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arXiv:0805.3180 · Physical Review A

Greenberger-Horne-Zeilinger and W entanglement witnesses for the noninteracting Fermi gas

Hessam Habibian, J. W. Clark, N. Behbood, Kurt Hingerl

Published Mar 3, 2010• 8 citations
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Abstract

The existence and nature of tripartite entanglement of a noninteracting Fermi gas (NIFG) is investigated. Three classes of parametrized entanglement witnesses (EWs) are introduced with the aim of detecting genuine tripartite entanglement in the three-body reduced density matrix and discriminating between the presence of the two types of genuine tripartite entanglement, $W\B$ and $\mathrm{GHZ}\W$ (the convex set of $B$ states is comprised of mixed states of product and biseparable states; that of $W$ states is comprised of mixed states of $B$ states and $W$-type pure entangled states; and the GHZ (Greenberger-Horne-Zeilinger) set contains generic mixtures of any kind for a tripartite system). By choosing appropriate EW operators, the problem of finding GHZ and $W$ EWs is reduced to linear programming. Specifically, we devise $W$ EWs based on a spin-chain model with periodic boundary conditions, and we construct a class of parametrized GHZ EWs by linearly combining projection operators corresponding to all the different state-vector types arising for a three-fermion system. A third class of EWs is provided by a GHZ stabilizer operator capable of distinguishing $W\B$ from $\mathrm{GHZ}\B$ entanglement, which is not possible with $W$ EWs. Implementing these classes of EWs, it is found that all states containing genuine tripartite entanglement are of $W$ type, and hence states containing $\mathrm{GHZ}\W$ genuine tripartite entanglement do not arise. Some genuine tripartite entangled states that have a positive partial transpose (PPT) with respect to some bipartition are detected. Finally, it is demonstrated that a NIFG does not exhibit ``pure'' $W\B$ genuine tripartite entanglement: three-party entanglement without any separable or biseparable admixture does not occur.

