Abstract:
We investigate the lower bound obtained from experimental data of a quantum state $\rho$, as proposed independently by G\"uhne et al. and Eisert et al. for mixed states of three qubits. The measure we consider is the convex-roof extended three-tangle. Our findings highlight an intimate relation to lower bounds obtained recently from so-called characteristic curves of a given entanglement measure. We apply the bounds to estimate the three-tangle present in recently performed experiments aimed at producing a three-qubit GHZ state. A non-vanishing lower bound is obtained if the GHZ-fidelity of the produced states is larger than 3/4.

Abstract:
This paper is concerned with all tests for continuous-variable entanglement that arise from linear combinations of second moments or variances of canonical coordinates, as they are commonly used in experiments to detect entanglement. All such tests for bi-partite and multi-partite entanglement correspond to hyperplanes in the set of second moments. It is shown that all optimal tests, those that are most robust against imperfections with respect to some figure of merit for a given state, can be constructed from solutions to semi-definite optimization problems. Moreover, we show that for each such test, referred to as entanglement witness based on second moments, there is a one-to-one correspondence between the witness and a stronger product criterion, which amounts to a non-linear witness, based on the same measurements. This generalizes the known product criteria. The presented tests are all applicable also to non-Gaussian states. To provide a service to the community, we present the documentation of two numerical routines, FULLYWIT and MULTIWIT, which have been made publicly available.

Abstract:
The concepts of separability, entanglement, spin-squeezing and Heisenberg limit are central in the theory of quantum enhanced metrology. In the current literature, these are well established only in the case of linear interferometers operating with input quantum states of a known fixed number of particles. This manuscript generalizes these concepts and extends the quantum phase estimation theory by taking into account classical and quantum fluctuations of the particle number. Our analysis concerns most of the current experiments on precision measurements where the number of particles is known only in average.

Abstract:
Spin-changing collisions in trapped Fermi gases may acquire a resonant character due to the compensation of quadratic Zeeman effect and trap energy. These resonances are absent in spinor condensates and pseudo-spin-1/2 Fermi gases, being a characteristic feature of high-spin Fermi gases that allows spinor physics at large magnetic fields. We analyze these resonances in detail for the case of lattice spinor fermions, showing that they permit to selectively target a spin-changing channel while suppressing all others. These resonances allow for the controlled creation of non-trivial quantum superpositions of many-particle states with entangled spin and trap degrees of freedom, which remarkably are magnetic-field insensitive. Finally, we show that the intersite tunneling may lead to a quantum phase transition described by an effective quantum Ising model.

Abstract:
We propose a unifying approach to the separability problem using covariance matrices of locally measurable observables. From a practical point of view, our approach leads to strong entanglement criteria that allow to detect the entanglement of many bound entangled states in higher dimensions and which are at the same time necessary and sufficient for two qubits. From a fundamental perspective, our approach leads to insights into the relations between several known entanglement criteria -- such as the computable cross norm and local uncertainty criteria -- as well as their limitations.

Abstract:
Bell inequalities, considered within quantum mechanics, can be regarded as non-optimal witness operators. We discuss the relationship between such Bell witnesses and general entanglement witnesses in detail for the Bell inequality derived by Clauser, Horne, Shimony, and Holt (CHSH). We derive bounds on how much an optimal witness has to be shifted by adding the identity operator to make it positive on all states admitting a local hidden variable model. In the opposite direction, we obtain tight bounds for the maximal proportion of the identity operator that can be subtracted from such a CHSH witness, while preserving the witness properties. Finally, we investigate the structure of CHSH witnesses directly by relating their diagonalized form to optimal witnesses of two different classes.

Abstract:
We investigate several problems in entanglement theory from the perspective of convex optimization. This list of problems comprises (A) the decision whether a state is multi-party entangled, (B) the minimization of expectation values of entanglement witnesses with respect to pure product states, (C) the closely related evaluation of the geometric measure of entanglement to quantify pure multi-party entanglement, (D) the test whether states are multi-party entangled on the basis of witnesses based on second moments and on the basis of linear entropic criteria, and (E) the evaluation of instances of maximal output purities of quantum channels. We show that these problems can be formulated as certain optimization problems: as polynomially constrained problems employing polynomials of degree three or less. We then apply very recently established known methods from the theory of semi-definite relaxations to the formulated optimization problems. By this construction we arrive at a hierarchy of efficiently solvable approximations to the solution, approximating the exact solution as closely as desired, in a way that is asymptotically complete. For example, this results in a hierarchy of novel, efficiently decidable sufficient criteria for multi-particle entanglement, such that every entangled state will necessarily be detected in some step of the hierarchy. Finally, we present numerical examples to demonstrate the practical accessibility of this approach.

Abstract:
In this paper we address the problem of detection of entanglement using only few local measurements when some knowledge about the state is given. The idea is based on an optimized decomposition of witness operators into local operators. We discuss two possible ways of optimizing this local decomposition. We present several analytical results and estimates for optimized detection strategies for NPT states of 2x2 and NxM systems, entangled states in 3 qubit systems, and bound entangled states in 3x3 and 2x4 systems.

Abstract:
We introduce a general method for the experimental detection of entanglement by performing only few local measurements, assuming some prior knowledge of the density matrix. The idea is based on the minimal decomposition of witness operators into a pseudo-mixture of local operators. We discuss an experimentally relevant case of two qubits, and show an example how bound entanglement can be detected with few local measurements.

Abstract:
In this paper we describe how three qubit entanglement can be analyzed with local measurements. For this purpose we decompose entanglement witnesses into operators which can be measured locally. Our decompositions are optimized in the number of measurement settings needed for the measurement of one witness. Our method allows to detect true threepartite entanglement and especially GHZ-states with only four measurement settings.