to a Lindblad equation for the dynamics of the s ator corresponding to the Lindbladian density matrix of the system. The exposition of the conceptual ←→ connection between measurement-induced quantum steering ρ (t ) = exp( L t ) · ρ (0). (25) and the dynamics being described by the Lindblad equation, s s often associated with dissipative dynamics, is an important The steady-state manifold of the dynamics is then given by result of this work. The Lindblad equation is a master equation the manifold of states corresponding to zero eigenvalues of

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←→ L Jt =0.0 Jt =0.1 Jt =1.0 Jt =10.0 . Hence, the uniqueness of the steady state can be deter- 1 mined by studying the degeneracy of the zero eigenvalues. S0 (a1) (a2) (a3) (a4) ←→ T Moreover, the gap in the spectrum of L between the zero + eigenvalue and the rest of the eigenvalues, {v}, defined as T0 T  = minv=0{|Re[v]|} determines the nature of the approach − 0 to the steady state. For a finite  in the thermodynamic limit, S0 T+ T0 T− S0 T+ T0 T− S0 T+ T0 T− S0 T+ T0 T− the system approaches the steady state exponentially fast in 1.0 (b) 8 (c) 100 (d)  time with a rate . −1 0.8 10 6 In some of the examples we study, the Lindblad equation S0|ρs(t)|S0 ) −2

. δt 10 that arises has a special form. Specifically, if there exists a 0 6 T+|ρs t |T+ ( ) ]( choice of basis {|α} such that all the jump operators are of v 4 −3 √ T0|ρs(t)|T0 10 γ |αβ| 0.4 Re[ the form αβ , then the matrix elements of the density T |ρ t |T − − s( ) − −4 D 2 10 1 matrix in the same basis follow a master equation given by 0.2 δt distance from singlet DF Δ( ) −5 10 ∼ e−Δt 1 0.0 0 ∂t ρs,αβ = δαβ γανρs,νν − ρs,αβ (γνα + γνβ ), (26) −1 0 1 eigenvalue index ν 2 ν 10 10 Jt 10 0 Jt25 50 which directly implies that the off-diagonal elements of FIG. 1. Steering of a pair of spins-1/2 to the singlet state using the density matrix decay exponentially in time. Moreover, three detector qubits coupled via the Hamiltonian in Eq. (28). (a) The the equations for the diagonal elements do not involve the density matrix of the system, ρs, in the basis spanned by the singlet off-diagonal elements and hence follow a classical master and the three triplets as a color map (for the absolute values of the equation. matrix elements) at different times t. It shows the decaying support on the Stot = 2 diagonal elements. (b) The evolution of the diagonal C. Steering a pair of spins-1/2 elements of ρs(t ) shows the steering to the singlet. (c) The spectrum We next illustrate these ideas using a two-spin problem. of the corresponding Lindbladian with a unique zero eigenvalue Consider a pair of spins-1/2, acted on by the set of Pauli (steady state) and a finite gap . (d) The distance of ρs(t ) from the {σ μ} {σ μ} singlet state |S S | as measured via the matrix norms in Eq. (18) matrices, 1 and 2 , which we would√ like to steer to 0 0  the singlet state |S =(|↑↓ − |↓↑)/ 2. The Hilbert space decays exponentially in time at a rate is correctly given by . For the 0 = δ = . of two spins-1/2 is four-dimensional with the subspace or- plots, J 1and t 0 1. thogonal to |S0 spanned by the three triplet√ states, |T+= Results for the evolution of the two-spin system with the |↑↑, |T−=|↓↓, and |T =(|↑↓ + |↓↑)/ 2. Hence, the 0 above steering protocol are shown in Fig. 1. All the elements simplest steering protocol uses three detector qubits, each for of density matrix in the basis spanned by the singlet and the steering the state out of one of the triplet states onto the singlet three triplet states decay to zero exponentially in time except state. At the start of every steering event, each of the detector for the diagonal element corresponding to the singlet, which qubits is prepared in a pure state fully polarized along the approaches unity. This shows that the system is indeed steered positive z axis such that to the singlet state exponentially in time. z z z I2 + σ I2 + σ I2 + σ Turning to the effective Lindblad dynamics corresponding ρ = d1 ⊗ d2 ⊗ d3 , (27) d 2 2 2 to the protocol, the three jump operators of the√ Lindblad μ equation can be read off from Eq. (28)asL = JU δt.The where {σ } denotes the set of Pauli matrices for the ith i i ←→ di spectrum of the so-obtained Lindbladian superoperator, L , detector. For this choice of ρd, the system-detector coupling Hamiltonian motivated from the form in Eq. (7) can be written is shown in Fig. 1(c). The zero eigenvalue is nondegenerate, as implying that the singlet state is the unique steady state of the dynamics. Moreover, the distance of ρ (t ) from the singlet 3 s − state obtained from the exact dynamics and measured via the Hs-d = J (σ ρd ⊗ Ui + H.c.), (28) di Frobenius or the trace norm [Eq. (18)] decays exponentially = i 1 in time with a rate given by  × δt, which is in excellent with agreement to the numerically obtained gap in the spectrum of the Lindbladian between the zero eigenvalue and the rest 1 z − − z =|  +|= + σ σ − σ + σ , U1 S0 T 3 1 1 2 1 1 2 (29a) of the eigenvalues. This shows that the effective Lindblad 2 2 equation is a valid description of the dynamics. =|  |= 1 − σ z σ + − σ + − σ z , Note further that the form of the jump operators in Eq. (29) U2 S0 T− 3 1 1 2 1 1 2 (29b) 22 leads the equation of the density matrix elements in the 1 σ z + σ z basis spanned by the singlet and the triplets to be of the U =|S T |= σ +σ − − σ −σ + + 1 2 . (29c) 3 0 0 2 1 2 1 2 2 form in Eq. (26). This explains the exponential decay of the off-diagonal elements. Additionally, the master equation † = = Note from the above equations that UiUj 0fori j for the diagonal elements possesses only the loss term for the as each of the triplet states are orthogonal to each other triplet states and only the gain terms for the singlet, resulting and also to the target singlet state, and hence the conditions the exponential decay and growth of the former and latter asserted for the system operators in Sec. II C are satisfied. respectively.

033347-7 ROY, CHALKER, GORNYI, AND GEFEN PHYSICAL REVIEW RESEARCH 2, 033347 (2020)

