Entanglement of Grassmannian Coherent States for Multi-Partite n-Level Systems

Abstract

Abstract. 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.

Multi-mode entangled states represented as Grassmannian polynomials

“…As was mentioned in Ref. [25], the weight function was not defined in a unique manner that can serve any given state in the previous studies, and no systematic way of defining a proper weight function was established. In this work, however, we observe that a proper definition of the weight function for the states can be established and subsequently one may construct MESs by integrating over tensor product of GCSs with a well-defined weight function.…”

“…In Refs. [20,25,26] construction of entangled states in the framework of GCSs was investigated. Accordingly, by considering a proper form of the linear combination of tensor product of GCSs as…”

“…On the other hand, in this work, we not only construct a given entangled states based on some well-defined weight function, but also present an appropriate Grassmannian representative polynomial of a given state which defines the state of the system and has interesting properties. This investigation, may open up new insight into entanglement of fermions and parafermions [20,25,26]. Furthermore, it can also make contact with various condensed matter systems, for example, study of spinons and holons on a one-dimensional lattice, and study of fractional statistical particles such as anyons [29][30][31][32][33].…”

“…Recently, a supersymmetric generalization of the qubit, the superqubit, was proposed using Grassmann variables [22,24]. In the same direction, entanglement of composite quantum system was studied based on GCSs [20,25,26], where maximally entangled states were constructed by integrating over tensor product of GCSs with some appropriate weight functions. These studies may open an avenue for further generalization of quantum information theory to super quantum information theory [22,24].…”

“…One may compare the approach of this work with our previous studies in Refs. [20,25,26], where in order to construct an entangled state, we applied integration over tensor product of GCSs over a weight function; however, in [20,25,26] the weight function was not defined in a way that can be applied for all given states and no systematic way of attributing a weight function for the states was given. On the other hand, in this work, we not only construct a given entangled states based on some well-defined weight function, but also present an appropriate Grassmannian representative polynomial of a given state which defines the state of the system and has interesting properties.…”