We present three different matrix bases that can be used to decompose density matrices of d-dimensional quantum systems, so-called qudits: the generalized Gell-Mann matrix basis, the polarization operator basis, and the Weyl operator basis. Such a decomposition can be identified with a vector -the Bloch vector, i.e. a generalization of the well known qubit case-and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We present a new method to decompose density matrices via so-called standard matrices, consider the important case of an isotropic twoqudit state and decompose it according to each basis. In case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin 1 measurements, which is appropriate for an experimental realization.

We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separable state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular 2-qubit and 2-qutrit states -and their generalizations to arbitrary dimensions -where we calculate the optimal entanglement witnesses explicitly.

We establish a relation between entanglement of a many-body system and its diffractive properties, where the link is given by structure factors. Based on these, we provide a general analytical construction of multiqubit entanglement witnesses. The proposed witnesses contain two-point correlations. They could be either measured in a scattering experiment or via local measurements, depending on the underlying physical system. For some explicit examples of witnesses we analyze the properties of the states that are detected by them. We further study the robustness of these witnesses with respect to noise.

We study entanglement witnesses that can be constructed with regard to the geometrical structure of the Hilbert-Schmidt space, i.e. we present how to use these witnesses in the context of quantifying entanglement and the detection of bound entangled states. We give examples for a particular three-parameter family of states that are part of the magic simplex of two-qutrit states.The determination whether a general quantum state is entangled or not is of utmost importance in quantum information. States of composite systems can be classified in Hilbert space as either entangled or separable. The geometric structure of entanglement is of high interest, particularly in higher dimensions. For 2 × 2 bipartite qubits, the geometry is highly symmetric and very well known. In higher dimensions, like 3 × 3 two-qutrit states, the structure of the corresponding Hilbert space is more complicated and less known. New phenomena like bound entanglement occur. In principle, all entangled states can be detected by the entanglement witness procedure. We construct in this Brief Report entanglement witnesses with regard to the geometric structure of the Hilbert-Schmidt space and explore a specific class of states, the three-parameter family of states. We discover by our method new regions of bound entanglement.Let us recall some basic definitions we need in our discussion. We consider a HilbertSchmidt space A = A 1 ⊗ A 2 ⊗ . . . ⊗ A n of operators acting on the Hilbert space of n composite quantum systems -it is of dimension[1] a new geometric entanglement measure (which we call Hilbert-Schmidt measure) based on the Hilbert-Schmidt distance between density matrices is presented -and discussed in Ref.[2] -that is an instance of a distance measure (see Refs. [3,4]). The entanglement of a state ρ can be quantified (or "measured") via the minimal Hilbert-Schmidt distance of the state to the convex and compact set of separable (disentangled) states S:It is defined on the Hilbert-Schmidt metric with a scalar product between operators that are elements of the Hilbert-Schmidt space A: A, B := TrA † B, A, B ∈ A and the norm A := A, A . Of course these definitions apply to density matrices (states of the quantum system), since they are operators of A with the properties ρ † = ρ, Trρ = 1 and ρ ≥ 0 (positive semidefinite operators).Because the norm is continuous and the set of separable states S is compact the minimum in Eq. (1) is attained for some separable state σ 0 , min σ∈S σ − ρ = σ 0 − ρ , which we call the nearest separable state to ρ. Clearly if ρ ∈ S then σ 0 = ρ and D(ρ) = 0. If ρ is entangled, ρ ent , then D(ρ) > 0 and σ 0 lies on the boundary of the set S. It is shown in Refs. [5,6,7,8] that an operator

We construct entanglement witnesses with regard to the geometric structure of the Hilbert-Schmidt space and investigate the geometry of entanglement. In particular, for a two-parameter family of two-qutrit states that are part of the magic simplex we calculate the Hilbert-Schmidt measure of entanglement. We present a method to detect bound entanglement which is illustrated for a three-parameter family of states. In this way, we discover new regions of bound entangled states. Furthermore, we outline how to use our method to distinguish entangled from separable states.