Before concluding the section, it is worth mentioning that P ( , +1) tot = where Stot=2 is a projector onto the five-dimensional S a protocol for steering a pair of spins-1/2 to the singlet using 2 subspace for the pair of spins-1 at sites and + 1. The only a single detector qubit can also be envisaged. Such a form of HAKLT in Eq. (32) implies that the ground state has the protocol involves steering the system out of any one of the special property that each bond has zero weight in the Stot = 2 triplet states onto the singlet state while also acting on it with sector. Hence, it is natural to imagine that a spin-1 chain can an intrinsic Hamiltonian which has matrix elements connect- be steered to the AKLT ground state by locally steering each ing the other two triplet states to the first one. Consequently, bond out of the Stot = 2 sector. Moreover, since the Stot = 2 the Hamiltonian unitarily rotates the states within the triplet sector of each bond is finite dimensional, such a steering sector and weight from the triplet sector keeps leaking into protocol would satisfy our physically motivated constraints: the singlet, eventually leading to the pure singlet state as the locality and only an extensive number of detectors. steady state. For details, see Appendix C. In Sec. IV A, we discuss the basic building block of the steering protocol, namely steering a pair of spins-1 out of the IV. SPIN-1 CHAIN Stot = 2 sector. In Sec. IV B, we use this building block to steer a spin-1 chain to the AKLT state. Let us now discuss the steering of a quantum many-body system, namely that of the spin-1 chain to the AKLT ground A. Pair of spins-1 state. The AKLT Hamiltonian is N Consider steering on the bond between a pair of spins-1 tot 1 1 1 2 labeled as and + 1. The S , + = 2 subspace is five HAKLT = + Sˆ · Sˆ +1 + (Sˆ · Sˆ +1) , (30) ( 1) = 3 2 6 dimensional and hence the simplest protocol involves five 1 tot = detector qubits, one to steer out of each of the S( , +1) 2 where Sˆ denotes spin-1 operators at site and the site = tot = states. Note that we only wish to steer out of the S( , +1) 2 N + 1 is identified with = 1 to impose periodic boundary subspace and there are no restrictions on what precise state(s) conditions. tot = tot = in the S( , +1) 1 and S( , +1) 0 subspaces we steer onto. It is useful to set up the notation for the rest of the section This offers a lot of freedom in choosing the steering protocol. z here. The projectors onto the three eigenstates of Sˆ (with This could be exploited to construct the simplest system- ± ± 0 eigenvalues 1 and 0 respectively), P and P ,aregivenby detector coupling Hamiltonian, or from a practical point of view, the one that it is easiest to implement. In the following, ± = z 2 ± z / 0 = − + − −. P S S 2; P I3 P P (31) we choose a particular protocol which steers the state onto the tot = tot,z The total spin on a bond between the sites and + 1 will S( , +1) 1 subspace, keeping the sign of S( , +1) the same. tot = + At the start of each steering event, the detector qubits [41] be denoted by S( , +1) S S +1. In what follows, it will be often convenient to use the simultaneous eigenstates of are prepared in a polarized pure state ˆ tot 2 ˆtot,z | tot, tot,z (S( , +1)) and (S( , +1)), denoted as S S ( , +1), where 5 + σ z I2 ( , +1) tot tot,z , + d S takes values 0, 1, and 2, and S takes all integer values ρ( 1) =⊗ i , (33) − tot  tot,z  tot d 2 in the range S S S . Additionally, we denote the i=1 AKLT ground state as |AKLT and the corresponding density ρ =|  | matrix as AKLT AKLT AKLT . and the system-detector coupling Hamiltonian is of the form The AKLT Hamiltonian with periodic boundaries has a unique valence-bond ground state which is most easily un- 5 ( , +1) − ( , +1) ( , +1) derstood by noting that HAKLT can be expressed as a sum of = σ ρ ⊗ + . ., Hs-d J ( , +1) d Ui H c (34) di local projectors i=1 N = P ( , +1), HAKLT Stot=2 (32) where =1 ( , +1) = | ,  , | = 1 − − − + + , U1 ( 1 1 2 2 )( , +1) [(S S +1)P P +1] (35a) 2 − + + − ( , +1) 1 S S +1 + 0 S S +1 0 + U = (|1, 12, 1|) , + = I − P P + − I − P P , (35b) 2 ( 1) 2 9 2 1 9 2 +1 − + + − 1 (S S )2 (S S )2 ( , +1) = | ,  , | = √ − +1 + − − − +1 − + + + − − − + 0 0 , U ( 1 0 2 0 )( , +1) I9 P P + I9 P P + (S S + S S + )P P + 3 2 3 4 1 4 1 1 1 1 (35c) + − − + ( , +1) 1 S S +1 − 0 S S +1 0 − U = (|1, −12, −1|) , + = I − P P + − I − P P , (35d) 4 ( 1) 2 9 2 1 9 2 +1

( , +1) 1 + + − − U = (|1, −12, −2|) , + = [(S − S )P P ]. (35e) 5 ( 1) 2 +1 +1

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Jt =0.1 Jt =1.0 Jt =10.0 Jt = 100.0 An important point to notice is for a pair of spins-1, the 1 (a1) (a2) (a3) (a4) above dynamics does not have a unique steady state unlike tot S =2 the case exemplified in Sec. III C. This is due to the fact that in the case of the spins-1, the steering is not onto a Stot =1 particular pure state but rather onto a subspace of states, and Stot =0 0 the specific steady state of the dynamics depends on the initial =1 =2 =0 =1 =2 =0 =1 =2 =0 =0 =1 =2 condition of the system. The nonuniqueness of the steady state tot tot tot tot tot tot tot tot tot tot tot tot S S S S S S S S S S S S for the pair of spins-1 is also manifested in the corresponding

(b) ,z |2, 2 |2, 0 |2, −2 |1, 0 |0, 0 Lindbladian having a degenerate zero-eigenvalue manifold. tot |2, 1 |2, −1 |1, 1 |1, −1 ,S 0.4 In the case of a spin-1 chain, as we will discuss in the next tot

S subsection, the uniqueness of the AKLT ground state leads | s ρ | to the uniqueness of the steady state reached under dynamics ,z 0.2 tot obtained by extending the approach described above to all the ,S