DOI
10.1103/physreva.81.032302

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International Journal of Quantum Information•Oct 1, 2017•1 citation
The existence and nature of tripartite entanglement of a noninteracting Fermi gas (NIFG) is investigated. Three classes of parametrized entanglement witnesses (EWs) are introduced with the aim of detecting genuine tripartite entanglement in the three-body reduced density matrix and discriminating between the presence of the two types of genuine tripartite entanglement, W\BW\BW\B and GHZ\W\mathrm{GHZ}\WGHZ\W (the convex set of BBB states is comprised of mixed states of product and biseparable states; that of WWW states is comprised of mixed states of BBB states and WWW-type pure entangled states; and the GHZ (Greenberger-Horne-Zeilinger) set contains generic mixtures of any kind for a tripartite system). By choosing appropriate EW operators, the problem of finding GHZ and WWW EWs is reduced to linear programming. Specifically, we devise WWW EWs based on a spin-chain model with periodic boundary conditions, and we construct a class of parametrized GHZ EWs by linearly combining projection operators corresponding to all the different state-vector types arising for a three-fermion system. A third class of EWs is provided by a GHZ stabilizer operator capable of distinguishing W\BW\BW\B from GHZ\B\mathrm{GHZ}\BGHZ\B entanglement, which is not possible with WWW EWs. Implementing these classes of EWs, it is found that all states containing genuine tripartite entanglement are of WWW type, and hence states containing GHZ\W\mathrm{GHZ}\WGHZ\W genuine tripartite entanglement do not arise. Some genuine tripartite entangled states that have a positive partial transpose (PPT) with respect to some bipartition are detected. Finally, it is demonstrated that a NIFG does not exhibit ``pure'' W\BW\BW\B genuine tripartite entanglement: three-party entanglement without any separable or biseparable admixture does not occur.
Abstract This chapter takes a closer look at entanglement in multipartite systems. An examination of pure tripartite systems serves a starting point from which we discover the equivalence classes of Greenberger-Horne-Zeilinger (GHZ) states and W states. We then continue with the mixed-state case and discuss the notions of full separability, partition-separability and biseparability versus genuine tripartite entanglement, before stating the GHZ theorem in the formulation following Mermin. For systems of three or more parties we then formulate the definitions for k-separability and genuine multipartite entanglement (GME), as well as for k-producibility and entanglement depth, followed by prominent examples for GME states such as the generalized GHZ states and Dicke states. We give an overview of the problem of detecting GME using various linear and non-linear GME witnesses, lifted witnesses, PPT mixers, before discussing challenges for the characterization and quantification of multipartite entanglement. We close by analysing the phenomenon of entangled entanglement.
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
We introduce a real-valued functional I(n⃗1,n⃗2)I(\vec{n}_1,\vec{n}_2)I(n1​,n2​) of four three-qubit correlations that turns the Greenberger-Horne-Zeilinger (GHZ) algebraic paradox into a quantitative witness of genuine tripartite entanglement. For every three-qubit state ρρρ and every pair of directions, ∣I(n⃗1,n⃗2;ρ)∣≤2|I(\vec{n}_1,\vec{n}_2;ρ)|\le 2∣I(n1​,n2​;ρ)∣≤2, with equality if and only if n⃗1⊥n⃗2\vec{n}_1\perp\vec{n}_2n1​⊥n2​ and ρρρ is locally unitarily equivalent to the GHZ state. On the five-parameter Acín family, I(x^,y^)I(\hat{x},\hat{y})I(x^,y^​) depends only on λ0λ4λ_0λ_4λ0​λ4​ and is maximised by the GHZ state; for the W state I(x^,y^)=0I(\hat{x},\hat{y})=0I(x^,y^​)=0 and max⁡∣IW∣=35/27\max|I_{\mathrm{W}}|=35/27max∣IW​∣=35/27. Maximising over independent local orthonormal frames yields a local-unitary invariant measure EGHZ∈[0,1]\mathcal{E}_{\mathrm{GHZ}}\in[0,1]EGHZ​∈[0,1], computable from four three-body Pauli correlators, that equals one exactly on the GHZ orbit and takes the values 35/5435/5435/54 on the W state and 1/21/21/2 on product states. We show that EGHZ\mathcal{E}_{\mathrm{GHZ}}EGHZ​ is not convex: an equal mixture of three biseparable states has EGHZ=14/27\mathcal{E}_{\mathrm{GHZ}}=14/27EGHZ​=14/27, larger than the average 1/21/21/2 of its components. We also relate EGHZ\mathcal{E}_{\mathrm{GHZ}}EGHZ​ to GHZ fidelity witnesses, discuss its behaviour under local operations and classical communication, and outline a generalisation to three qudits via Heisenberg-Weyl operators.
We study genuine tripartite entanglement shared among the spins of three localized fermions in the noninteracting Fermi gas at zero temperature. First, we prove analytically with the aid of entanglement witnesses that in a particular configuration the three fermions are genuinely tripartite entangled in spin. Then various three-fermion configurations are investigated in order to quantify and calculate numerically the amount of genuine tripartite entanglement present in the system. Further we give a lower and an upper limit to the maximum diameter of the three-fermion configuration below which genuine tripartite entanglement exists and find that this distance is comparable with the maximum separation between two entangled fermions. The upper and lower limit turn out to be very close to each other indicating that the applied witness operator is well suited to reveal genuine tripartite entanglement in the collection of noninteracting fermions.
We introduce a classification of mixed three-qubit states, in which we define the classes of separable, biseparable, W, and Greenberger-Horne-Zeilinger states. These classes are successively embedded into each other. We show that contrary to pure W-type states, the mixed W class is not of measure zero. We construct witness operators that detect the class of a mixed state. We discuss the conjecture that all entangled states with positive partial transpose (PPTES) belong to the W class. Finally, we present a new family of PPTES "edge" states with maximal ranks.
In this paper, entanglement classification shared among the spins of localized fermions in the noninteracting Fermi gas is studied. It is proven that the Fermi gas density matrix is block diagonal on the basis of the projection operators to the irreducible representations of symmetric group [Formula: see text]. Every block of density matrix is in the form of the direct product of a matrix and identity matrix. Then it is useful to study entanglement in every block of density matrix separately. The basis of corresponding Hilbert space are identified from the Schur–Weyl duality theorem. Also, it can be shown that the symmetric part of the density matrix is fully separable. Then it has been shown that the entanglement measure which is introduced in Eltschka et al. [New J. Phys. 10, 043104 (2008)] and Guhne et al. [New J. Phys. 7, 229 (2005)], is zero for the even [Formula: see text] qubit Fermi gas density matrix. Then by focusing on three spin reduced density matrix, the entanglement classes have been investigated. In three qubit states there is an entanglement measure which is called 3-tangle. It can be shown that 3-tangle is zero for three qubit density matrix, but the density matrix is not biseparable for all possible values of its parameters and its eigenvectors are in the form of W-states. Then an entanglement witness for detecting non-separable state and an entanglement witness for detecting nonbiseparable states, have been introduced for three qubit density matrix by using convex optimization problem. Finally, the four spin reduced density matrix has been investigated by restricting the density matrix to the irreducible representations of [Formula: see text]. The restricted density matrix to the subspaces of the irreducible representations: [Formula: see text], [Formula: see text] and [Formula: see text] are denoted by [Formula: see text], [Formula: see text] and [Formula: see text], respectively. It has been shown that some highly entangled classes (by using the results of Miyake [Phys. Rev. A 67, 012108 (2003)] for entanglement classification) do not exist in the blocks of density matrix [Formula: see text] and [Formula: see text], so these classes do not exist in the total Fermi gas density matrix.