Commutativity and the third Reidemeister movement

Abstract

In quantum information theory, for $a,b$ two positive operators living in $B(\mathcal{H})$, where $\mathcal{H}$ is a separable Hilbert space, the quantum fidelity is denoted by $a*b =(b^{1/2}ab^{1/2})^{1/2}$. One of the aim of this let ter is to interpret the quantum fidelity as an algebraic law. We remark that if $a,b,c$ are three positive operators whi ch commute pairwise, the law * becomes self-distributive, i.e. the third Reidemeister movement in knot theory is verif ied. We study the converse. Let three positive operators be given, does the fact that the third Reidemeister movement between them is possible implie that they commute pairwise ? Though in general we only conjecture it for the moment, we prove it in some par ticular but important cases. Should this movement be not possible, we interpret it as an obstruction to comm utativity. We give also new examples of quandle algebras and left distributive systems and study the generalisation of Ito maps.