We introduce a family of criteria to detect quantum non-Gaussian states of a harmonic oscillator, that is, quantum states that can not be expressed as a convex mixture of Gaussian states. In particular we prove that, for convex mixtures of Gaussian s
tates, the value of the Wigner function at the origin of phase space is bounded from below by a non-zero positive quantity, which is a function only of the average number of excitations (photons) of the state. As a consequence, if this bound is violated then the quantum state must be quantum non-Gaussian. We show that this criterion can be further generalized by considering additional Gaussian operations on the state under examination. We then apply these criteria to various non-Gaussian states evolving in a noisy Gaussian channel, proving that the bounds are violated for high values of losses, and thus also for states characterized by a positive Wigner function.
We study the possibility of taking bosonic systems subject to quadratic Hamiltonians and a noisy thermal environment to non-classical stationary states by feedback loops based on weak measurements and conditioned linear driving. We derive general ana
lytical upper bounds for the single mode squeezing and multimode entanglement at steady state, depending only on the Hamiltonian parameters and on the number of thermal excitations of the bath. Our findings show that, rather surprisingly, larger number of thermal excitations in the bath allow for larger steady-state squeezing and entanglement if the efficiency of the optimal continuous measurements conditioning the feedback loop is high enough. We also consider the performance of feedback strategies based on homodyne detection and show that, at variance with the optimal measurements, it degrades with increasing temperature.
Controllability -- the possibility of performing any target dynamics by applying a set of available operations -- is a fundamental requirement for the practical use of any physical system. For finite-dimensional systems, as for instance spin systems,
precise criterions to establish controllability, such as the so called rank criterion, are well known. However most physical systems require a description in terms of an infinite-dimensional Hilbert space whose controllability properties are poorly understood. Here, we investigate infinite-dimensional bosonic quantum systems -- encompassing quantum light, ensembles of bosonic atoms, motional degrees of freedom of ions, and nano-mechanical oscillators -- governed by quadratic Hamiltonians (such that their evolution is analogous to coupled harmonic oscillators). After having highlighted the intimate connection between controllability and recurrence in the Hilbert space, we prove that, for coupled oscillators, a simple extra condition has to be fulfilled to extend the rank criterion to infinite dimensional quadratic systems. Further, we present a useful application of our finding, by proving indirect controllability of a chain of harmonic oscillators.
We address realistic schemes for the generation of non-Gaussian states of light based on conditional intensity measurements performed on correlated bipartite states. We consider both quantum and classically correlated states and different kind of det
ection, comparing the resulting non Gaussianity parameters upon varying the input energy and the detection efficiency. We find that quantum correlations generally lead to higher non Gaussianity, at least in the low energy regime. An experimental implementation feasible with current technology is also suggested.
We introduce a novel measure to quantify the non-Gaussian character of a quantum state: the quantum relative entropy between the state under examination and a reference Gaussian state. We analyze in details the properties of our measure and illustrat
e its relationships with relevant quantities in quantum information as the Holevo bound and the conditional entropy; in particular a necessary condition for the Gaussian character of a quantum channel is also derived. The evolution of non-Gaussianity (nonG) is analyzedfor quantum states undergoing conditional Gaussification towards twin-beam and de-Gaussification driven by Kerr interaction. Our analysis allows to assess nonG as a resource for quantum information and, in turn, to evaluate the performances of Gaussification and de-Gaussification protocols.
We address truncated states of continuous variable systems and analyze their statistical properties numerically by generating random states in finite-dimensional Hilbert spaces. In particular, we focus to the distribution of purity and non-Gaussianit
y for dimension up to d=21. We found that both quantities are distributed around typical values with variances that decrease for increasing dimension. Approximate formulas for typical purity and non-Gaussianity as a function of the dimension are derived.
Entanglement does not correspond to any observable and its evaluation always corresponds to an estimation procedure where the amount of entanglement is inferred from the measurements of one or more proper observables. Here we address optimal estimati
on of entanglement in the framework of local quantum estimation theory and derive the optimal observable in terms of the symmetric logarithmic derivative. We evaluate the quantum Fisher information and, in turn, the ultimate bound to precision for several families of bipartite states, either for qubits or continuous variable systems, and for different measures of entanglement. We found that for discrete variables, entanglement may be efficiently estimated when it is large, whereas the estimation of weakly entangled states is an inherently inefficient procedure. For continuous variable Gaussian systems the effectiveness of entanglement estimation strongly depends on the chosen entanglement measure. Our analysis makes an important point of principle and may be relevant in the design of quantum information protocols based on the entanglement content of quantum states.
We address the issue of quantifying the non-Gaussian character of a bosonic quantum state and introduce a non-Gaussianity measure based on the Hilbert-Schmidt distance between the state under examination and a reference Gaussian state. We analyze in
details the properties of the proposed measure and exploit it to evaluate the non-Gaussianity of some relevant single- and multi-mode quantum states. The evolution of non-Gaussianity is also analyzed for quantum states undergoing the processes of Gaussification by loss and de-Gaussification by photon-subtraction. The suggested measure is easily computable for any state of a bosonic system and allows to define a corresponding measure for the non-Gaussian character of a quantum operation.
