Separability criteria and entanglement witnesses for symmetric quantum states

Tóth, G.; Gühne, Otfried

doi:10.1007/s00340-009-3839-7

Cited by 40 publications

(68 citation statements)

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“…Therefore, Eq. (27) We already know that for there exist PPT entangled symmetric states consisting of more than three qubits [8,17,18] and thus indecomposable entanglement witnesses detecting them. Consequently, the above method does not apply in general for even N ≥ 8.…”

Section: Lemma 3 Consider An Equationmentioning

confidence: 99%

“…The so-called symmetric states have recently been attracting much attention [11][12][13][14][15][16][17][18]. In particular, the underlying symmetry allowed for the use of the Majorana representation [19] for an identification of SLOCC classes of multipartite symmetric states [14] (see also Refs.…”

Section: Introductionmentioning

confidence: 99%

“…Then, examples of five or six-qubit PPT entangled symmetric states were found in Refs. [17,18]. Recently, the remaining case of N = 4 has been studied in Ref.…”

Section: Introductionmentioning

confidence: 99%

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Entangled symmetric states ofNqubits with all positive partial transpositions

et al. 2012

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From both theoretical and experimental points of view symmetric states constitute an important class of multipartite states. Still, entanglement properties of these states, in particular those with positive partial transposition (PPT), lack a systematic study. Aiming at filling in this gap, we have recently affirmatively answered the open question of existence of four-qubit entangled symmetric states with positive partial transposition and thoroughly characterized entanglement properties of such states [J. Tura et al., Phys. Rev. A 85, 060302(R) (2012)] With the present contribution we continue on characterizing PPT entangled symmetric states. On the one hand, we present all the results of our previous work in a detailed way. On the other hand, we generalize them to systems consisting of arbitrary number of qubits. In particular, we provide criteria for separability of such states formulated in terms of their ranks. Interestingly, for most of the cases, the symmetric states are either separable or typically separable. Then, edge states in these systems are studied, showing in particular that to characterize generic PPT entangled states with four and five qubits, it is enough to study only those that assume few (respectively, two and three) specific configurations of ranks. Finally, we numerically search for extremal PPT entangled states in such systems consisting of up to 23 qubits. One can clearly notice regularity behind the ranks of such extremal states, and, in particular, for systems composed of odd number of qubits we find a single configuration of ranks for which there are extremal states.

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Section: Lemma 3 Consider An Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Entangled symmetric states ofNqubits with all positive partial transpositions

et al. 2012

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“…For example, the authors of [14] and [10] considered matrices invariant under multiplication by the flip operator and matrices invariant under partial transposition, respectively. This paper is arranged as follows.…”

Section: Introductionmentioning

confidence: 99%

Completely Reducible Maps in Quantum Information Theory

Cariello¹

2016

IEEE Trans. Inform. Theory

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Abstract. In order to compute the Schmidt decomposition of A ∈ M k ⊗M m , we must consider an associated self-adjoint map. Here, we show that if A is positive under partial transposition (PPT) or symmetric with positive coefficients (SPC) or invariant under realignment then its associated self-adjoint map is completely reducible. We give applications of this fact in Quantum Information Theory. We recover some theorems recently proved for PPT and SPC matrices and we prove these theorems for matrices invariant under realignment using theorems of Perron-Frobenius theory. We also provide a new proof of the fact that if C k contains k mutually unbiased bases then C k contains k + 1. We search for other types of matrices that could have the same property.We consider a group of linear transformations acting on M k ⊗ M k , which contains the partial transpositions and the realignment map. For each element of this group, we consider the set of matrices in M k ⊗ M k ≃ M k 2 that are positive and remain positive, or invariant, under the action of this element. Within this family of sets, we have the set of PPT matrices, the set of SPC matrices and the set of matrices invariant under realignment. We show that these three sets are the only sets of this family such that the associated self-adjoint map of each matrix is completely reducible. We also show that every matrix invariant under realignment is PPT in M 2 ⊗ M 2 and we present a counterexample in M k ⊗ M k , k ≥ 3.

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“…Entanglement witnesses are helpful tools in the experimental characterization of entanglement in composite quantum systems [5,7,17,18]. For a composite system consisting of two parts labeled by A and B respectively, let H A be the state space of the subsystem A and H B be the state space of the subsystem B, dim H A ⊗ H B ≤ +∞.…”

Section: Introductionmentioning

confidence: 99%

Constructing Entanglement Witnesses for States in Infinite-Dimensional Bipartite Quantum Systems

Hou

Guo

2010

Int J Theor Phys

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In this paper, two approaches of constructing entanglement witnesses for finiteor infinite-dimensional bipartite quantum systems are presented. Let H A and H B be complex Hilbert spaces and {E k } and {F k } be sequences of self-adjoint Hilbert-Schmidt operators on H A and H B , respectively, such that Tr(E † kIf ρ is an entangled state and τ 0 is the nearest separable state to ρ under the Hilbert-Schmidt norm, then W = c 0 I + τ 0 − ρ with c 0 = Tr[τ 0 (ρ − τ 0 )] is an entanglement witness.

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Separability criteria and entanglement witnesses for symmetric quantum states

Cited by 40 publications

References 40 publications

Entangled symmetric states ofNqubits with all positive partial transpositions

Entangled symmetric states ofNqubits with all positive partial transpositions

Completely Reducible Maps in Quantum Information Theory

Constructing Entanglement Witnesses for States in Infinite-Dimensional Bipartite Quantum Systems

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