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arXiv:0803.1463 · Physical Review A

Preparation of entangled states by quantum Markov processes

Barbara Kraus, Hans Peter Büchler, Sebastian Diehl, Adrian Kantian, Andrea Micheli, P. Zoller

Published Oct 8, 2008• 761 citations
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Abstract

We investigate the possibility of using a dissipative process to prepare a quantum system in a desired state. We derive for any multipartite pure state a dissipative process for which this state is the unique stationary state and solve the corresponding master equation analytically. For certain states, such as the cluster states, we use this process to show that the jump operators can be chosen quasilocally, i.e. they act nontrivially only on a few, neighboring qubits. Furthermore, the relaxation time of this dissipative process is independent of the number of subsystems. We demonstrate the general formalism by considering arbitrary matrix-product states or projected entangled pair states. In particular, we show that the ground state of the Affleck-Kennedy-Lieb-Tasaki model can be prepared employing a quasi-local dissipative process.

DOI
10.1103/physreva.78.042307

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We investigate the possibility of using a dissipative process to prepare a quantum system in a desired state. We derive for any multipartite pure state a dissipative process for which this state is the unique stationary state and solve the corresponding master equation analytically. For certain states, such as the cluster states, we use this process to show that the jump operators can be chosen quasilocally, i.e. they act nontrivially only on a few, neighboring qubits. Furthermore, the relaxation time of this dissipative process is independent of the number of subsystems. We demonstrate the general formalism by considering arbitrary matrix-product states or projected entangled pair states. In particular, we show that the ground state of the Affleck-Kennedy-Lieb-Tasaki model can be prepared employing a quasi-local dissipative process.
Preparing interacting quantum systems in low-energy or ground states is a fundamental task in quantum simulation and quantum information processing. In realistic settings, dissipation and active cooling can typically be engineered only on a limited subset of the system. We study the dissipative dynamics of excitation-number conserving quantum systems governed by a GKLS master equation with local jump operators acting on a subset of qubits. We establish sufficient conditions under which such localized dissipation drives the full system to a unique globally attractive zero-excitation state. In particular, we prove that if the Hamiltonian generates excitation transfer described by a graph for which the dissipative subsystem forms a zero forcing set, then the zero-excitation state is the unique globally attractive stationary state. When this state coincides with a ground state of the Hamiltonian, the same mechanism realizes ground-state cooling. Our results provide a graph-theoretic criterion for global state preparation from localized dissipation, which we illustrate using a nearest-neighbor Heisenberg spin chain. For this model, a reduction to the single-excitation sector further yields a scaling estimate with the length of the chain which indicates efficient cooling.
We characterize time-independent Markovian dynamics that drive a finite-dimensional multipartite quantum system into a target (pure) entangled steady state, subject to physical locality constraints. New control schemes are introduced in situations where the desired stabilization task {\em cannot} be attained solely based on quasi-local dissipative means, as considered in previous analysis. The new schemes either allow for Hamiltonian control or, if the latter is not an option, suitably restrict the set of admissible initial states. In both cases, we provide explicit algorithms for constructing a Markovian master equation that achieves the intended objective and show how this genuinely extends the manifold of stabilizable states. In particular, we present dissipative quasi-local control protocols for deterministically engineering multipartite GHZ ``cat'' states and W states on nnn qubits. For GHZ states, we show that no scalable procedure exists for achieving stabilization from arbitrary initial states, whereas this is possible for a target W state by a suitable combination of a two-body Hamiltonian and dissipators. Interestingly, for both entanglement classes, we show that quasi-local stabilization may be {\em scalably} achieved conditional to initialization of the system in a large, appropriately chosen subspace.
We investigate to what extent a suitably chosen system Hamiltonian can counteract local dissipative processes and preserve entanglement in the stationary state. The results determine prospects and limitations of dissipative state preparation schemes based on natural dissipative processes---in contrast to engineered, typically nonlocal dissipative schemes. As an exemplary case, we determine the stationary state of two spontaneously decaying two-level systems with optimal entanglement properties. The corresponding system Hamiltonian is derived, and its possible experimental implementation is discussed in detail. Finally, we discuss generalizations for NNN qubits.
Highly multipartite entangled states play an important role in various quantum computing tasks. We investigate the dissipative generation of a complex entanglement structure as in a cluster state through engineered Markovian dynamics in the spin systems coupled via Ising interactions. Using the Lindblad master equation, we design a projection based dissipative channel that drives the system toward a unique pure steady state corresponding to the desired cluster state. This is done by removing the contribution of the orthogonal states. By explicitly constructing the Liouvillian superoperator in the full 2N2^N2N-dimensional Hilbert space, we compute the steady-state density matrix, the Liouvillian spectral gap, entanglement witness and the fidelity with respect to the ideal cluster state. The results demonstrate that the cluster state emerges as the steady state when the engineered Liouvillian dissipation dominates over the local Ising interaction between spins. Moreover, we find that the fidelity and Liouvillian spectral gap is relatively insensitive to the system size once the saturation dissipation has been achieved that scales linearly with the qubit number. This analysis illustrates a physically realizable path towards steady-state entanglement generation in the spin systems using engineered dissipation.
This paper shows that an arbitrary Gaussian pure state can be deterministically generated in a dissipative open system that has quasilocal interactions between the subsystems and couples to the surrounding environment in a local manner. A quasilocal interaction, which means that the interaction occurs among only a few subsystems, is a crucial requirement for practical engineering of a dissipative system. The key idea is that first an auxiliary system having a local interaction with the environment is prepared and then that auxiliary system is coupled to the underlying target system via a set of two-body Hamiltonians in such a way that a desired pure state is generated. Moreover, we show that even with a simple single-mode auxiliary system, the deterministic generation of an arbitrary approximate Gaussian cluster state is possible by devising an appropriate switching scheme. We discuss in a specific example how much a dissipation-induced pure Gaussian state can be perturbed by decoherence and parameter error.
The controlled preparation of quantum many-body states is a central challenge for quantum technologies. While preparing target states coherently is possible in principle, it is often fragile and experimentally demanding. Dissipative state preparation, where engineered environments drive the system toward desired steady states, provides a complementary and powerful paradigm. This thesis explores dissipative methods for preparing complex many-body states along three distinct directions. First, we investigate the dissipative stabilization of injective matrix product states (MPS). Injective MPS are particularly well suited for dissipative preparation because they arise as ground states of frustration-free parent Hamiltonians, allowing for tailored jump operators to steer the system directly into the target state. We showcase the surprising result that such state preparation works even in the dilute limit where steering only acts on a single link. Moreover, we introduce a formalism for engineering additional coherent interactions, extending the set of accessible states. Second, and complementary to steering, we develop and analyze a universal cooling scheme capable of preparing ground states even in frustrated quantum systems. In contrast to steering methods, universal cooling can overcome the glass floor that forbids steering methods to prepare such frustrated ground states. A key insight here is that the presented cooling scheme is completely system-agnostic making it extremely robust. Finally, we pursue dressing of the system eigenstates by means of either classical drives or coherent interactions as a means to influence the action of dissipation on the system. Focusing first on time-dependent systems, we propose a generalized numerical implementation of non-adiabatic master equations suitable for the use of optimal control tools. Additionally, we investigate how the collective coupling between a spin ensemble and a cavity can steer the system into a spin-squeezed steady state. Together, these three avenues illustrate how tailored dissipative dynamics can extend the frontier of many-body state preparation, from stabilizing injective MPS to cooling complex frustrated systems and generating non-classical steady states in the dressed framework.