KrausDarstellung

KrausDarstellung, also known as the Kraus representation, is a formalism in quantum information theory for describing the evolution of open quantum systems. It expresses a completely positive, trace preserving (CPTP) map, which models irreversible processes such as decoherence and measurement, as a sum over a finite set of operators called Kraus operators. These operators act on the state space of a quantum system and capture the effect of an environment on that system.

Mathematically, a CPTP map \(\mathcal{E}\) acting on a density operator \(\rho\) is written as \(\mathcal{E}(\rho)=\sum_{k} K_k \,\rho\, K_k^\dagger\). The Kraus operators \(K_k\) satisfy the completeness condition \(\sum_{k} K_k^\dagger K_k = \mathbb{I}\) to ensure trace preservation. The representation is not unique; different sets of Kraus operators related by a unitary transformation in the index space can represent the same map, reflecting the freedom in choosing a basis for the environment degrees of freedom.

The Kraus representation is widely used in the study of quantum noise models, such as depolarizing, dephasing, and amplitude-damping channels. It allows efficient simulation of open-system dynamics and forms the basis for error correction protocols, where the Kraus operators describe errors that must be detected and corrected. Additionally, the formalism underlies the concept of quantum operations and channels in the wider mathematical framework of completely positive maps, linking physical processes to mathematical structures used in quantum computing and information theory.