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Dynamics of quantum quadratic Volterra operators

Farrukh Maksutovich Mukhamedov, Abduaziz Abduganiev

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

In the present paper we introduce a new class of quantum quadratic operators which are called Volterra operators. We study dynamics of such kind of quadratic operators. Note
that these operators act on the state space of the algebra of2 by 2 matrices

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The present paper deals with a connection between quantum quadratic operators (QQOs) and quasi QQOs on 𝕄2(ℂ). We show that QQOs and quasi QQOs on 𝕄2(ℂ) coincide in the class of Volterra type of operators. To establish this result, we first describe these two kind of operators on the commutative part of 𝕄2(ℂ). Furthermore, in the last section, we introduce a quantum analogue of Volterra operators and provide concrete examples of such kind of operators. It is established that the considered examples also satisfy quasiness condition as well. The obtained results will allow to produce (with explicit conditions) a class of unital, but not trace-preserving positive maps of 𝕄2(ℂ).
In the present paper we study nonlinear dynamics of quantum quadratic operators (q.q.o) acting on the algebra of 2×22\times 22×2 matrices $\bm_2(\bc)$. First, we describe q.q.o. with Haar state as well as quadratic operators with the Kadison-Schwarz property. By means of such a description we provide an example of q.q.o. which is not the Kadision-Schwarz operator. Then we study stability of dynamics of q.q.o.
In this paper we study quasi-quantum quadratic operators (quasi-QQO) acting on the algebra of 2 × 2 matrices 𝕄2(ℂ). It is known that a channel is called pure if it sends pure states to pure ones. In this paper, we introduce a weaker condition for the channel called q-purity. To study q-pure channels, we concentrate on quasi-QQO acting on 𝕄2(ℂ). We describe all trace-preserving quasi-QQO on 𝕄2(ℂ), which allows us to prove that if a trace-preserving symmetric quasi-QQO is such that the corresponding quadratic operator is linear, then its q-purity implies its positivity. If a symmetric quasi-QQO has a Haar state τ, then its corresponding quadratic operator is nonlinear, and it is proved that such q-pure symmetric quasi-QQO cannot be positive. We think that such a result will allow one to check whether a given mapping from 𝕄2(ℂ) to 𝕄2(ℂ) ⊗ 𝕄2(ℂ) is pure or not. On the other hand, our study is related to the construction of pure quantum nonlinear channels. Moreover, we also indicate that nonlinear dynamics associated with pure quasi-QQO may have different kind of dynamics, i.e. it may behave chaotically or trivially.