Few Remarks on Quasi Quantum Quadratic Operators on 𝕄₂(ℂ)

The present paper deals with a connection between quantum quadratic operators (QQOs) and quasi QQOs on 𝕄 2 (ℂ). We show that QQOs and quasi QQOs on 𝕄 2 (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 𝕄 2 (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 𝕄 2 (ℂ).

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