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arXiv:1212.5675

Entanglement of Multipartite Fermionic Coherent States for Pseudo Hermitian Hamiltonians

G. Najarbashi, M. A. Fasihi, Mikio Nakahara, Фороузан Мирмасоуди, Sevda Mirzaei

Published Dec 22, 2012• 0 citations
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Abstract

In this paper the entanglement of multi-qubit fermionic pseudo Hermitian coherent states (FPHCS) described by anticommutative Grassmann numbers is studied. The pseudo-Hermitian versions of the well known maximally entangled pure states such as Bell and GHZ, W and biseparable states are introduced through integrating over the tensor products of FPHCSs with suitable choice of Grassmannian weight functions. Meanwhile as an illustration, the method is applied to tensor product of 2 and 3 qubit pseudo Hermitian systems. Then the measures of concurrence and average entropy are applied to quantify the entanglement of the pseudo two and three qubit states respectively.

DOI
10.48550/arxiv.1212.5675

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In this paper the entanglement of multi-qubit fermionic pseudo Hermitian coherent states (FPHCS) described by anticommutative Grassmann numbers is studied. The pseudo-Hermitian versions of the well known maximally entangled pure states such as Bell and GHZ, W and biseparable states are introduced through integrating over the tensor products of FPHCSs with suitable choice of Grassmannian weight functions. Meanwhile as an illustration, the method is applied to tensor product of 2 and 3 qubit pseudo Hermitian systems. Then the measures of concurrence and average entropy are applied to quantify the entanglement of the pseudo two and three qubit states respectively.
We study the entanglement of multiqubit fermionic pseudo-Hermitian coherent states (FPHCSs) described by anticommutative Grassmann numbers. We introduce pseudo-Hermitian versions of well-known maximally entangled pure states, such as Bell, GHZ, Werner, and biseparable states, by integrating over the tensor products of FPHCSs with a suitable choice of Grassmannian weight functions. As an illustration, we apply the proposed method to the tensor product of two- and three-qubit pseudo-Hermitian systems. For a quantitative characteristic of entanglement of such states, we use a measure of entanglement determined by the corresponding concurrence function and average entropy.
In this paper we investigate the entanglement of multi-qubit fermionic coherent states described by anticommutative Grassmann numbers. Choosing an appropriate weight function, we show that it is possible to construct some entangled pure states, consisting of {\bf{GHZ}}, {\bf{W}}, Bell and biseparable states, by tensor product of fermion coherent states. Moreover a comparison with maximal entangled bosonic coherent states is presented and it is shown that in some cases they have fermionic counterpart which are maximal entangled after integration with suitable weight functions.
In this paper, we investigate the entanglement of multi-partite Grassmannian coherent states (GCSs) described by Grassmann numbers for n > 2 degree of nilpotency.Choosing an appropriate weight function, we show that it is possible to construct some wellknown entangled pure states, consisting of GHZ, W, Bell, cluster type and bi-separable states, which are obtained by integrating over tensor product of GCSs.It is shown that for three level systems, the Grassmann creation and annihilation operators b and b † together with b z form a closed deformed algebra, i.e., SU q (2) with q = e 2πi 3 , which is useful to construct entangled qutrit-states.The same argument holds for three level squeezed states.Moreover combining the Grassmann and bosonic coherent states we construct maximal entangled super coherent states.
The purpose of this paper is to generalize fermionic coherent states for two-level systems described by pseudo-Hermitian Hamiltonian (Cherbal et al 2007 J. Phys. A: Math. Theor. 40 1835 ) to n -level systems. Central to this task is the expression of the coherent states in terms of generalized Grassmann variables. These kinds of Grassmann coherent states satisfy the bi-overcompleteness condition instead of the overcompleteness one, as it is reasonably expected because of the biorthonormality of the system. Choosing an appropriate Grassmann weight function, the resolution of identity is examined. Moreover Grassmannian coherent and squeezed states of the deformed group SU q (2) for the three-level pseudo-Hermitian system are presented.
In this paper, q-deformed oscillator for pseudo-Hermitian systems is investigated and pseudo-Hermitian appropriate coherent and squeezed states are studied.Also, some basic properties of these states is surveyed.The over-completeness property of the para-Grassmannian pseudo-Hermitian coherent states (PGPHCSs) examined, and also the stability of coherent and squeezed states discussed.The pseudo-Hermitian supercoherent states as the product of a pseudo-Hermitian bosonic coherent state and a para-Grassmannian pseudo-Hermitian coherent state introduced, and the method also developed to define pseudo-Hermitian supersqueezed states.It is also argued that, for q-oscillator algebra of k + 1 degree of nilpotency based on the changed ladder operators, defined in here, we can obtain deformed SU q 2 (2) and SU q 2k (2) and also SU q 2k (1, 1).Moreover, the entanglement of multilevel para-Grassmannian pseudo-Hermitian coherent state will be considered.This is done by choosing an appropriate weight function, and integrating over tensor product of PGPHCSs.
We propose two classes of the generalized concurrence vectors of the multipartite systems consisting of qubits. Making use of them, we are able to, respectively, describe and quantify GHZ-class and W-class entanglement both in total and between arbitrary two partite in the multipartite system consisting of qubits. In the case of pure state of three qubits that one partite is separable, it is shown to exactly back to the usual Wootters' concurrence after introduce a whole concurrence vector. In principle, our method is applicable to any NNN-partite systems consisting of NNN qubits.