Krausoperatorer

Krausoperatorer, also known as Kraus operators or operation elements, are mathematical constructs used in quantum mechanics to describe the evolution of quantum states in open quantum systems. They provide a framework for representing quantum operations, including measurement processes and interactions with an environment, in the language of density matrices.

In the formalism of quantum operations, a system's state is represented by a density matrix ρ. Kraus operators {K_i} are a set of linear operators that satisfy the completeness relation ∑_i K_i† K_i = I, where I is the identity operator and K_i† denotes the Hermitian conjugate of K_i. The transformation of the state under a quantum operation 𝒩 is given by ρ' = ∑_i K_i ρ K_i†. This representation ensures that the output state remains a valid density matrix, preserving positivity and trace.

Kraus operators are instrumental in describing processes such as decoherence, measurement, and quantum channels. They generalize unitary evolution, which is reversible, to include non-unitary, irreversible processes encountered in realistic quantum systems. The formalism provides a complete and convenient way to model noisy channels, quantum noise, and the effects of environment-induced decoherence.

The concept was introduced by Karl Kraus in 1971 as part of the operator-sum representation. Its applications span quantum information theory, quantum computing, and quantum communication, forming a foundational element in understanding how quantum states evolve under various operations and interactions with the environment.