Para-Grassmannian Coherent and Squeezed States for Pseudo-Hermitian q-Oscillator and their Entanglement

Abstract

Abstract. 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 paraGrassmannian 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 pseudoHermitian coherent state introduced, and the method also developed to define pseudoHermitian 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.

Multi-mode entangled states represented as Grassmannian polynomials

“…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.…”

“…Naturally, the generalization of these anti-commuting nilpotent variables is utilized in the study of multi-level quantum mechanical systems which are described in arbitrary but finite-dimensional Hilbert spaces [11,13,17,18]. Among many applications of Grassmann variables, the notion of Grassmannian coherent state (GCS) has found enormous applications in various branches of physics [17,[19][20][21]. However, the entanglement of fermions and parafermions, which are justified by Grassmannian states in quantum field theory, have remained a challenging task for quantum information science [22,23].…”

A concise review of pseudobosons, pseudofermions, and their relatives

“…We should also mention that similar deformations of the CCR, CAR etc. have been considered along the years by several other authors, see [4,5] just to cite two such contributions.…”

“…This means that, for all f and g in G, 5) which can be seen as a weak form of the resolution of the identity, restricted to G. Of course, if f ∈ G is orthogonal to all the ϕ n 's, or to all the Ψ n 's, then (2.5) implies that f = 0. Hence F ϕ and F Ψ are complete in G.…”