tot bonds of the chain. S

. 0 0 −1 0 1 2 10 10 Jt 10 10 B. Steering to the AKLT state FIG. 2. Steering of a pair of spins-1 out of the Stot = 2 subspace Let us now discuss in detail the many-body case of a chain using the system-detector coupling Hamiltonian of Eq. (34). (a) The of spins-1. The protocol for steering the system to the AKLT ρ | tot, tot,z density matrix of the system, s in the S S basis as a color map ground state is a scaled-up version of the protocol described (for the absolute values of the matrix elements) at different times t: in the previous subsection where each bond is steered out of It shows the decaying and growing support on the Stot = 2 and the the Stot = 2 subspace and as such there are 5N detector qubits Stot = 0 and 1 subspaces respectively. The blocks demarcated by the at play, N being the size of the spin-1 chain. Formally, the tot = blue, red, and gray dashed lines respectively denote the S 2, 1, initial state of the detectors at the start of each steering event ρ and 0 sectors. (b) The evolution of the diagonal elements of s(t ) is expressed as shows the steering out of the Stot = 2 subspace, all the five diagonal = δ = . N 5 + σ z elements of which decay to zero. For the plots, J 1and t 0 1. I2 ( , +1) d ρ =⊗ i , (37) d 2 =1 i=1 As in Sec. III, the jump operators for the corresponding Lindblad equation in the time-continuum limit can be identi- and the system-detector coupling Hamiltonian is of the form fied straightforwardly as N 5 − ( , +1) √ H = J σ , + ρ ⊗ U + H.c., (38) , + , + s-d d( 1) d i ( 1) = ( 1) δ . i Li JUi t (36) =1 i=1 ( , +1) In the rest of this subsection, we only discuss the pair of where Ui is the same as in Eq. (35). spins-1 and hence drop the labels and + 1. We also refer Let us first argue that the protocol described by Eqs. (37) to the density matrix of the two spins as ρs(t ). The dynamics and (38) does posses the AKLT ground state at least as of ρs(t ) under the steering protocol with the system-detector one of its steady states. Note that each term in the system- coupling Hamiltonian in Eq. (34) starting from a random detector coupling Hamiltonian in Eq. (38)isoftheform tot tot,z σ − ⊗| ,  , |+ . . ρ ⊗ mixed state is shown in Fig. 2. Expressed in the |S , S d 1 s1 2 s2 H c . Hence, a state of the form d tot ρ σ +ρ = = basis, the matrix elements of ρs(t )intheS = 2 block and AKLT is annihilated by the Hamiltonian as d d 0 the off-diagonal blocks connected to it decay exponentially |1, s12, s2| ρAKLT, implying that ρAKLT is indeed a steady in time. At long enough times, ρs(t ) is supported entirely state of the dynamics. However, compared to the “ideal” on the subspace spanned by the Stot = 1 and Stot = 0 states, steering protocol described in Sec. III, a number of difficulties and thus is steered out of the Stot = 2 subspace. With regard arise for many-body systems. The difficulties arise because to the corresponding Lindblad equation, note that the jump the Hilbert space for a many-body system is exponentially operators, Eq. (36), for the system-detector coupling Hamil- large in system size. The protocol of Sec. III uses a separate tonian, Eq. (35), result in an equation of motion of the form term in Hs-d to steer to the target state from each other state. tot tot,z of Eq. (26) for the matrix elements of ρs(t )inthe|S , S Such an approach is not practical for a many-body system, basis. Several features of the dynamics can be inferred from where one requires a limit of at most an extensive number tot tot this observation. For any state |α in the S = 1orS = of terms in Hs-d. In the following, we discuss these issues, 0 sector, γνα = 0forν = α which renders the off-diagonal and present numerical results for the AKLT chain which show matrix elements within the Stot = 1 and Stot = 0 subspace successful steering in this many-body setting. invariant in time [see also Figs. 2(a1)–2(a4)]. This also implies In the derivation of the strong inequality for steering in the that the master equations for the diagonal elements within this presence of multiple detectors (see Sec. II C and Appendix B), (m) (n)† = (m) (n) subspace have no loss terms and only gain terms, contrary to we had required that Us Us 0ifUs and Us have those in the Stot = 2 subspace, which only have loss terms. overlapping spatial supports. For the system-detector cou- One can then immediately infer that the support of ρs(t )onthe plings with Eq. (35) for the AKLT chain, it can be shown tot = ( , +1) (r,r+1)† = | − |= S 2 subspace decays exponentially, whereas it grows on that Ui Uj 0if r 1. In other words, the the Stot = 1 and Stot = 0 subspace before saturating to unity. assumption made in the derivation is not satisfied by the

033347-9 ROY, CHALKER, GORNYI, AND GEFEN PHYSICAL REVIEW RESEARCH 2, 033347 (2020) system-detector couplings on adjacent bonds but it is satisfied Jt =0.1 Jt =1.0 Jt =10.0 Jt = 100.0 1 by all the other pairs of couplings. The noncommutativity (a1) (a2) (a3) (a4) tot of the steering operators on adjacent bonds may also lead S =2 to nonmonotonic time dependence for E (t ). Steering a AKLT Stot bond, say between sites and + 1, affects the state of three =1 Stot bonds: the one that is steered and the two adjacent to it. While =0 0 =1 =2 =0 =1 =2 =0 =1 =2 the energy of the bond being steered necessarily goes down as =0 =0 =1 =2 tot tot tot tot tot tot tot tot tot ( , +1) tot tot tot S S S S S S S S S S P ρ S S steering decreases Tr[ Stot=2 s(t )], the same need to be the ,z D ( , +1) |2, 2 |2, −1 |1, 0 0 1 P ρ = ± tot case for Tr[ tot s(t )] with 1. For this reason, 10 S =2 DF ,S |2, 1 |2, −2 |1, −1 . AKLT 0 4 −Δt the total energy of the state need not decrease. tot Ψ −1 ∼ e | , | , | , | S 2 0 1 1 0 0 The difficulty associated with the noncommutativity of the | 10 (2) s ρ steering operators can be partially removed by considering a | −2 ,z 0.2 10 protocol in which each steering step is divided into multiple tot

,S −3 tot steps, in one of which only the system-detector couplings (b) distance from 10 (c) S 0.0 on a particular bond acts. In this case, at any time only one −1 0 1 2 10 10 Jt 10 10 0 Jt 200 bond is steered. For the operators acting on the same bond, ( , +1) ( , +1)† = tot = Ui Uj 0 as the different S 2 states on each FIG. 3. Steering of a spin-1 chain to the AKLT ground state. bond are orthogonal to each other. Hence, the inequality is (a) The evolution of the reduced density matrix of two adjacent spins, satisfied in each of the steps. The evolution of the diago- ρ(2) s , shown as a color map (for the absolute values of the matrix nal element of the system’s density matrix corresponding to elements) in the |Stot, Stot,z basis. The support on Stot = 2 sector | =| s⊕ AKLT is then described by decays to zero and so do the off-diagonal elements in this basis. (b) The evolution of the diagonal elements of ρ(2). The horizontal  |ρ + + δ | s s⊕ s(t ( 1) t ) s⊕ / / ρ(2) dashed lines denote the values 1 3and29, obtained from AKLT. = |ρ + δ | ρ ρ s⊕ s(t t ) s⊕ (c) The distance between s(t )and AKLT as measured via the matrix norms DF and D1 decays exponentially to zero with time. The dashed 5  +  | ( , +1)ρ + δ ( , +1)†| 2 δ , line represents an exponential decay with a rate obtained from the s⊕ Ui s(t t )Ui s⊕ sin t spectral gap of the corresponding Lindbladian. For the plots, N = 5, i=1 J = 1, and δt = 0.1. Q (39) be expressed to leading order in δt as (see Appendix D for where the equation corresponds to the specific case of steering details) the bond between sites and + 1.The sum on the right-hand  =−δ 2 Eˆ ρ , side of Eq. (39) is over all the detector qubits, acting only on EAKLT(t ) ( t ) Tr[ s(t )] (41) + ( , +1) ( , +1)† = the bond between sites and 1. Since Ui Uj where Eˆ is a four-site operator involving sites − 1,..., + 0, Eq. (39) follows from the derivation of Eq. (B5) 2. Since the spectrum of the operator Eˆ includes eigenvalues  (Appendix B). Note that Q 0 for all as for Q in Eq. (16). of both signs, monotonic decay of EAKLT(t ) is not guaranteed. In the time-continuum limit δt → 0, the set of equations in On the other hand, we find that for a choice of ρ (t ) which s (39) for all give leads to an initial growth of Eˆ , the distance from the target  |ρ + δ | = |ρ | + δ 2 , measured via DF (t )orD1(t ) nevertheless decays monoton- s⊕ s(t N t ) s⊕ s⊕ s(t ) s⊕ ( t ) Q (40) ically. A detailed study of the monotonicity of steering, or with Q = Q . lack thereof, depending on the distance measure, is presented Since Q  0, Eq. (40) ensures that the state is never steered in Appendix D. The upshot of this study is that one needs further away from the target AKLT state if Q is employed as to show that the target state is the unique steady state of the a measure of distance from the target state. At the same time, dynamics. As any state of the spin-1 chain which is not the Q = 0 does not necessarily ensure that the system is in the AKLT state will have some overlap on the Stot = 2 sector target state. As an illustration, consider an initial pure state on one or more bond, the form of the operators in Eq. (35) that has excitations on multiple bonds. A single steering step ensures that such a state is not a stationary state. of the type we have described removes one excitation without To demonstrate that the AKLT state is indeed the unique generating overlap with the target state, and so Q = 0 for this state and also to show the validity of results away from the step. So while the state changes, its overlap with the AKLT time-continuum limit, we simulate numerically the dynamics state does not. of a spin-1 chain starting from a random mixed state subjected In fact, for our steering protocol it can be shown explicitly to the steering protocol with all the bonds steered simulta- that there can arise ρs(t ) for which the change in EAKLT(t ) neously. We calculate two quantities: (i) the reduced density ρ(2) [defined in Eq. (19)] is positive. Using EAKLT(t ) as a measure matrix of a pair of adjacent spins-1 which we denote as s , of distance, one would then conclude that the state is steered and (ii) the distance of ρs(t ) from ρAKLT as a function of time, further away from the AKLT ground state. To show this, we measured via D1 and DF . ρ(2) use the fact that steering a bond affects only three bonds or The results are shown in Fig. 3. The evolution of s (t ) tot equivalently the four sites involving them. Since HAKLT is a shows that each bond is indeed steered out of the S = sum of projectors for each bond, the change in energy can 2 subspace. However, a drastic difference between the

