Noncommutative Ricci flow in a matrix geometry

We study noncommutative Ricci flow in a finite-dimensional representation of a noncommutative torus. It is shown that the flow exists and converges to the flat metric. We also consider the evolution of entropy and a definition of scalar curvature in terms of the Ricci flow.National Research Foundation of South Africahttp://iopscience.iop.org/0305-4470hb201

Quantum statistical mechanics, KMS states and Tomita-takesaki theory

Please read the abstract in the section 00front of this documentDissertation (MSc (Mathematics))--University of Pretoria, 2006.Mathematics and Applied Mathematicsunrestricte

The general structure and ergodic properties of quantum and classical mechanics: A unified C*-algebraic approach

Please read the abstract in the section 00front of this documentThesis (PhD (Mathematics))--University of Pretoria, 2005.Mathematics and Applied Mathematicsunrestricte

Fermionic quantum detailed balance and entanglement

A definition of detailed balance, tailored to a system of indistinguishable fermions, is suggested and studied using an entangled fermionic state. This is done in analogy to a known characterization of standard quantum detailed balance with respect to a reversing operation.The National Research Foundation of South Africa.https://jphysplus.iop.org/category/journal-of-physics-a-mathematical-and-theoretical2019-08-01hj2018Physic

Relatively independent joinings and subsystems of W*-dynamical systems

Relatively independent joinings ofW*-dynamical systems are constructed. This is intimately related to subsystems of W*-dynamical systems, and therefore we also study general properties of subsystems, in particular fixed point subsystems and compact subsystems. This allows us to obtain characterizations of weak mixing and relative ergodicity, as well as of certain compact subsystems, in terms of joinings.The National Research Foundation of South Africahttp://journals.impan.gov.pl/sm

Ergodicity and mixing of W*-dynamical systems in terms of joinings

We study characterizations of ergodicity, weak mixing and strong mixing of W*-dynamical systems in terms of joinings and subsystems of such systems. Ergodic joinings and Ornstein’s criterion for strong mixing are also discussed in this context.The National Research Foundation of South Africahttp://ijm.math.illinois.edu/nf201

Optimal quantum channels

A method to optimize the cost of a quantum channel is developed. The goal is to determine the cheapest channel that produces prescribed output states for a given set of input states. This is essentially a quantum version of optimal transport. To attach a clear conceptual meaning to the cost, channels are viewed in terms of what we call elementary transitions, which are analogous to point-to-point transitions between classical systems. The role of entanglement in optimization of cost is emphasized. We also show how our approach can be applied to theoretically search for channels performing a prescribed set of tasks on the states of a system, while otherwise disturbing the state as little as possible.http://pra.aps.orgam2022Physic

Detailed balance and entanglement

We study a connection between quantum detailed balance, which is a concept of importance in statistical mechanics, and entanglement. We also explore how this connection ts into thermo eld dynamics.National Research Foundation of South Africa.http://iopscience.iop.org0305-44702016-04-30hb201

Relative weak mixing of W∗-dynamical systems via joinings

A characterization of relative weak mixing in W∗-dynamical systems in terms of a relatively independent joining is proven.Partially supported by the National Research Foundation of South Africa.http://journals.impan.gov.pl/smhj2019Mathematics and Applied MathematicsPhysic

Disjointness of C-dynamical systems

We study ergodic theorems for disjoint C*- and W*-dynamical systems, where disjointness here is a noncommutative version of the concept introduced by Furstenberg for classical dynamical systems. We also consider speci c examples of disjoint W*-dynamical systems. Lastly we use unique ergodicity and unique ergodicity relative to the xed point algebra to give examples of disjoint C*-dynamical systems.The National Research Foundation of South Africahttp://www.math.uh.edu/~hjm/am2016Mathematics and Applied MathematicsPhysic