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Physical Review A

Continuous-variable entanglement mediated by a thermal oscillator

Vojtěch Kupčík, Radim Filip

Published Aug 21, 2015• 5 citations
Publisher

Abstract

We discuss the deterministic Gaussian entangling procedure between continuous variables of two directly noninteracting oscillators. The oscillators can only couple to the same mediating oscillator, which is initially at arbitrary noisy state and can be lost after the procedure. Unitary idealization of the procedure can generate large entanglement for any initial thermal noise of the mediator, thanks to the quantum version of the geometric phase. The generation of entanglement may considerably change when the procedure contains a weak dissipation. A robustness of this entangling procedure for weak dissipation is therefore a basic test of feasibility, especially in the limit of noisy mediator. We theoretically confirm the basic robustness of entanglement generation. Moreover, we suggest a nontrivial numerical optimization of the entangling procedure to approach an absolute robustness. For the weak dissipation, the optimization allows complete avoiding of the impact of the mediator noise. The suggested entangling procedure is therefore verified to be sufficiently robust for further development of quantum transducers.

DOI
10.1103/physreva.92.022346

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Sep 17, 2009•0 citations
We discuss the deterministic Gaussian entangling procedure between continuous variables of two directly noninteracting oscillators. The oscillators can only couple to the same mediating oscillator, which is initially at arbitrary noisy state and can be lost after the procedure. Unitary idealization of the procedure can generate large entanglement for any initial thermal noise of the mediator, thanks to the quantum version of the geometric phase. The generation of entanglement may considerably change when the procedure contains a weak dissipation. A robustness of this entangling procedure for weak dissipation is therefore a basic test of feasibility, especially in the limit of noisy mediator. We theoretically confirm the basic robustness of entanglement generation. Moreover, we suggest a nontrivial numerical optimization of the entangling procedure to approach an absolute robustness. For the weak dissipation, the optimization allows complete avoiding of the impact of the mediator noise. The suggested entangling procedure is therefore verified to be sufficiently robust for further development of quantum transducers.
We propose a setup allowing to entangle two directly non-interacting radiation modes applying four sequential pulsed quantum resonant interactions with a noisy vibrational mode of a mechanical oscillator which plays the role of the mediator. We analyze Gaussian entanglement of the radiation modes generated by the transducer and confirm that the noisy mechanical mode can mediate generation of entanglement. The entanglement, however, is limited if the interaction gains are not individually optimized. We prove the robustness of the transducer to optical losses and the influence of the mechanical bath and propose the ways to achieve maximal performance through the individual optimization.
We consider two separated oscillators initially in equilibrium and continuously interacting with thermal environments and propose a way to entangle them using a mediating qubit. An appropriate interaction allows for an analytic treatment of the open system, removes the necessity of fine-tuning interaction times, and results in a high tolerance of the entanglement to finite temperature. The entanglement thus produced between the oscillators can be verified either through a Bell inequality relying on oscillator parity measurements or through conditional extraction of the entanglement on two mutually noninteracting qubits. The latter process also shows that the generated mixed-entangled state of the oscillators is an useful resource for entangling qubits. By allowing for influences from environments, taking feasible qubit-oscillator interactions and measurement settings, this scheme should be implementable in a variety of experimental setups. The method presented for the solution of the master equation can also be adapted to a variety of problems involving the same form of qubit-oscillator interaction
We propose a feasible Gaussian version of the Schr\"odinger-cat state. We consider it as a robust Gaussian state of an oscillator prepared in a low-energy Gaussian quantum state entangled with another oscillator which was initially in a quantum state at thermal equilibrium with an arbitrary large energy (macroscopic oscillator). To generate it, we suggest two-mode squeezing as the best candidate. We test that the Gaussian version of the Schr\"odinger-cat state can be also used to remotely prepare a squeezed state of the macroscopic oscillator.
A meaningful description of the low-energy dynamics of multipartite entanglement is provided\nfor harmonic oscillator systems in general dissipative scenarios. Without doing any\ncentral approximation, this mainly relies on a set of reasonable physical statements on the\nenvironment and system-environment interaction consistent with a linear analysis of the\nopen-system dynamics.\nWe rst investigate the inseparability properties of arbitrary entangled states, which\ngenerally entails an optimization procedure of certain functional de ned in the in nitedimensional\nHilbert space of continuous-variable systems. Using a Gaussian-like assumption\nanalogous to that used in the derivation of Gaussian entanglement of formation, we\nderive a computational-e cient form of such functional which considerable simpli es the\noptimization task. This consists on a hierarchy of separability criteria that permit in a\nuni ed way to characterize k-partite entanglement of broad classes of Gaussian and non-\nGaussian states. The strength of the criteria is proven by showing that they satisfactorily\nreproduce previous results from the theory of entanglement, like PPT criterion applied to\narbitrary two-mode and pure three-mode Gaussian states.\nThe separability criteria allow to monitor the transient evolution of multipartite entanglement\nunder the environmental in\nuence, this permitted to show that non-Gaussian\nentanglement may be as robust against harmful dissipative e ects and thermal noise as\nGaussian one. We devote special attention to the stationary properties of tripartite entanglement\nwhen the system oscillators are in contact with either a common environment\nor independent environments at initial di erent temperatures. For the former dissipative\nscheme, it is shown that the environment mediates an e ective many-party interaction with\na spatial long-range feature between system oscillators, which is able to generate tripartite\nentanglement whereas two-mode entanglement is degraded. Regarding the second scheme,\nit is illustrated that thermal non-equilibrium conditions result in a rich variety of quantum\ncorrelations, however a temperature gradient eventually destroys the entanglement shared\nby the system oscillators as a consequence of the growth of thermal noise. Finally, the\nstationary entanglement and energy current across a nite harmonic chain are studied in\ndetail by doing an extensive numerical analysis of both phenomena for a broad range of the\nparameters (i.e., temperatures, oscillator frequencies, and oscillator coupling strengths).\nOur numerical ndings are discussed in terms of the derived energy current expressions,\nthese show an explicit dependence on the two-time correlation functions (between oscillator\noperators) which carry quantum correlations.
We study the task of distilling entanglement by a coherent superposition operation $t\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}+r{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}$ applied to a continuous-variable state under a thermal noise. In particular, we compare the performances of two different strategies; i.e., the non-Gaussian operation $t\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}+r{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}$ is applied before or after the noisy Gaussian channel. This is closely related to a fundamental problem of whether Gaussian or non-Gaussian entanglement can be more robust under a noisy channel and also provides a useful insight into the practical implementation of entanglement distribution for a long-distance quantum communication. We specifically look into two entanglement characteristics, the logarithmic negativity as a measure of entanglement and the teleportation fidelity as a usefulness of entanglement, for each distilled state. We find that the non-Gaussian operation after (before) the thermal noise becomes more effective in the low- (high-) temperature regime.
Using the theory of open systems based on completely positive quantum dynamical semigroups, we describe the dynamics of entanglement for a system consisting of two uncoupled harmonic oscillators interacting with a thermal reservoir. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for a Gaussian input state. For some values of the temperature of environment, the state keeps for all times its initial type - separable or entangled. In other cases, entanglement generation, entanglement sudden death or a repeated collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of quantum entanglement.