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ρ(2) L dynamics of s and that of a single pair of spins-1 is that (a) 0 (b) ED of . approach to SS the former has a unique steady state which is diagonal in the ) 0 4 | tot, tot,z  , |ρ(2) →∞| , = / | − S S basis with 1 s1 s (t ) 1 s1 2 9 and 2 (2) 0, 0|ρ (t →∞)|0, 0=1/3. This highlights the collective AKLT 0.3

s Ψ |

) − N =5 t 4 effect of the many-body yet local steering protocol, as each ( Δ of the system-detector couplings are such that they keep the ψ N =6 0.2 N off-diagonal elements within the Stot = 1 and 0 subspace −| −6 =7 N =8 0.1 invariant. Moreover, note from the terms in Eq. (35) that they log(1 N =9 do not change the diagonal element corresponding Stot = 0. −8 N =10 0.0 On the other hand, in the many-body case, the matrix element 0 10 20 0.0 0.1 0.2 ρ(2) / Jt /N of s on a particular bond goes to its steady state value of 1 3 1 purely due to the interplay of the steering on the bond with that on the other bonds, further highlighting the collective effects FIG. 4. Scaling of steering rate with system size for the spin-1 of the many-body steering protocol. chain. (a) The exponential approach of the spin-1 chain to the AKLT ρ(2) state as obtained from the wave-function Monte Carlo method for The fact that s approaches a diagonal form in the |Stot, Stot,z basis with the diagonal elements taking the above- solving the Lindblad dynamics for different system sizes N.Asthe mentioned specific values is crucial because ρ(2) has exactly method involves evolving pure states, we measure the distance from AKLT the AKLT state via log(1 − ψ(t )| ) where the bar denotes the this feature. For the reduced density matrix of a subsystem of AKLT average over the wave-function trajectories. The black dashed lines the spin-1 chain comprising the contiguous sites 1 through  are fits to the data used to extract . (b) The rate of the exponential n, it can be easily argued that its eigenvectors corresponding approach to the steady state (SS) obtained from panel (a) or equiv- to nonzero eigenvalues are the four degenerate AKLT ground alently the gap in the spectrum of the Lindbladian obtained from states of the spin-1 chain of length n with open boundary con- L / , + exact diagonalization of shown as a function of 1 N. The intercept = n P ( 1) ditions described the Hamiltonian Hsubsys = tot= .  1 S 2 indicates a finite in the thermodynamic limit. The data for panel n 2 At the same time, Hsubsys commutes with ( = Sˆ ) and (a) were averaged over 20 000 trajectories and the error bars were 1 n z obtained via standard bootstrap with 500 resamplings. Errors in the = Sˆ . These eigenstates are simultaneous eigenstates of 1 extrapolated value of  introduced via the errors in  for different n 2 n z H ,( Sˆ ) , and Sˆ .Forn = 2, it directly subsys = 1 = 1 N as well as the fitting procedure are also shown but the error bars ρ(2) | tot, tot,z implies that s is diagonal in the S S basis. Moreover, are smaller than the point size. it has been shown that the four nonzero eigenvalues are (1 + 3(−3)−n )/4 and (1 − (−3)−n )/4 with a threefold degeneracy = / / [42], which for n 2are13 and 2 9 respectively. Since N  8. In order to extract  for slightly larger sizes, N  10, every bond of the spin-1 chain is steered to a state that we solve for the dynamics of the Lindblad equation using the corresponds to the AKLT ground state of the entire chain, ρ wave-function Monte Carlo method, which involves evolving it is natural that the many-body state s(t ) is also steered pure states with stochastic quantum jumps and then averaging to ρAKLT. This confirmed by the exponential decay of the ρ ρ over the so-obtained trajectories [43,44]. With this method, distances DF and D1 between s(t ) and AKLT with time as we find again that the spin-1 chain approaches the AKLT shown in Fig. 3(c). ρ ground state exponentially quickly in time and hence the rate So far we have argued that AKLT is a steady state of the  can be extracted; see Fig. 4(a).The obtained from the steering protocol and have shown numerically that an arbitrary dynamics for smaller system sizes agrees very well with the initial state of a finite spin-1 chain evolved with the steering ←→ L protocol approaches the AKLT ground state exponentially fast one obtained from diagonalizing ; see Fig. 4(b).Thetwo  in time. The natural question to ask then is what happens methods together allow us to study the scaling of with N to the steering protocol in the N →∞limit, with regard to and from the scaling shown in Fig. 4(b), we conclude both the uniqueness of the steady state and the timescale for steering to it. We find it most convenient to answer these (N) = ∞ + c/N . (42) questions in terms of the Lindbladian corresponding to the steering protocol. Following the discussion in Sec. III B,the ←→ jump operators, of which there are√ 5 × N, can be immediately Hence, the gap in the the spectrum of L or equivalently the ( , +1) = ( , +1) δ timescale required to steer the spin-1 chain to the AKLT state identified√ as Li JUi t. For convenience, we will set J δt = 1 in the following. As discussed in Sec. III B, stays finite in the thermodynamic limit. the uniqueness of the steady state manifests itself in the Combining all the results presented in this section, we ←→ conclude that the map of Eq. (2), with the detectors’ initial nondegeneracy of the zero eigenvalue of L . The rate of the states and the system-detector coupling described by Eqs. (37) approach to the steady state is given by the gap  of the rest of and (38) respectively, constitutes a protocol for steering a the spectrum from the zero eigenvalue. Hence, it is instructive many-body system to a strongly correlated state, in this case a to study the scaling of  with N. ←→ spin-1 chain to the AKLT ground state. The protocol satisfies L By exactly diagonalizing for finite sizes, we find that the constraints of having an extensive number of detector the zero-energy eigenvalue is indeed nondegenerate and the qubits coupled only locally to the system and of having a unique steady state corresponding to it is identical to ρ . ←→ AKLT steering timescale that does not diverge in the thermodynamic The dimension of L is 32N , which restricts calculations to limit.

