Separability Condition Research Articles

Tracking Performance in LoRaWAN-Like Systems and Equivalence of a Class of Distributed Learning Algorithms

Considering a low power wide area random access system (like LoRaWAN) where the individual transmitters are mobile, we 1) propose a way of quantifying the performance of tracking of the mobile devices, and 2) design a distributed algorithm to achieve a target tracking performance. The insights gained are then used to provide an analysis of a family of target-achieving reinforcement-learning algorithms used in the literature to learn the optimal (Nash Equilibrium) random access probabilities. By mapping the payoff function in the equivalent game to the performance metric, we establish that, under some general conditions of inter-node parameter separability, the algorithm convergence is independent of the payoff function used. The mapping from the desired performance metric to the success probability in random access can be used in the algorithm to achieve a target success probability.

On the disentanglement of Gaussian quantum states by symplectic rotations

We show that every Gaussian mixed quantum state can be disentangled by conjugation with a unitary operator corresponding to a symplectic rotation via the metaplectic representation of the symplectic group. The main tools we use are the Werner–Wolf condition for separability on covariance matrices and the symplectic covariance of Weyl pseudo-differential operators.

Empirical Underdetermination for Physical Theories in C* Algebraic Setting: Comments to an Arageorgis's Argument

In this paper, I reconstruct an argument of Aristidis Arageorgis against empirical underdetermination of the state of a physical system in a C*-algebraic setting and explore its soundness. The argument, aiming against algebraic imperialism, the operationalist attitude which characterized the first steps of Algebraic Quantum Field Theory, is based on two topological properties of the state space: being T1 and being first countable in the weak*-topology. The first property is possessed trivially by the state space while the latter is highly non-trivial, and it can be derived from the assumption of the algebra of observables’ separability. I present some cases of classical and of quantum systems which satisfy the separability condition, and others which do not, and relate these facts to the dimension of the algebra and to whether it is a von Neumann algebra. Namely, I show that while in the case of finite-dimensional algebras of observables the argument is conclusive, in the case of infinite-dimensional von Neumann algebras it is not. In addition, there are cases of infinite-dimensional quasilocal algebras in which the argument is conclusive. Finally, I discuss Porrmann's construction of a net of local separable algebras in Minkowski spacetime which satisfies the basic postulates of Algebraic Quantum Field Theory.

Separation of variables, Lax-integrable systems and [formula omitted]-valued classical [formula omitted]-matrices

In the present paper we consider a problem of separation of variables for Lax-integrable hamiltonian system generated by general non-skew-symmetric gl(2)⊗gl(2)-valued classical r-matrix r(u,v) with spectral parameters. We specify the general separability condition of Dubrovin and Skrypnyk (2018) on the components of the r-matrix and consider two new classes of examples of classical non-skew-symmetric r-matrices r(u,v) for which the separability condition is satisfied and the whole construction produces a complete set of canonical coordinates. For each of these r-matrices we consider in details the corresponding classical Gaudin-type models with N spins. In the case N=2 we explicitly find coordinates and momenta of separation, reconstruction formulae and Abel–Jacobi quadratures. Some potential physical applications of the considered models are discussed.

Separability of heterogeneous multipartite quantum systems using Bloch Vectors

Refining recent results, we propose an upper bound for the norms of Bloch Vectors for heterogeneous quantum systems containing an arbitrary number of qudits with possibly different numbers of levels. We also generalize recent approaches for tripartite and four-partite heterogeneous systems and for multipartite homogeneous systems. Our results imply a necessary algebraic condition for separability of a general multipartite heterogeneous quantum system under arbitrary partitioning.

General mapping of multiqudit entanglement conditions to nonseparability indicators for quantum-optical fields

We show that any multi-qudit entanglement witness leads to a non-separability indicator for quantum optical fields, which involves intensity correlations. We get, e.g., necessary and sufficient conditions for intensity or intensity-rate correlations to reveal polarization entanglement. We also derive separability conditions for experiments involving multiport interferometers, now feasible with integrated optics. We show advantages of using intensity rates rather than intensities, e.g., a mapping of Bell inequalities to ones for optical fields. The results have implication for studies of non-classicality of "macroscopic" systems of undefined or uncontrollable number of "particles".

Separability criteria based on Bloch representation of density matrices

We study separability criteria in multipartite quantum systems of arbitrary dimensions by using the Bloch representation of density matrices. We first derive the norms of the correlation tensors and obtain the necessary conditions for separability under partition of tripartite and four-partite quantum states. Moreover, based on the norms of the correlation tensors,we obtain the separability criteria by matrix method. Using detailed examples, our results are seen to be able to detect more entangled states than previous studies. Finally, necessary conditions of separability for multipartite systems are given under arbitrary partition.

Non-Gaussian states with strong positive partial transpose

We consider the problem of entanglement criteria in non-Gaussian systems, providing a necessary and sufficient condition of separability of non-Gaussian states based on strong positive partial transpose (SPPT) of the corresponding density matrices. The separability conditions of states generated by photon subtraction from vacuum squeezed states are given.

Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations

Painlev\'e metrics are a class of Riemannian metrics which generalize the well-known separable metrics of St\"ackel to the case in which the additive separation of variables for the Hamilton-Jacobi equation is achieved in terms of groups of independent variables rather than the complete orthogonal separation into ordinary differential equations which characterizes the St\"ackel case. Our goal in this paper is to carry out for Painlev\'e metrics the generalization of the analysis, which has been extensively performed in the St\"ackel case, of the relation between separation of variables for the Hamilton-Jacobi and Helmholtz equations, and of the connections between quadratic first integrals of the geodesic flow and symmetry operators for the Laplace-Beltrami operator. We thus obtain the generalization for Painlev\'e metrics of the Robertson separability conditions for the Helmholtz equation which are familiar from the St\"ackel case, and a formulation thereof in terms of the vanishing of the off-block diagonal components of the Ricci tensor, which generalizes the one obtained by Eisenhart for St\"ackel metrics. We also show that when the generalized Robertson conditions are satisfied, there exist $r<n$ linearly independent second-order differential operators which commute with the Laplace-Beltrami operator and which are mutually commuting. These operators admit the block-separable solutions of the Helmholtz equation as formal eigenfunctions, with the separation constants as eigenvalues. Finally, we study conformal deformations which are compatible with the separation into blocks of variables of the Helmholtz equation for Painlev\'e metrics, leading to solutions which are $R$-separable in blocks. The paper concludes with a set of open questions and perspectives.

Preparation of bipartite bound entangled Gaussian states in quantum optics

The positivity of the partial transpose is in general only a necessary condition for separability. There exist quantum states that are not separable, but nevertheless are positive under partial transpose. States of this type are known as bound entangled states meaning that these states are entangled but they do not allow distillation of pure entanglement by means of local operations and classical communication (LOCC). We present a parametrization of a class of $2\times 2$ bound entangled Gaussian states for bipartite continuous-variable quantum systems with two modes on each side. We propose an experimental protocol for preparing a particular bound entangled state in quantum optics. We then discuss the robustness properties of this protocol with respect to the occupation number of thermal inputs and the degrees of squeezing.

Classical mechanics of the Hydrogen atom perturbed by Van der Waals potential interacting with combined electric and magnetic fields

The hydrogen atom perturbed by Van der Waals potential interacting with combined electric and magnetic fields is a non-integrable system, except for some cases, one of which is separable in elliptical coordinates. A detailed study of the classical mechanics of the system is presented giving a complete description of the real phase space structure of the system, we also determine all the generic bifurcations of Liouville tori. This study is supported by numerical investigations via the Poincare surface of section and the phase space trajectories method. The dynamic character of this system depends on the van der Waals potential and the intensity of the magnetic field. Numerical calculations show that if the separability condition is verified, the classical dynamics is regular. However, with a small change in the condition, the dynamic property of the hydrogen atom begins to change. Since the condition is far from integrable behavior, almost all phase space trajectories are chaotic. On the other hand, the intensity of the electric field has no influence on the system.

Random access networks with separable schemes

Consider a time slotted multiuser random access network where M users have access to a single channel and each user attempts to send packets in a particular time slot with a certain probability, according to a predetermined access scheme. This forms an interacting system of queues and in this study, the authors consider generic access schemes that satisfy a separability condition and obtain bounds for the stability region, directly using a Lyapunov-type analysis. They apply their results to the case of random access networks with partial conflicts where any particular user has collision conflicts with only a subset of the other users. Using conflict graphs to quantify the degree of such conflicts, they obtain bounds for the stability region for both deterministic and random conflict graphs that model, respectively, static and mobile networks.

Single-Channel Signal Separation Using Spectral Basis Correlation with Sparse Nonnegative Tensor Factorization

A novel approach for solving the single-channel signal separation is presented the proposed sparse nonnegative tensor factorization under the framework of maximum a posteriori probability and adaptively fine-tuned using the hierarchical Bayesian approach with a new mixing mixture model. The mixing mixture is an analogy of a stereo signal concept given by one real and the other virtual microphones. An “imitated-stereo” mixture model is thus developed by weighting and time-shifting the original single-channel mixture. This leads to an artificial mixing system of dual channels which gives rise to a new form of spectral basis correlation diversity of the sources. Underlying all factorization algorithms is the principal difficulty in estimating the adequate number of latent components for each signal. This paper addresses these issues by developing a framework for pruning unnecessary components and incorporating a modified multivariate rectified Gaussian prior information into the spectral basis features. The parameters of the imitated-stereo model are estimated via the proposed sparse nonnegative tensor factorization with Itakura–Saito divergence. In addition, the separability conditions of the proposed mixture model are derived and demonstrated that the proposed method can separate real-time captured mixtures. Experimental testing on real audio sources has been conducted to verify the capability of the proposed method.

Bloch vector norms of separable multi-partite quantum systems

In this letter, we find an upper bound for the norms of Bloch vectors for quantum systems comprised of an arbitrary number of qudits. We generalize a recent result of Li et al. for four-partite quantum systems. We apply our result to provide an upper bound on the entanglement measure given by the Bloch vector norm and we provide a necessary algebraic condition for separability of a general multi-partite quantum system under any arbitrary partition function. Finally we show that the finer the partitioning is, the smaller our upper bound becomes.

