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arXiv:1004.3703

Entanglement in multi-qubit pure fermionic coherent states

G. Najarbashi, Yusef Maleki

Published Apr 21, 2010• 3 citations
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Abstract

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.

DOI
10.48550/arxiv.1004.3703

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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-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 characterization of entanglement is a fundamental issue for Quantum Information Theory. But the definition of entanglement depends on the notion of locality, and thus on the tensor product structure of the state space of the composite system. This notion is affected by the presence of superselection rules that restrict the accessible Hilbert space to a direct sum of subspaces. Indistinguishability of particles imposes one such restriction, namely to totally symmetric or totally antisymmetric states. The entanglement can in this case be defined with respect to partitions of modes in the second quantization formalism. For fermionic systems the Fock space of m modes is isomorphic to the space of m qubits, but the action of creation and annihilation operators is not local, due to their anticommutation. Conservation of the parity of fermion number imposes another relevant superselection rule. It requires that local physical observables commute with the local parity operator. Taking into account the considerations above, it is possible to define the set of separable states or equivalently the concept of entanglement for fermionic systems in a number of ways. Here we analyze systematically these possibilities and the relation among the various sets of separable states. We also discuss the behavior of the different classes when taking several copies of the state, as well as the characterization of the sets in terms of the usual criteria regarding the tensor product.