We show that quantum circuits where the initial state and all the following quantum operations can be represented by positive Wigner functions can be classically efficiently simulated. This is true both for continuousvariable as well as discrete variable systems in odd prime dimensions, two cases which will be treated on entirely the same footing. Noting the fact that Clifford and Gaussian operations preserve the positivity of the Wigner function, our result generalizes the Gottesman-Knill theorem. Our algorithm provides a way of sampling from the output distribution of a computation or a simulation, including the efficient sampling from an approximate output distribution in case of sampling imperfections for initial states, gates, or measurements. In this sense, this work highlights the role of the positive Wigner function as separating classically efficiently simulatable systems from those that are potentially universal for quantum computing and simulation, and it emphasizes the role of negativity of the Wigner function as a computational resource.What renders a quantum computer a superior computational device? Where is the precise boundary between classically efficiently simulable problems and ones for which this is no longer possible? Despite a significant research effort and partial progress [1][2][3][4][5][6][7][8][9][10], these questions are still largely open. Quantum correlations surely play a role in one way or the other in quantum computers and simulators outperforming their classical counterpart. For example, if the entanglement is -in a precise sense -too low in a pure-state computation with respect to any bi-partite split, then one can classically efficiently simulate the dynamics [2-6]. In measurement-based computing specifically [11,12], where the resource character of entanglement is particularly manifest, states can be too little entangled [2], but also in a sense too entangled [7]. Possibly the singularly most important result on classical simulatability of quantum computers is the Gottesman-Knill theorem, stating that stabilizer circuits consisting of Clifford gates only can be classically efficiently simulated [1,8,13]. Though, adding almost any further gate will render the Clifford gate set universal for quantum computing. Similarly, Gaussian operations for continuous-variable systems can be efficiently simulated [16,17].Here we present a generalization of the Gottesman-Knill theorem, stating that one can efficiently classically sample from quantum circuits starting from product states with a positive Wigner function, applying quantum gates that have a positive Wigner function (in a sense made precise below) and performing measurements associated with positive Wigner functions. This result holds true both for discrete variable systems where the constituents have odd prime dimension (it can easily be generalized to arbitrary odd dimension) as well as for continuous-variable systems so common in quantum optics. In fact, these two situations can be treated on exactly the same footing -since at...

We perform an analysis of the optomechanical entanglement between the experimentally detectable output field of an optical cavity and a vibrating cavity end-mirror. We show that by a proper choice of the readout (mainly by a proper choice of detection bandwidth) one can not only detect the already predicted intracavity entanglement but also optimize and increase it. This entanglement is explained as being generated by a scattering process owing to which strong quantum correlations between the mirror and the optical Stokes sideband are created. All-optical entanglement between scattered sidebands is also predicted and it is shown that the mechanical resonator and the two sideband modes form a fully tripartite-entangled system capable of providing practicable and robust solutions for continuous variable quantum communication protocols.

We introduce a framework of opto-mechanical systems that are driven with a mildly amplitude-modulated light field, but that are not subject to classical feedback or squeezed input light. We find that in such a system one can achieve large degrees of squeezing of a mechanical micromirror -signifying quantum properties of opto-mechanical systems -without the need of any feedback and control, and within parameters reasonable in experimental settings. Entanglement dynamics is shown of states following classical quasi-periodic orbits in their first moments. We discuss the complex time-dependence of the modes of a cavity-light field and a mechanical mode in phase space. Such settings give rise to certifiable quantum properties within experimental conditions feasible with present technology.Periodically driven quantum systems exhibit a rich behavior and display non-equilibrium properties that are absent in their static counterparts. By appropriately exploiting time-periodic driving, strongly correlated Bose-Hubbard-type models can be dynamically driven to quantum phase transitions [1], systems can be dynamically decoupled from their environments to avoid decoherence in quantum information science [2], and quite intriguing dynamics of Rydberg atoms strongly driven by microwaves [3] can arise. It has also been muted that such time-dependent settings may give rise to entanglement dynamics in oscillating molecules [4]. A framework of such periodically driven systems is provided by the theory of linear differential equations with periodic coefficients or inhomogeneities, including Floquet's theorem [5].In this work, we aim at transferring such ideas to describe a new and in fact quite simple regime of opto-mechanical systems, of micromirrors as part of a Fabry-Perot cavity [6,7,8,9]: So to one of the settings [10,11,12,13,14] that are the most promising candidates in the race of exploring certifiable quantum effects involving macroscopic mechanical modes. This is an instance of a regime of driving with mildly amplitude-modulated light. We find that in this regime, high degrees of squeezing below the vacuum noise level can be reached, signifying genuine quantum dynamics. More specifically, in contrast to earlier descriptions of opto-mechanical systems with a periodic time-dependence in some aspect of the description, we will not rely on classical feedback based on processing of measurement-outcomes -a promising idea in its own right in a continuous-measurement perspective [15,16] -or resort to driving with squeezed light. Instead, we will consider the plain setting of a time-periodic amplitude modulation of an input light. The picture developed here, based in the theory of differential equations, gives rise to a framework of describing such situations. We find that large degrees of squeezing can be reached (complementing other very recent non-periodic approaches based on cavity-assisted squeezing using an additional squeezed light beam [17]). It is the practical appeal of this work that such quantum signatures can be reached...