Entanglement condition for W type multimode states and schemes for experimental realization

We derive a class of inequality relations, using both the sum uncertainty relations of su(2) algebra operators and the Schrodinger–Robertson uncertainty relation of partially transposed su(1,1) algebra operators, to detect the three-mode entanglement of non-Gaussian states of electromagnetic field. These operators are quadratic in mode creation and annihilation operators. The inseparability condition obtained using su(2) algebra operators is shown to guarantee the violation of Schrodinger–Robertson uncertainty relation of partially transposed su(1,1) algebra operators. The obtained inseparability condition incorporates all the required bipartite entanglement conditions to detect the W type three-mode entangled states and hence it is shown to be a necessary condition for W type three-mode entangled states. A general form for a family of such inseparability conditions is provided. The results derived for three-mode systems are generalized to multimode systems. Experimental schemes using symmetric multiport beam splitters and multi-Λ EIT system are proposed to test the violation of multimode separability condition.

Nonnegative Matrix Factorization Via Archetypal Analysis

Given a collection of data points, nonnegative matrix factorization (NMF) suggests expressing them as convex combinations of a small set of “archetypes” with nonnegative entries. This decomposition is unique only if the true archetypes are nonnegative and sufficiently sparse (or the weights are sufficiently sparse), a regime that is captured by the separability condition and its generalizations. In this article, we study an approach to NMF that can be traced back to the work of Cutler and Breiman [(1994), “Archetypal Analysis,” Technometrics, 36, 338–347] and does not require the data to be separable, while providing a generally unique decomposition. We optimize a trade-off between two objectives: we minimize the distance of the data points from the convex envelope of the archetypes (which can be interpreted as an empirical risk), while also minimizing the distance of the archetypes from the convex envelope of the data (which can be interpreted as a data-dependent regularization). The archetypal analysis method of Cutler and Breiman is recovered as the limiting case in which the last term is given infinite weight. We introduce a “uniqueness condition” on the data which is necessary for identifiability. We prove that, under uniqueness (plus additional regularity conditions on the geometry of the archetypes), our estimator is robust. While our approach requires solving a nonconvex optimization problem, we find that standard optimization methods succeed in finding good solutions for both real and synthetic data. Supplementary materials for this article are available online

Cross-spectral purity of nonstationary light

The concept of cross-spectral purity of light, established by Mandel for stationary optical fields, is extended to nonstationary fields. Separability conditions that ensure the cross-spectral purity of nonstationary light are derived in the space-frequency and space-time domains. It is also shown that the property of cross-spectral purity is not strictly preserved on propagation of fields with any appreciable spectral bandwidth. Further, we introduce a method for generating cross-spectrally pure (stationary and nonstationary) fields from spatially incoherent light sources by means of achromatic Fourier-transform systems.

Causality in the Social Sciences: a structural modelling framework

There is no unified theory of causality in the sciences and in philosophy. In this paper, we focus on a particular framework, called structural causal modelling (SCM), as one possible perspective in quantitative social science research. We explain how this methodology provides a fruitful basis for causal analysis in social research, for hypothesising, modelling, and testing explanatory mechanisms. This framework is not based on a system of equations, but on an analysis of multivariate distributions. In particular, the modelling stage is essentially distribution-free. Adopting an SCM approach means endorsing a particular view on modelling in general (the hypothetico-deductive methodology), and a specific stance on exogeneity (namely as a condition of separability of inference), on the one hand, and in interpreting marginal–conditional decompositions (namely as mechanisms), on the other hand.

Pareto Optimality on Compact Spaces in a Preference-Based Setting under Incompleteness

We characterize the existence of Pareto optimal elements for a family of not necessarily total preorders on a compact topological space. We identify a rather general semi-continuity assumption, called weak upper semicontinuity, under which there exist Pareto optimal elements. We also show that weak upper semicontinuity of each individual preorder is a necessary and sufficient condition for determining the Pareto optimal elements by solving the classical multi-objective optimization problem in case that each function is upper semicontinuous and order-preserving for the respective preorder, and each preorder satisfies a condition of weak separability.

SDO/HMI observations of the average supergranule are not compatible with separable flow models

Aims. Despite extensive studies carried out since its discovery half a century ago, the nature of supergranulation remains an open question in solar physics. Separability of flow models is a common assumption made in the literature to shed light on the properties of supergranules. This paper studies the ability of separable mass-conserving flow models to reproduce photospheric observations from the Helioseismic Magnetic Imager (HMI) on board of the Solar Dynamics Observatory (SDO) spacecraft corresponding to an average supergranule. Methods. For a steady mass-conserving separable flow model to be compatible with the observations, there is an integral relation between the horizontal and vertical components of the flow. We test this relation directly on observations and compare the results with the proportionality relationship for a separable model. Results. Observations of an average supergranule do not satisfy the condition for separability. Selecting a narrower range of horizontal scales of supergranules when performing the average does not change this result. Separable models are not consistent with observations of an average supergranule.