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V. DISCUSSION Second, the experiments make use of postselected readouts. Third, the specifics of the experiment rely on knowledge of the In this work, we have developed a general protocol that system’s initial state. A further experiment [32,50] focuses on uses entanglement to induce steering of a quantum system to the study of a three-level system, where the evolution of a two- a target state. We have shown how our protocol can be applied level subsystem is conditioned on the system not being in the both to steer systems with a few degrees of freedom and to third level. This can be represented using modified Lindblad prepare strongly correlated states in many-body systems. The dynamics leading to non-Hermitian Hamiltonian dynamics. form of steering that we study here is rooted in but distinct Alternatively, the dynamics can be described in terms of a from steering of the type discussed originally in the context set of null weak measurements with post-selected readout of the EPR paradox [1,2]. Specifically, our protocol shares sequence [51,52]. Either picture differs substantially from our with earlier discussions of quantum steering the fact that it protocol. However, adjusting these protocols to our paradigm exploits the entanglement between the system of interest and of steering should be rather straightforward. Generalizations another system. Crucially, however, it differs in offering a of these protocols could work for a detector coupled to two or general framework to steer a quantum system into a specific more qubits [53]. and predesignated state. The second possible realization of superconducting qubits involves the use of state-of-the-art fluxonium qubits. These A. Experimental realisations offer coherence times comparable to more common transmon Experimental implementations of the ideas presented in qubits, with increased anharmonicity, reduced cross-talk, and this work require realizations of qubits which satisfy several design flexibility. Such devices could be implemented to conditions. (i) The relaxation time of the qubit due to un- construct both our many-body system as well as the detector desired coupling to external degrees of freedom should be qubits [54–57]. Concretely, two fluxonium qubits could be long. (ii) Likewise, the coherence times should be long. (iii) coupled with variable strengths. One fluxonium plays the It should be possible to design system-detector couplings as role of the qubit to be steered, while the other fluxonium, required for versatile steering to arbitrary target states. (iv) inductively coupled to a readout resonator [56], plays the The system should be scalable to multiple qubits and should role of detector on which dispersive projective readout can allow for multiqubit interactions (for example, a three-qubit be performed with high fidelity. The coupling between the interaction term in the system-detector Hamiltonian involving qubits can be implemented using shared capacitors or shared one detector qubit and two system qubits). inductors. With the recent availability of compact inductors There are several promising candidate platforms for realis- with low nonlinearity, using disordered superconductors, such ing our protocol, which include superconducting qubits, cold as granular aluminum [58]orTiN[59], inductive coupling ions, nitrogen-vacancy (NV) centers, macroscopic mechanical appears more appealing, as it minimizes cross-talk between objects with entangled quantum degrees of freedom, and the qubit to be steered and the readout resonator. quantum optics systems. These are platforms that have all been tested, or will be tested, as far as the steering of a 2. Cold ions single qubit is concerned. The extent to which more complex systems, comprising multiple quantum degrees of freedom, Another prospective platform for implementing may be steered relies on the extent to which conditions listed measurement-induced steering is based on cold ions trapped earlier can be satisfied. In the following, we briefly discuss by magnetic field and lasers. Qubits are implemented in some of these platforms. electronic states of each ion and the ions in a common trap can communicate with each other via intrinsic long-range interactions and external, laser-induced couplings. Below, 1. Superconducting qubits we will describe an example of such a platform consisting The most prospective experimental avenue for our proposal of Sr+ ions, which are essentially hydrogen-like atoms with is based on superconducting qubits. There are two realiza- one outer electron; the lowest two levels represent the qubit. tions of qubits which appear to be the leading candidates Coupling to the ion with light can effect arbitrary unitary for constructing steering platforms. The first are transmons. rotations. Qubit operations are well developed in this area, but In previous experimental work, weak measurements were steering of a single qubit is only under way, while a challenge constructed via entangling operations between two supercon- for applications to steering is to develop multiple-qubit ducting qubits, and subsequent projective measurement on the operations. detector qubit [45], which is exactly the measurement struc- Generalizing from single-qubit to two-qubit measurements ture we propose in this work. Another approach, implemented is highly challenging. The more straightforward protocol primarily for the steering of a single qubit, relies on dispersive would involve shining light of a wavelength larger than the interaction between the qubit and a coupling to a waveg- distance between ions, so that each ion can absorb and emit uide cavity. Existing implementations of weak measurement a photon, ensuring coherence between the two ions [60–63]. backaction on such a systems [46–49] have three major dif- The detection of an emitted photon in a given direction de- ferences from our steering protocol. First, the sequence of termines the backaction. To achieve versatility of the system- weak measurements in the experiments amounts to a strong detector Hamiltonian, one needs to go one step further and measurement when integrated over time (i.e., when consider- designate one ion as a detector that measures two other ions ing a sequence of many weak measurements), which projects (one on the left and the other on the right of it). Every two the initial state onto a final state with probabilistic outcomes. neighboring ions can be made to interact through terms of the

033347-12 MEASUREMENT-INDUCED STEERING OF QUANTUM … PHYSICAL REVIEW RESEARCH 2, 033347 (2020) system-detector Hamiltonian that flip the pair of qubits. If it platforms, with the promise that other types of manipulations is possible to couple and decouple the detector and the left are, in principle, also possible. Going to the arena of many- spin on demand, and separately the detector and the right spin, body physics, one needs to engineer scalable multiqubit states then by performing unitary transformations on the right spin by our steering protocols. This is feasible with platforms detector and/or the left spin detector a broad class of effective based on superconducting qubits and NV centers, where flexi- three-qubit Hamiltonians can be achieved. ble (and, in principle, arbitrarily complex) few-qubit-detector It is also worth noting that dynamical maps governing couplings can be engineered, as demonstrated for two-qubit nonunitary time evolution of quantum systems have also been realizations. Implementation of many-body steering on these experimentally realized using trapped 40Ca+ ions [64]. platforms requires experimental efforts toward long coherence times, circuit connectivity, and quality control of system- 3. Nitrogen-vacancy (NV) centers detector coupling and detector reinitiation. Importantly, even when some specific system-detector couplings are still hard to NV centers in diamonds are point defects which realize design for technological reasons, the shake-and-steer protocol electronic spins localized at atomic scales. In these setups, may resolve this issue. We expect our work to stimulate the detector could be the NV center and the qubits could be experiments based on existing and future platforms in the 13 nuclei spins. The interaction between them is of a dipole- C direction of steering and manipulating many-body states. dipole type, which may be controlled with high precision [65]. Direct optical measurement can only be preformed on the NV center. The control is made with microwaves [66]. Another B. Connection to previous work possibility is to use NV centers to control external spins The use of external degrees of freedom as a designed [67]. In principle, one could aim at a multispin measurement environment for manipulating and controlling quantum states or control. The level structure for such systems has been has been discussed previously in other settings [23,29,30,80]. analyzed extensively [68]. Various aspects of the NV readout Treatment of the environment as Markovian or the measure- and polarization mechanism [69] as well as control at weak ments as nonselective leads to a Lindblad equation, similar to coupling have been studied (cf. Refs. [70] and [71] for theory one derived in the present work. Much of the earlier focus has and experiment respectively). We note that the control can be been either on the formal aspects of such nonunitary dynam- much improved at low temperatures [72]. ics, or on applications involving few-body quantum systems. Our work constitutes a generalization of such quantum control 4. Macroscopic mechanical objects and steering to macroscopic many-body systems. Measuring and controlling macroscopic mechanical ob- In the context of open quantum systems, the creation of jects that exhibit quantum coherence is a challenge realized in nontrivially correlated or topological many-body states has recent years [73]. The next step, accomplished very recently been proposed using so called drive-and-dissipation schemes [74], is to scale this up to two macroscopic mechanical [17–19,22,24–26,81]. In such schemes, the system is driven degrees of freedom. Unfortunately, achieving a high level of or excited using a time-dependent Hamiltonian while a dis- control and measurement may expose the system to undesired sipative channel, often a Markovian environment, working interactions with its environment, a problem that becomes simultaneously relaxes the system. This interplay of drive more pronounced at the macroscopic scale. Using a super- and dissipation is then used to engineer nontrivial states. conducting electromechanical circuit and a pulsed microwave From a theoretical point of view, the Markovian nature of protocol, a system of two mechanical drumheads has been the environment naturally lends itself to a treatment via the steered toward an entangled state, where entanglement of Lindblad equation, hence yielding a formal connection to our continuous degrees of freedom satisfies the criteria formulated measurement protocol in the time-continuum limit alluded to [75,76]. The strength of the measurement in these systems previously. In fact, the jump operators postulated in Ref. [18] can be tuned from strong to weak, and coherence may be in the context of the AKLT chain also bear formal similarities maintained over long timescales. However, given the immense with those derived in our case from the measurement protocol. experimental difficulties here, the issues of versatility of Further related work is described in Ref. [20], where aux- system-detector Hamiltonian, and the question of scalability iliary qubits are exploited to drive a Rydberg-atom realization to systems consisting of multiple degrees of freedom are all of Kitaev’s toric code to its ground state. Excited states of the likely to remain a challenge in the near future. toric code contain spatially localized excitations on plaquettes Finally, one could adopt these ideas to the field of quantum or vertices of the system, which can annihilate in pairs. The optics [77]. Photon polarization has been used as a qubit, for protocol of Ref. [20] takes advantage of this special feature example, in the process of storing and then retrieving quantum of the toric code. To cool this system to its ground state, information in cold-atom platforms [78]. More directly related an implementation of Lindblad jump operators is proposed to our ideas are two polarization-based qubits (one serving as that generates stochastic motion of excitations, so that pairs system and the other as detector), allowing for back-action meet and annihilate. An advantage of our steering protocol control [79]. for the AKLT chain is that we directly steer each local degree In summary, experimental implementations of our ideas of freedom of the system (a bond between two sites in the should be assessed according to the complexity of the re- AKLT chain) toward the ground state. Moreover, our general quired platform. As far as a single-qubit steering is concerned, protocol is independent of special features of the steered reliable implementations are achievable with all platforms. system. The local unit which is steered could have a large Entangling two qubits has already been done with some excited state manifold; steering would still be possible using

