In this paper, we analyze selected evolution models of N-level open quantum systems to find the minimal number of observables (Hermitian operators) such that their expectation values at some time instants determine the accurate representation of the quantum system. The assumption that lies at the foundation of this approach to quantum tomography claims that time evolution of an open quantum system can be expressed by the Kossakowski–Lindblad equation of the form
ρ˙=Lρ , which is the most general type of Markovian and time-homogeneous master equation which preserves trace and positivity. We consider the cases when the generator of evolution can be presented by means of two or more double commutators. Determining the minimal number of observables required for quantum tomography can be the first step towards optimal tomography models for N-level quantum systems.
We consider the behavior of quantum states under stochastic local quantum operations and classical communication (SLOCC) for an arbitrary fixed number of qubits. We use a real (Lorentz) group to describe the action of SLOCC operations on n-qubit states. We discuss the natural quantum Lorentz-group group-invariant length for an arbitrary number of qubits. We relate this approach to that based on local operations and classical communication and provide an example of how the invariant length can be used to describe entanglement. We also note that this invariant length is the Minkowskian analog to the quantum state purity, which is the corresponding Euclidean length.