Measurement-based quantum correlation mimics several characteristics of multipartite quantum correlations and at the same time, it reduces the parent system to a smaller subsystem. On the other hand, genuine multipartite entanglement measures can capture certain features of a multisite composite system that are inaccessible via bipartite quantum correlation quantifiers. We merge these two concepts by introducing localizable genuine multimode entanglement for continuous variable systems, both for Gaussian and non-Gaussian multimode parent states. We report a compact form of localizable generalized geometric measure for multimode Gaussian states when Gaussian measurements are performed in some of the modes. We show that non-Gaussian measurements can concentrate more genuine multimode entanglement compared to the Gaussian ones. For non-Gaussian states with non-Gaussian measurements, we find that although four-mode squeezed vacuum state has permutation symmetry with respect to the exchange of first and third modes as well as the second and the fourth modes, the symmetry can be broken by performing measurements in one of the modes in case of addition while for subtraction, such symmetry is preserved, thereby providing a method for distinguishing multimode photon-added and -subtracted states via localizations.
Measurement-based quantum correlation mimics several characteristics of multipartite quantum correlations and at the same time, it reduces the parent system to a smaller subsystem. On the other hand, genuine multipartite entanglement measures can capture certain features of a multisite composite system that are inaccessible via bipartite quantum correlation quantifiers. We merge these two concepts by introducing localizable genuine multimode entanglement for continuous variable systems, both for Gaussian and non-Gaussian multimode parent states. We report a compact form of localizable generalized geometric measure for multimode Gaussian states when Gaussian measurements are performed in some of the modes. We show that non-Gaussian measurements can concentrate more genuine multimode entanglement compared to the Gaussian ones. For non-Gaussian states with non-Gaussian measurements, we find that although four-mode squeezed vacuum state has permutation symmetry with respect to the exchange of first and third modes as well as the second and the fourth modes, the symmetry can be broken by performing measurements in one of the modes in case of addition while for subtraction, such symmetry is preserved, thereby providing a method for distinguishing multimode photon-added and -subtracted states via localizations.
Genuine multimode entanglement in continuous variable systems can be quantified by exploring the geometry of the state space, namely, via the generalized geometric measure (GGM). It is defined by the shortest distance of a given multimode state from a nongenuinely multimode entangled state. For the multimode Gaussian states, we derive a closed-form expression of GGM in terms of the symplectic eigenvalues of the reduced states. Following that prescription, the characteristics of GGM for typical three- and four-mode Gaussian states are investigated. In the non-Gaussian paradigm, we compute GGM for photon-added as well as -subtracted states having three and four modes and find that both addition and subtraction of photons enhance the genuine multimode entanglement of the state compared to its Gaussian counterpart. Our analysis reveals that, when an initial three-mode vacuum state is evolved according to an interacting Hamiltonian, photon addition is more beneficial in increasing GGM compared to photon subtraction, while the scenario reverses when one considers the four mode non-Gaussian states. Specifically, subtracting photons from four-mode squeezed vacuum states almost always results in higher multimode entanglement content than that of photon addition to both single as well as multimode and constrained as well as unconstrained operations. Furthermore, we observe that GGM freezes under subtraction of photons involving multiple modes, in some specific cases. This feature does not appear while adding photons. Finally, we relate the enhancements of GGM with the distance-based non-Gaussianity measure.
The characterisation of quantum correlations, also known as entanglement, is central to quantum information theory. Operationally, entanglement gives rise to the notion of quantum non-locality, which challenges the ‘reality’ of quantum mechanics (adopting the terminology of Einstein, Podolski and Rosen) and arguably represents its most distinctive feature, in the sense that it calls for a departure from certain foundational tenets of classical physics. It should thus not come as a surprise that entanglement turns out to be a key resource for the implementation of a number of quantum communication protocols and also necessary to achieve computational speed-ups with quantum hardware, in the sense that accessing regions of the Hilbert space with entangled states allows one to shorten the computational depth of certain algorithms. Not surprisingly, generating and maintaining entanglement is generally difficult in practice, as it requires strong interactions between selected components of a system that must at the same time be isolated from the environment. Thanks to the techniques developed in quantum optical set-ups to achieve squeezed light, entanglement is, however, comparatively accessible with optical quantum continuous variables. As we shall see, in fact, continuous variable entanglement is closely related to the notion of squeezing. It is therefore of the utmost importance to be able to qualify and quantify the entanglement of quantum continuous variables, which is the subject matter of the present chapter. The first part of the chapter is devoted to techniques and criteria based on second moments to detect the entanglement of Gaussian states and also to the description of possible ways to quantify such quantum correlations, bearing in mind that the determination of entanglement monotones of clear operational significance is still an open problem, even in the restricted arena of Gaussian states. Although the criteria based on second moments are sufficient, but not necessary, for the detection of non-Gaussian entanglement too, we will push our analysis further by introducing more general criteria based on higher order statistical moments, which will be put to use to detect, on paper, the quantum correlations of an entangled cat-like state. We will also address the issues of entanglement distillation and quantum non-locality tests, in whose contexts we shall encounter other non-Gaussian operations and states, such as photon-subtracted states.
