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arXiv:1211.5788 · Physical Review A

Deterministic generation of Gaussian pure states in a quasilocal dissipative system

Yusuke Ikeda, Naoki Yamamoto

Published Mar 1, 2013• 25 citations
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Abstract

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.

DOI
10.1103/physreva.87.033802

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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.
This paper provides some necessary and sufficient conditions for a general Markovian Gaussian master equation to have a unique pure steady state. The conditions are described by simple matrix equations; thus the so-called environment engineering problem for pure-Gaussian-state preparation can be straightforwardly dealt with in the linear algebraic framework. In fact, based on one of those conditions, for an arbitrary given pure Gaussian state, we obtain a complete parametrization of the Gaussian master equation having that state as a unique steady state; this leads to a systematic procedure for engineering a desired dissipative system. We demonstrate some examples including Gaussian cluster states.
Recently, the complete characterization of a general Gaussian dissipative system having a unique pure steady state was obtained. This result provides a clear guideline for engineering an environment such that the dissipative system has a desired pure steady state such as a cluster state. In this paper, we describe the system in terms of a quantum stochastic differential equation (QSDE) so that the environment channels can be explicitly dealt with. Then, a physical meaning of that characterization, which cannot be seen without the QSDE representation, is clarified; more specifically, the nullifier dynamics of any Gaussian system generating a unique pure steady state is passive. In addition, again based on the QSDE framework, we provide a general and practical method to implement a desired dissipative Gaussian system, which has a structure of quantum state transfer.
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.
We study the dissipative preparation of many-body entangled Gaussian states in bosonic lattice models which could be relevant for quantum technology applications. We assume minimal resources, represented by systems described by particle-conserving quadratic Hamiltonians, with a single localized squeezed reservoir. We show that in this way it is possible to prepare, in the steady state, the wide class of pure states which can be generated by applying a generic passive Gaussian transformation on a set of equally squeezed modes. This includes nontrivial multipartite entangled states such as cluster states suitable for measurement-based quantum computation.
When only local observables of a many-body quantum system are of interest, it is desirable to formulate a reduced description within the Hilbert space of the corresponding subsystem, with the remaining degrees of freedom traced out and acting as an environment. Assuming initially uncorrelated states and Gaussian environments, we develop a framework for reconstructing the local dynamical generator of noninteracting quantum chains, with polynomial computational complexity. As an application, we consider two representative models: a bipartitioned Kitaev chain and a Kitaev chain boundary-coupled to a fully connected free-fermion environment. In both models, strong subsystem-environment coupling leads to non-Markovian dynamics characterized by ballistic spreading of the Lindblad dissipator support within the subsystem. On the other hand, weak coupling to a fully connected environment yields predominantly boundary-localized, Markovian dissipation. Our work highlights the implications of subsystem-environment correlations on the generator of local dynamics, beyond the conventional weak-coupling approximations.
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.