A special simplex in the state space for entangled qudits

Baumgartner, Bernhard; Hiesmayr, Beatrix C.; Narnhofer, Heide

doi:10.1088/1751-8113/40/28/s03

Cited by 64 publications

(135 citation statements)

References 21 publications

Supporting

Mentioning

133

Contrasting

Unclassified

Order By: Relevance

“…the structure of PPT Bell diagonal states in C n ⊗ C n depends upon the parity of 'n' (cf. [11]). It should be clear that the structure of tori T m is related with the properties of orthogonal group considered in the previous section.…”

Section: Using Lemma 22 One Easily Finds That Formula (51) Reproducesmentioning

confidence: 99%

A class of symmetric Bell diagonal entanglement witnesses—a geometric perspective

Chruściński¹

2014

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Abstract. We provide a class of Bell diagonal entanglement witnesses displaying an additional local symmetry -a maximal commutative subgroup of the unitary group U (n). Remarkably, this class of witnesses is parameterized by a torus being a maximal commutative subgroup of an orthogonal group SO(n − 1). It is shown that a generic element from the class defines an indecomposable entanglement witness. The paper provides a geometric perspective for some aspects of the entanglement theory and an interesting interplay between group theory and block-positive operators in C n ⊗ C n .

show abstract

Section: Using Lemma 22 One Easily Finds That Formula (51) Reproducesmentioning

confidence: 99%

A class of symmetric Bell diagonal entanglement witnesses—a geometric perspective

Chruściński¹

2014

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…Refs. [11,15,18,19]), in particular, to create a basis of d 2 maximally entangled qudit states [18,20,21].…”

Section: Weyl Operator Basismentioning

confidence: 99%

“…This eight-dimensional simplex has a very interesting geometry concerning separability and entanglement (see Refs. [11,15,16] …”

Section: Region IImentioning

confidence: 99%

“…Two qutrits are states on the 3 × 3 dimensional Hilbert space of bipartite quantum systems. In analogy to the familiar two-qubit case, which we discuss at the beginning, we introduce a two-and three-parameter family of two-qutrit states which are part of the magic simplex of states [11,15,16] and determine geometrical properties of the states: For the two-parameter family we quantify the entanglement via the Hilbert-Schmidt measure, for the three-parameter family we discover bound entangled states in addition to known ones in the simplex [9,15]. Finally, we give a sketch of how to use our method to construct the shape of the separable states for the three-parameter family.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Entanglement witnesses and geometry of entanglement of two-qutrit states

Bertlmann

Krammer

2009

Annals of Physics

View full text Add to dashboard Cite

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.

show abstract

“…Viewing the indices nm as points in a discrete phase space the magic simplex reveals a high symmetry that makes its geometry much more evident (see Refs. [16,17,18]). …”

mentioning

confidence: 99%

Geometric entanglement witnesses and bound entanglement

Bertlmann

Krammer

2008

Phys. Rev. A

View full text Add to dashboard Cite

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

show abstract

A special simplex in the state space for entangled qudits

Cited by 64 publications

References 21 publications

A class of symmetric Bell diagonal entanglement witnesses—a geometric perspective

A class of symmetric Bell diagonal entanglement witnesses—a geometric perspective

Entanglement witnesses and geometry of entanglement of two-qutrit states

Geometric entanglement witnesses and bound entanglement

Contact Info

Product

Resources

About