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arXiv:1306.2403 Β· Open Systems & Information Dynamics

On Pure Quasi-Quantum Quadratic Operators of 𝕄2(β„‚)

Farrukh Maksutovich Mukhamedov, Abduaziz Abduganiev

Published Nov 25, 2013β€’ 10 citations
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Abstract

In this paper we study quasi-quantum quadratic operators (quasi-QQO) acting on the algebra of 2 Γ— 2 matrices 𝕄2(β„‚). 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(β„‚). We describe all trace-preserving quasi-QQO on 𝕄2(β„‚), 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 Ο„, 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(β„‚) to 𝕄2(β„‚) βŠ— 𝕄2(β„‚) 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.

DOI
10.1142/s1230161213500182

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In this paper we study quasi-quantum quadratic operators (quasi-QQO) acting on the algebra of 2 Γ— 2 matrices 𝕄2(β„‚). 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(β„‚). We describe all trace-preserving quasi-QQO on 𝕄2(β„‚), 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 Ο„, 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(β„‚) to 𝕄2(β„‚) βŠ— 𝕄2(β„‚) 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.
In this paper we study quasi quantum quadratic operators (QQO) acting on the algebra of [Formula: see text] matrices [Formula: see text]. We consider two kinds of quasi QQO the corresponding quadratic operator maps from the unit circle into the sphere and from the sphere into the unit circle, respectively. In our early paper we have defined a q-purity of quasi QQO. This notion is equivalent to the invariance of the unit sphere in [Formula: see text]. But to check this condition, in general, is tricky. Therefore, it would be better to find weaker conditions to check the q-purity. One of the main results of this paper is to provide a criterion of q-purity of quasi QQO in terms of the unit circles. Moreover, we are able to classify all possible kinds of quadratic operators which can produce q-pure quasi QQO. We think that such result will allow one to check whether a given mapping is a pure channel or not. This finding suggests us to study such a class of nonpositive mappings. Correspondingly, the complement of this class will be of potential interest for physicist since this set contains all completely positive mappings.
The present paper deals with a connection between quantum quadratic operators (QQOs) and quasi QQOs on 𝕄2(β„‚). We show that QQOs and quasi QQOs on 𝕄2(β„‚) 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(β„‚). 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(β„‚).
In the present paper we study nonlinear dynamics of quantum quadratic operators (q.q.o) acting on the algebra of 2Γ—22\times 22Γ—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.