Gaussian quantum resource theories

Abstract

We develop a general framework to assess capabilities and limitations of the Gaussian toolbox in continuous variable quantum information theory. Our framework allows us to characterize the structure and properties of quantum resource theories specialized to Gaussian states and Gaussian operations, establishing rigorous methods for their description and yielding a unified approach to their quantification. We show in particular that, under a few intuitive and physically motivated assumptions on the set of free states, no Gaussian quantum resource can be distilled with free Gaussian operations, even when an unlimited supply of the resource state is available. This places fundamental constraints on state manipulations in all such Gaussian resource theories. We discuss in particular the applications to quantum entanglement, where we extend previously known results by showing that Gaussian entanglement cannot be distilled even with Gaussian operations preserving the positivity of the partial transpose, as well as to other Gaussian resources such as steering and optical nonclassicality. A comprehensive semidefinite programming representation of all these resources is explicitly provided. *

All phase-space linear bosonic channels are approximately Gaussian dilatable

“…where: E is an m-mode optical system; U AE is a Gaussian unitary on the bipartite system AE (obtained by combining arbitrary displacements on A and E with symplectic unitaries on AE [46, section 5.1.2]); and E s is an arbitrary state of the system E. For a pictorial representation of equation (20), see figure 1. From a practical point of view, remember that the Gaussian unitary U AE can be implemented by means of multimode interferometers (passive optics) and single-mode squeezers [51].…”

“…If we want to specify that an m-mode ancillary state suffices to implement a Gaussian dilation, we say that the channel is Gaussian dilatable on m modes. The requirement that U SE is a Gaussian unitary here is crucial; in fact, by Stinespringʼs dilation theorem [54] every quantum channel can be represented as in equation (20) for some unitary U AE and some ancillary state E s .…”

“…This is a consequence of the completion theorem for symplectic bases [55, theorem 1.15]. As we mentioned before, Gaussian channels are always Gaussian dilatable, and the corresponding state E s of equation (20) can always be chosen to be Gaussian [40,56]. Another important subclass of Gaussian dilatable channels is that composed of passive dilatable channels [4,57] 2.3.…”

“…In this context, corollary 9 should be compared with the results of [40]. There the authors give an upper bound on the number of modes that are needed in order to construct a Gaussian dilation of any given Gaussian channel (with the ancillary state in equation (20) being also Gaussian). For instance, for a Gaussian additive noise channel, i.e.…”

“…that in spite of the rich structure Gaussian entanglement exhibits [12][13][14][15][16], it turns out that it can not be distilled using Gaussian operations alone [17][18][19]. More generally, a similar no-go result holds for generic state conversion tasks in arbitrary Gaussian resource theories [20]. On the other hand, non-Gaussian operations are provably necessary to realise universal quantum computation [21] and many other quantum information processing tasks [22][23][24][25][26][27][28][29][30].…”