# Tensorial dynamics on the space of quantum states

## TL;DR

This paper presents a geometric tensorial framework for describing the dynamics of quantum states and their evolution under Markovian processes, enabling analysis of nonlinear quantities like entropy and concurrence.

## Contribution

It introduces a novel geometric formulation using tensor fields to model quantum state evolution and Markovian dynamics in a unified, natural way.

## Key findings

- Tensorial formulation describes Markovian evolution as a vector field on state space
- Allows analysis of nonlinear physical quantities such as entropy and concurrence
- Identifies limits leading to contraction of Jordan and Lie products

## Abstract

A geometric description of the space of states of a finite-dimensional quantum system and of the Markovian evolution associated with the Kossakowski-Lindblad operator is presented. This geometric setting is based on two composition laws on the space of observables defined by a pair of contravariant tensor fields. The first one is a Poisson tensor field that encodes the commutator product and allows us to develop a Hamiltonian mechanics. The other tensor field is symmetric, encodes the Jordan product and provides the variances and covariances of measures associated with the observables. This tensorial formulation of quantum systems is able to describe, in a natural way, the Markovian dynamical evolution as a vector field on the space of states. Therefore, it is possible to consider dynamical effects on non-linear physical quantities, such as entropies, purity and concurrence. In particular, in this work the tensorial formulation is used to consider the dynamical evolution of the symmetric and skew-symmetric tensors and to read off the corresponding limits as giving rise to a contraction of the initial Jordan and Lie products.