The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.
The q -deformed quantum harmonic oscillator model is applied to the description of the untrusted seller–buyer relationship. We assumed that the demand law by the buyer is described through the square modulus of the wavefunctions of the q -oscillator model in the position representation, whereas the supply law by the seller is described through the square modulus of the wavefunctions in the momentum representation. Thus, their untrusted relationship is described through the q -oscillator Wigner function of the stationary states in terms of the Al Salam-Chihara polynomials. As an example, we study the first excited state and show that the finite-difference differentiation parameter h or the deformation parameter q generated through it can be considered as a parameter defining the level of the bluffing phenomenon in this untrusted relationship. JEL Classifications: C02, D01, D11, D46, D58, D81
In this work the machinery of q-deformed algebras are used to enhance cryptographic security during teleportation. We use q-deformed harmonic oscillator states to develop a novel method of teleportation. The deformed states can be expressed in terms of standard oscillator states and the expressions contain certain arbitrary functions of
q. It is the presence of these arbitrary functions that allows an enhancement of cryptographic security. The specifics are :
(a) q-deformed Bell-like states are constructed which reduce to the usual Bell states when the deformation parameter
q→1. These deformed states form an orthonormal basis for q-deformed entangled bipartite states when certain arbitrary functions of
q satisfy a constraint.
(b) We discuss the generalisation of the usual teleportation protocol with non-maximally entangled states. This generalisation is then employed to construct two new protocols using q-deformed non-maximally entangled states. These states have additional parameters and these have to be shared for decryption after teleportation. Consequently, the cryptographic security is improved.