Krausoperatorit

Krausoperatorit, in the plural form often referred to as Kraus-operatorit or Kraus-opeerat, describe a representation of quantum operations as a set of linear operators on a system’s Hilbert space. If a system is in a density operator ρ, a quantum operation E can be written as E(ρ) = ∑_i K_i ρ K_i†, where the K_i are the Kraus operators. This operator-sum representation captures the effect of noise, general measurements, and open-system dynamics in a basis-independent way.

The Kraus representation is closely tied to complete positivity. A quantum operation is completely positive and, when it is trace-preserving, a quantum channel. In that case the Kraus operators satisfy the completeness relation ∑_i K_i† K_i = I. If the map is not trace-preserving, for example when probability is lost to an environment, the relation is ∑_i K_i† K_i ≤ I. The number of Kraus operators needed can be at most d^2, where d is the dimension of the system’s Hilbert space, and the representation is not unique.

Kraus operators arise naturally from modeling system-environment interactions, for instance via a unitary on a joint system followed by partial tracing over the environment (Stinespring dilation). Different sets of Kraus operators can describe the same map, related by unitary mixing: if {K_i} is a valid representation, then {∑_j U_{ij} K_j} is another valid set for any unitary matrix U.

Applications of Kraus operators abound in quantum information and open quantum systems, including modeling noise channels (amplitude damping, depolarizing, phase damping) and describing generalized measurements. The concept, named after Karl Kraus, provides a compact and versatile framework for describing physical evolutions of quantum states.