033347-13 ROY, CHALKER, GORNYI, AND GEFEN PHYSICAL REVIEW RESEARCH 2, 033347 (2020) the shake-and-steer protocol (see Appendix C) with a finite questions call for a detailed and comprehensive study of the number of locally coupled detectors. consequences of errors in steering protocols and possible In contrast to the drive-and-dissipation protocols, our ways of correcting them, which we leave for a future work. measurement-based protocol does not require a system Specifically, one could ask under these conditions whether the Hamiltonian to induce dynamics in the system. Moreover, system has a steady state, and if so how its distance from the unlike an environment which is to a large extent uncontrolled, target state scales with the strength and probability of errors in the detectors that play the role of the dissipative channel are the steering protocol. Furthermore, one might look for specific well-controlled simple quantum systems that can be tuned in error correction protocols akin to a stabilizer formalism [84], time. As on-demand jump operators, they represent a partic- but within the setting of a steering protocol. ularly a useful form of quantum manipulation and control. It is reasonable to anticipate that a finite gap in the spec- Moreover, as outlined above, using the measurement out- trum of the Linbladian in the thermodynamic limit (as we comes for active decision making, or having a measurement find in our study of the AKLT state) will ensure that the protocol which varies with time and has a memory kernel, steady state is robust to small random errors in the steering takes us to realms that seemingly cannot be addressed by protocol that are local in space and time. Robustness of drive-and-dissipation techniques. the protocol to time-independent deformations is a separate issue. A specific instance concerns the interplay of a system Hamiltonian (which we have so far omitted) with the system- C. Future directions detector coupling Hamiltonian. This would amount to having There is a wide range of possible future directions and a nonzero V in Eq. (23). It is clear that if the Lindbladian open questions arising from the work presented here, and in the absence of the system Hamiltonian shares an invariant we close by outlining some of these. While the protocol we subspace with the system Hamiltonian, the steering protocol have described is arguably the simplest version, it is also is robust to the latter. However, if that is not the case, the quite robust in the sense that the probability of reaching the apparent tension between the system Hamiltonian and the target state is unity irrespective of the initial state of the system-detector coupling Hamiltonians could lead to a new system. One can also envisage a wide variety of general- steady state which deviates continuously with the strength of izations, in which instead of simply decoupling the detector the system Hamiltonian from the original dark state. Alter- qubits from the system of interest at the end of each steering natively, in the many-body case there could be a dynamical step, a projective measurement is made of the final detector phase transition in the nature of the steady state. states. In this broader context, the protocol we have described In fact, measurements on an otherwise random unitary corresponds to the special case of blind measurements in circuit have been shown to induce an entanglement phase which an unbiased average is made over all such measurement transition in the steady state as a function of the density or outcomes. A natural direction is to ask whether making use strengths of measurements [85–94]. Most of these works use of the measurement outcomes can allow optimization of the local projective measurements or generic positive operator- protocol. More precisely, the initial state of the detector, the valued measurements which manifestly decrease the entangle- system-detector coupling, and the duration of the coupling ment. However, our work shows that measurement protocols at each step can be made to depend on the measurement can be constructive in the sense that they can be designed outcome of the previous step or on an extended history of so that steady state is a strongly correlated state. Thus, a the measurement outcomes. Active error correction could be natural question arises as to the extent to which measurement viewed as an example of a broader class of protocols involving protocols and their interplay with system Hamiltonians can be such decision making. Generalized protocols of this type are classified in terms of entanglement properties of the resulting manifestly not Markovian and hence are not describable in steady states. terms of a Lindblad equation with time-independent jump An intriguing future direction would be to study measure- operators. In the context of quantum control [80], there have ment protocols for which the emergent Lindbladian has a dark been proposals to utilize the measurement outcomes as a space spanned by several dark states. An example is provided feedback to the dynamics using the so-called quantum fil- by the AKLT state in an open chain, for which this space is tering equations [29,31]. Another direction for taking non- four dimensional. In such a scenario, one can envision a closed Markovian effects into account is via the quantum collision adiabatic trajectory (in the protocol parameter space) that models [82]. We reserve the generalization of our protocol to could be used to induce a unitary transformation in the dark include such effects for future work. space. This could be harnessed to induce adiabatic rotations There are many possible motivations for such generaliza- in a degenerate many-body space and may even give rise to a tions. One would be to optimize the timescale over which many-body non-Abelian geometric phase [95,96]. Similarly, the system reaches a predefined vicinity of the target state. the presence of multiple stationary states of Lindbladians has Another would be to maximize the purity of the quantum state been used theoretically to realize dissipative time crystals of the system throughout the protocol. This would become [97–100], and these could in principle also be generated via particularly important if one extends the objective of the steer- a measurement protocol. ing protocol from preparation of a target state to manipulation of a state using a sequence of weak measurements [83]. ACKNOWLEDGMENTS Another set of open questions concerns the robustness of the protocol to deformations of the initial states of the E. Karimi, K. Murch, R. Ozeri, I. Pop, A. Retzker, and detector qubits or of the system-detector couplings. These K. Snizhko have made insightful comments on experimental

