TY - JOUR
T1 - On pure quasi-quantum quadratic operators of M2(C)
AU - Mukhamedov, Farrukh
AU - Abduganiev, Abduaziz
N1 - Funding Information:
The authors acknowledge the MOHE grants FRGS11-022-0170, ERGS13-024-0057, the IIUM grant EDW B13-019-0904, and the Junior Associate scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Finally, the authors would also like to thank the anonymous referee whose useful suggestions and comments improved the contents of the paper.
PY - 2013/12
Y1 - 2013/12
N2 - 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) ⊗ M 2(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.
AB - 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) ⊗ M 2(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.
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U2 - 10.1142/S1230161213500182
DO - 10.1142/S1230161213500182
M3 - Article
AN - SCOPUS:84888624962
SN - 1230-1612
VL - 20
JO - Open Systems and Information Dynamics
JF - Open Systems and Information Dynamics
IS - 4
M1 - 1350018
ER -