Genuine multipartite entanglement underlies correlation experiments corroborating quantum mechanics and it is an expedient empowering many quantum technologies. One of many counterintuitive facets of genuine multipartite entanglement is its ability to exhibit an emergent character. That is, one can infer its presence in some multipartite states merely from a set of its separable marginals. Here we show that the effect can also be found in the context of Gaussian states of bosonic systems. Specifically, we construct examples of multimode Gaussian states carrying genuine multipartite entanglement which can be verified solely from separable nearest-neighbor two-mode marginals. The key tool of our construction is an entanglement witness acting only on some two-mode reductions of the global covariance matrix, which we find by a numerical solution of a semidefinite program. We also propose an experimental scheme for preparation of the simplest three-mode state, which requires interference of three correlatively displaced squeezed beams on two beam splitters. Besides revealing the concept of emergent genuine multipartite entanglement in the Gaussian scenario and bringing it closer to experimentally testable form, our results pave the way to effective diagnostics methods of global properties of multipartite states without complete tomography.
Genuine multipartite entanglement underlies correlation experiments corroborating quantum mechanics and it is an expedient empowering many quantum technologies. One of many counterintuitive facets of genuine multipartite entanglement is its ability to exhibit an emergent character, that is, one can infer its presence in some multipartite states merely from a set of its separable marginals. Here, we show that the effect can be found also in the context of Gaussian states of bosonic systems. Specifically, we construct examples of multimode Gaussian states carrying genuine multipartite entanglement which can be verified solely from separable nearest-neighbour two-mode marginals. The key tool of our construction is a genuine multipartite entanglement witness acting only on some two-mode reductions of the global covariance matrix, which we find by a numerical solution of a semi-definite programme. We also propose an experimental scheme for preparation of the simplest three-mode state, which requires interference of three correlatively displaced squeezed beams on two beam splitters. Besides revealing the concept of emergent genuine multipartite entanglement in the Gaussian scenario and bringing it closer to experimentally testable form, our results pave the way to effective diagnostics methods of global properties of multipartite states without complete tomography.
Multipartite entanglement is an essential resource for quantum information tasks, but characterizing entanglement structures in continuous-variable systems remains challenging, especially in multimode non-Gaussian scenarios. In this Letter, we introduce an efficient method for detecting multipartite entanglement structures in continuous-variable states. Based on the quantum Fisher information, we propose a systematic approach to identify an encoding operator that can efficiently capture the quantum correlations in multimode non-Gaussian states. We demonstrate the effectiveness of our method on over 10^{5} randomly generated multimode-entangled quantum states, achieving a very high success rate in entanglement detection. Additionally, the robustness of our method can be considerably enhanced against losses by expanding the set of accessible operators. This Letter provides a general framework for characterizing entanglement structures in diverse continuous-variable systems, enabling a number of experimentally relevant applications.
A quantum correlation N F G , A for ( n + m ) -mode continuous-variable systems is introduced in terms of local Gaussian unitary operations performed on Subsystem A based on Uhlmann fidelity F. This quantity is a remedy for the local ancilla problem associated with the geometric measurement-induced correlations; is local Gaussian unitary invariant; is non-increasing under any Gaussian quantum channel performed on Subsystem B;and is an entanglement monotone when restricted to pure Gaussian states in the ( 1 + m ) -mode case. A concrete formula for ( 1 + 1 ) -mode symmetric squeezed thermal states (SSTSs) is presented. We also compare N F G , A with other quantum correlations in scale, such as Gaussian quantum discord and Gaussian geometric discord, for two-mode SSTSs, which reveals that N F G , A has some advantage in detecting quantum correlations of Gaussian states.