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demonstrate that this is not a requirement and show that our 0.8 protocol can be used to steer the system qubit to an arbitrary target state. Let us suppose that we always prepare the detector . 0 6 spin in its |↑d state, but we want to steer the system spin to a pure state of the form 0.4 | = θ/ |↑ + θ/ |↓ . s⊕ cos( 2) s sin( 2) s (A1) . ↑ |ρ |↑ 0 2 s s s This is easily achieved using a system detector coupling ↑s |ρs|↓s Hamiltonian of the form 0.0 0 1 = σ − ⊗|  | − σ y + . ., 10 10 Hs-d J d s⊕ s⊕ ( i s ) H c (A2) Jt − σ y | | where i s s⊕ is the state orthogonal to s⊕ and out of FIG. 5. Steering of a single spin-1/2 using the system detector which we want to steer. In terms of the spin operators, the coupling in Eq. (A3) with θ = π/4. The different lines show the Hamiltonian can be written as different matrix elements of the system spin’s density matrix. The J horizontal dashed lines show the values that the matrix elements H = (sin θσz − cos θσx − iσ y ) ⊗ σ − + H.c. (A3) s-d 2 s s s d should take for ρs =|s⊕ s⊕ |.

The resulting dynamics of ρs under the protocol with the realizations. We have benefited from useful discussions with system-detector coupling Hamiltonian as above is shown Y. Herasymenko, P. Kumar, and K. Snizhko. We acknowledge in Fig. 5 for θ = π/4. The different lines correspond financial support from EPSRC Grants No. EP/N01930X/1 to the matrix elements of the time-evolving density ma- ↑ |ρ →∞|↑= 2 θ/ and No. EP/S020527/1, from the Israel Science Foundation, trix of the spin with s(t ) cos ( 2) and ↑ |ρ →∞|↓= θ/ θ/ from the Leverhulme Trust Grant No. VP1-2015-005, and s(t ) cos( 2)sin( 2) at long times show- ρ →|  | from the Deutsche Forschungsgemeinschaft (DFG) Projekt- ing that in the steady state s s⊕ s⊕ . nummer 277101999, TRR 183 (Project C01), and Grants No. EG 96/13-1 and No. GO 1405/6-1 and an NSF-BSF grant APPENDIX B: DERIVATION OF STEERING INEQUALITY within the EAGER program. WITH MULTIPLE DETECTOR QUBITS In this Appendix, we present some detail of the derivation APPENDIX A: STEERING A SINGLE SPIN-1/2TOAN of the steering inequality in the presence of multiple detectors. ARBITRARY STATE Partitioning the composite Hilbert space of the system and In the toy example of a single spin-1/2 presented in the detectors into ρd ⊗ Hs and (Dd ) ⊗ Hs as in Sec. III A, Sec. II D, the target state of the system qubit was the same the system-detector coupling Hamiltonian of Eq. (7) can be as the initial state of the detector. In this Appendix, we expressed as

| ··· (1) | ··· (n) | ··· ⎛ d Od d Od d ⎞ | 00 U † ··· U † ··· d ⎜ 1 n ⎟ . ⎜ ⎟ . ⎜ ··· ··· ··· ⎟ .00⎜ 0 ⎟ ⎜ . . . . . ⎟ O(1) | ⎜ U ...... ⎟ = d d ⎜ 1 ⎟, Hs-d . ⎜ ...... ⎟ (B1) . ⎜ ...... ⎟ ⎜ . . ⎟ (n) | ⎜ ...... ⎟ Od d ⎝ Un . . . . . ⎠ ...... 0...... 0

{ (n) | } † where the labels Od d on the rows and columns of the matrix denote the positions of nonzero entries, Un and Un . The entry th labeled by n corresponds to the coupling of the system with the n detector. With this notation, the operator W = exp (−iHs-dδt ) takes the form ⎛ ⎞ X k −iδt X k[···U † ···U † ···] ⎜ ⎡ ⎤ 2k+1 1 n ⎟ ⎜ 0 ⎟ ∞ ⎜ ⎢U ⎥ ⎟ (−iδt )2k ⎜ ⎢ 1⎥ ⎟ = ⎜ ⎢ . ⎥ ⎟, W ⎜ −iδt k⎢ . ⎥ k ⎟ (B2) (2k)! ⎜ Y ⎢ . ⎥ Y ⎟ k=0 ⎜ 2k+1 ⎢ ⎥ ⎟ ⎝ ⎣Un⎦ ⎠ . .

033347-15 ROY, CHALKER, GORNYI, AND GEFEN PHYSICAL REVIEW RESEARCH 2, 033347 (2020) where ⎡ ⎤ . ⎢ . ⎥ ⎢ ⎥ ⎢U1⎥ ⎢ ⎥ = † = ⎢ . ⎥ ⊗ ··· † ··· † ··· . X Un Un; Y ⎢ . ⎥ [ U1 Un ] (B3) i ⎢ ⎥ ⎣Un⎦ . . Using the form of the operator W from Eq. (B1) in the map described by Eqs. (1) and (2), we obtain the time-evolved density matrix of the system as ∞ ∞ (−iδt )2k (iδt )2l (−iδt )2k+1(iδt )2l+1 ρ (t + δt ) = X kρ (t )X l + U X kρ (t )X lU †. (B4) s (2k)!(2l)! s (2k + 1)!(2l + 1)! n s n k,l=0 k,l=0 n  |ρ | (m) (n)† = The quantity of interest is s⊕ s(t ) s⊕ , which can be obtained from Eq. (B4) under the condition that either Us Us 0 = (m) (n)† for n m or Us and Us commute, in the form  |ρ + δ | = |ρ | +  | (n)ρ (n)†| 2 δ , s⊕ s(t t ) s⊕ s⊕ s(t ) s⊕ s⊕ Us s(t )Us s⊕ sin ( t ) (B5) n

which is the same as equation as Eq. (16) with Q given with Hs-d and Hs given by Eqs. (C1) and (C2) respectively. by Eq. (17). This concludes our derivation of the steering Simulating the protocol described above again shows steering inequality for the case of multiple detectors. to pure singlet state; see Fig. 6(a). The Lindblad equation describing the dynamics in the time-continuum limit now also has a unitary part in addition to the dissipative part. The APPENDIX C: SHAKE-AND-STEER PROTOCOL spectrum of the corresponding Lindbladian has a unique zero In all the previously discussed cases, the number of system- eigenvalue; see Fig. 6(b). detector couplings used locally is equal to the dimension of Such protocols can be easily generalized to the case of the the subspace orthogonal to the target state. The shake-and- AKLT state. For a pair of spins-1, a single system-detector steer protocol permits steering with just one system-detector coupling steers weight out of one of the Stot = 2 states and a coupling. An additional Hamiltonian acting on the system Hamiltonian rotates the system within the Stot = 2 subspace unitarily rotates the state of the system within the orthogonal in a sufficiently general fashion. subspace. Thus, the weight from the orthogonal subspace keeps leaking into the target state via one channel correspond- ing to the system-detector coupling, while the weight from the APPENDIX D: MONOTONICITY OF STEERING rest of the orthogonal subspace is shaken into the dissipative TO THE AKLT STATE channel via the system Hamiltonian. Let us illustrate the The numerical results in Sec. IV B show that the AKLT protocol using the steering of two spins-1/2 to the singlet ground state is the unique steady state of our measurement state, similar to the one in Sec. III C. dynamics and steering to it is guaranteed via our protocol. Out of the three system-detector couplings used there, However, the approach to the steady state is not necessarily Eq. (29), let us consider only one of them, such that monotonic and can depend on the distance measure used. In = σ − ⊗ + . . . Hs-d J( d U3 H c ) (C1) . The above coupling steers the state of the system to the 1 0 (a) (b) singlet only from the |T0 state and leaves the |T± subspace 0.8 2 S0|ρs|S0 untouched. However, the state can be shaken out of the |T±

