Geometric entanglement witnesses and bound entanglement

Abstract

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

Geometry of the generalized Bloch sphere for qutrits

“…) as may be seen from (19). But the eigenvalues of n · λ corresponding to boundary points on the inner sphere |n| = 1/2 (i.e., µ = (2…”

Entanglement detection

“…The normalization is chosen in such a way that its expectation value for ̺ e equals (up to the sign) the distance to the separable states [151,152]. Such geometric entanglement witnesses have been applied to different situations [153,154,155,156]. (v) The criteria based on variances and covariance matrices also result in certain types of entanglement witnesses.…”

Entanglement witnesses and geometry of entanglement of two-qutrit states

“…But we can use our geometric entanglement witness (81) to detect bound entanglement (see Refs. [31,32]). …”