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 illustrate 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 analyzed for 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. PACS numbers: 03.67.-a, 03.65.Bz, 42.50.DvIntroduction-The use of Gaussian states and operations allows the implementation of relevant quantum information protocols including teleportation, dense coding and quantum cloning [1]. Indeed, the Gaussian sector of the Hilbert space plays a crucial role in quantum information processing with continuous variables (CV), especially for what concerns quantum optical implementations [2]. On the other hand, quantum information protocols required for long distance communication, as for example entanglement distillation and entanglement swapping, require nonG operations [3]. Besides, it has been demonstrated that using nonG states and operations teleportation [4,5,6] and cloning [7] of quantum states may be improved. Indeed, de-Gaussification protocols for singlemode and two-mode states have been proposed [4,5,6,8,9] and realized [10]. From a more theoretical point of view, it should be noticed that any strongly superadditive and continuous functional is minimized, at fixed covariance matrix (CM), by Gaussian states. This is crucial to prove extremality of Gaussian states and Gaussian operations [11,12] for various quantities such as channel capacities [13], multipartite entanglement measures [14] and distillable secret key in quantum key distribution protocols. Overall, nonG appears to be a resource for CV quantum information and a question naturally arises on whether a convenient measure to quantify the nonG character of a quantum state may be introduced. Notice that the notion of nonG already appeared in classical statistics in the framework of independent component analysis [15].
We develop a resource theory for continuous-variable systems grounded on operations routinely available within current quantum technologies. In particular, the set of free operations is convex and includes quadratic transformations and conditional coarse-grained measurements. The present theory lends itself to quantify both quantum non-Gaussianity and Wigner negativity as resources, depending on the choice of the free-state seti.e., the convex hull of Gaussian states or the states with positive Wigner function, respectively. After showing that the theory admits no maximally resourceful state, we define a computable resource monotone -the Wigner logarithmic negativity. We use the latter to assess the resource content of experimentally relevant states -e.g., photon-added, photon-subtracted, cubic-phase, and cat states -and to find optimal working points of some resource concentration protocols. We envisage applications of this framework to sub-universal and universal quantum information processing over continuous variables. * francesco.albarelli@unimi.it † marco.genoni@fisica.unimi.it ‡ matteo.paris@fisica.unimi.it § a.ferraro@qub.ac.uk arXiv:1804.05763v2 [quant-ph] 4 Dec 2018 1. M(ρ) = 0 ∀ρ ∈ G (resp. W + ).
We address the quantification of non-Gaussianity of states and operations in continuous-variable systems and its use in quantum information. We start by illustrating in details the properties and the relationships of two recently proposed measures of non-Gaussianity based on the Hilbert-Schmidt (HS) distance and the quantum relative entropy (QRE) between the state under examination and a reference Gaussian state. We then evaluate the non-Gaussianities of several families of non-Gaussian quantum states and show that the two measures have the same basic properties and also share the same qualitative behaviour on most of the examples taken into account. However, we also show that they introduce a different relation of order, i.e. they are not strictly monotone each other. We exploit the non-Gaussianity measures for states in order to introduce a measure of non-Gaussianity for quantum operations, to assess Gaussification and de-Gaussification protocols, and to investigate in details the role played by non-Gaussianity in entanglement distillation protocols. Besides, we exploit the QRE-based non-Gaussianity measure to provide new insight on the extremality of Gaussian states for some entropic quantities such as conditional entropy, mutual information and the Holevo bound. We also deal with parameter estimation and present a theorem connecting the QRE non-Gaussianity to the quantum Fisher information. Finally, since evaluation of the QRE non-Gaussianity measure requires the knowledge of the full density matrix, we derive some experimentally friendly lower bounds to non-Gaussianity for some class of states and by considering the possibility to perform on the states only certain efficient or inefficient measurements.
Phase estimation, at the heart of many quantum metrology and communication schemes, can be strongly affected by noise, whose amplitude may not be known, or might be subject to drift. Here we investigate the joint estimation of a phase shift and the amplitude of phase diffusion at the quantum limit. For several relevant instances, this multiparameter estimation problem can be effectively reshaped as a two-dimensional Hilbert space model, encompassing the description of an interferometer probed with relevant quantum statessplit single-photons, coherent states or N00N states. For these cases, we obtain a trade-off bound on the statistical variances for the joint estimation of phase and phase diffusion, as well as optimum measurement schemes. We use this bound to quantify the effectiveness of an actual experimental set-up for joint parameter estimation for polarimetry. We conclude by discussing the form of the trade-off relations for more general states and measurements.
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