0.6 T−|ρs|T− ] subspace onto the |T0 via a system-Hamiltonian of the form v T |ρ |T 0 0.4 0 s 0 Im[ T |ρ |T Hs = θ1 · σ1 + θ2 · σ2, (C2) + s + 0.2 −2 where θ / can be arbitrary vectors [101]. 1 2 0.0 The time-evolution map of the density matrix of the system 100 101 102 103 −0.3 −0.2 −0.1 0.0 can be written explicitly as Jt Re[v] δ t −iH δt iH δt FIG. 6. Shake-and-steer protocol for steering a pair of spins-1/2 ρ t + = Tr e s-d 2 ρ ⊗ ρ (t )e s-d 2 , (C3a) s 2 d d s to the singlet state. (a) The dynamics of the diagonal elements of the density matrix under the protocol described in Eq. (C3) with θ1 = δt δt δt −iHs iHs θ = , , ρs(t + δt ) = e 2 ρs t + e 2 , (C3b) 2 J(1 1 1). (b) The spectrum of the corresponding Lindbladian 2 with a unique zero-eigenvalue corresponding to the singlet state.

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. . 1 2 0 05 1.4 D1 DF . . 1.2 (3,4) 1 0 0 00 Tr[PStot=2ρs(t)]

. 0.8 −0.05 1 0 )] t (

s 0.8 . ρ

0 6 − . ˆ AKLT 0 10 E [ E

Tr 0.6 . 0 4 −0.15 0.4 . 0 2 −0.20 0.2 . 0 0 −0.25 0 1 2 3 0.0 10 10 10 10 steering events 100 101 102 103 steering events FIG. 7. Evolution of Tr[Eˆρs(t )] (red, orange) and EAKLT (blue, cyan) for an initial state which is the eigenstate corresponding to FIG. 8. Evolution of the global and local measures of distance the most negative eigenvalue of Eˆ. Results are shown for a N = 4 from the AKLT ground state for the initial state used in Fig. 7 spin-1 chain. The dots correspond to data points after steering each (the eigenstate corresponding to the most negative eigenvalue of Eˆ). bond whereas the squares correspond to data points after steering The Frobenius distance from the target state has a monotonic decay, each bond of the system once. whereas the trace distance at early times does not change, although it does eventually decay to zero. The reduced density matrix of a particular bond also gets steered to that of the AKLT ground state this section, we show explicitly that such a situation arises but in a nonmonotonic fashion. Results are shown for a N = 4 spin-1 using EAKLT [Eq. (19)] as the distance measure. We then chain. Similar to Fig. 7, the dots correspond to data points after steering each bond, whereas the squares correspond to data points compare the results with those of other distance measures. = P ( , +1) after steering each bond of the system once. Since HAKLT Stot=2 is a sum of projectors for each bond, we would like to understand the change in , + Tr[P ( 1)ρ (t )] when the bond between site and + 1 where Stot=2 s is steered. In the following, we will use the shorthand P = † A = PP + P P − 2 U P U , (D4a) P ( , +1) P = P ( , +1) = ( , +1) n n Stot=2 , Stot=2 , and Ui Ui . n Using Eq. (B4), it can be shown that B = PP P − U †P U . (D4b) n n P ρ + δ = P ρ + †P ρ 2 δ n Tr[ s(t t )] Tr[ s(t )] Tr [Un Un (t )] sin t n The change in energy of the system measured with respect

to H can be obtained by summing Eq. (D3) over , which − Tr[(PP + P P)ρ (t )](1 − cos δt ) AKLT s gives 2 + Tr[PP Pρs(t )](1 − cos δt ) . 2 EAKLT(t ) =−2sin (δt/2)Tr[(2P + A− + A+)ρs(t )] (D1) 4 + 4sin (δt/2)Tr[(P + B− + B+)ρs(t )] , The above equation has three different cases: (D5) (i) = : In this case, Eq. (D1) reduces to where the operators A± and B± are defined in Eq. (D4) with = ± 1. To leading order in δt, a sufficient condi- Pρ + δ = Pρ 2 δ , Tr[ s(t t )] Tr[ s(t )] cos t (D2) tion for the steering to be monotonic would be that four- site operator Eˆ = P + (A+ + A−)/2 has only non-negative which shows that the bond ( , + 1) is steered toward the eigenvalues. However, it turns out that the spectrum of Eˆ AKLT state. has both positive and negative eigenvalues. Denoting the

(ii) and are such that the two bonds do not share eigenvalues and eigenvectors of Eˆ as w and |w, the condition P ρ + δ = P ρ ρ Eˆρ = a common site. In this case, Tr[ s(t t )] Tr[ s(t )] on s(t )forEAKLT(t ) to decay (grow) is given by Tr[ s(t )] since the steering has no effect on the reduced density matrix w w|ρ (t )|w ≷ 0. w s of the bond ( , + 1). This observation allows us to specifically construct density

(iii) and are adjacent such that the two bonds in matrices which show nonmonotonic steering as measured question share a common site. In this case, by EAKLT. For example, consider the initial density matrix as ρs(t = 0) =|w−w−|, where w− is the most negative 2 Tr[P (ρs(t + δt ) − ρs(t ))] =−2sin (δt/2)Tr[Aρs(t )] eigenvalue of Eˆ. For such a state, Tr[Eˆρs(t )] is trivially 4 negative. Numerical results for such a situation are shown in + 4sin (δt/2)Tr[Bρs(t )], Fig. 7, where EAKLT(t ) initially grows while Tr[Eˆρs(t )] < 0 (D3) and starts decaying only when Tr[Eˆρs(t )] > 0.

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For such an initial state, it is also interesting compare times. This is analogous to the situation described in the main the behavior of other distance measures, both spatially lo- text where the overlap of the state with the AKLT ground P ( , +1)ρ state stays zero but the state keeps evolving. The Frobenius cal such as Tr[ Stot=2 s(t )] and global such as D1(t ) and DF (t ). The results are shown in Fig. 8. The distance of a distance, DF (t ), on the other hand, decays monotonically to P ( , +1)ρ zero. local reduced density matrix measured via Tr[ Stot=2 s(t )] shows a nonmonotonic behavior. This is a manifestation of The above results show that for many-body systems, mono- the phenomenon that when the bond adjacent to the one being tonicity of steering and the exact condition for convergence to monitored is steered, the former is steered away from the the target state can depend on the distance measure employed. target as such the total energy grows. Crucially, however, these results along with the numerical Turning to global measures of distance, the trace distance, results in Sec. IV B show that the system is guaranteed to be D1(t ), at early times is flat and it decays to zero only at late steered to the AKLT ground state.

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