The Lie Algebra of S-unitary Matrices, Twisted Brackets and Quantum Channels

Abstract

A dimension formula was given in [1] in order to partially classify the Lie algebras of $S$-unitary type. The natural question of when $\mathfrak{u}_{S}$ and $\mathfrak{u}_{T}$ are isomorphic is left unanswered. In this article, we will give an answer to this question using the notion of quantum channels and their Kraus representation. In line with this, we will also discuss linearly twisted versions of the usual commutator bracket and its relation to the standard Lie algebra structure on $M_{n}(\mathbb{C})$. Finally, we will mention some problems that are still unanswered in relation to $S$-unitary type matrices and twisted brackets.