Quantum operations arise naturally in many fields of quantum information theory and quantum computing. One of the simplest example of quantum operation is the von Neumann-Lüders measurement. Destruction of states in quantum mechanics can be treated as a supplement to the von Neumann-Lüders measurement [P. Caban, J. Rembieliński, K. A. Smoliński and Z. Walczak, J. Phys. A 35, 3265 (2002)]. We show that destruction of states in one-qudit system is a quantum operation by finding its Kraus representation.PACS numbers: 03.67.−a, 03.65.−w A general state change in quantum mechanics is described by the quantum operations formalism [1] (see also [2] and references therein) arising naturally in many fields of quantum information theory and quantum computing. In this formalism an input state ρ of a system is connected with an output state E(ρ) by a linear, trace decreasing and completely positive map E. Trace decreasing means that tr(E(ρ)) 1 for all normalized density operators ρ. Complete positivity means that a map E ⊗ I is positive for all trivial extensions of a system (a system plus an ancilla). A quantum operation E can be represented (not uniquely) in a form known as the Kraus representation or the operator-sum representationwhere the Kraus operators or operation elements {E i } satisfy the following conditionMoreover, a map E which can be written in the form (1) is a quantum operation. If the Kraus representation contains only one operator we say that a quantum operation E is pure. The unitary evolution of density operator is an example of a pure quantum operation. Note that for a trace preserving operation the Kraus operators {E i } satisfy a completeness relationThe von Neumann-Lüders measurement with no selection is an example of a trace preserving quantum operation. In this paper, we show that a destruction of states [3] in a one-qudit system is a quantum operation by finding its Kraus representation.Denote the Hilbert space of a system consisting of onequdit (a d-dimensional quantum system, with d finite) by is orthogonal to any vector from H 1 and for all observableŝ Λ acting on H 1 we adopt thatΛ|vac = 0. The vector |vac spans the one-dimensional vacuum space H 0 ∼ = C. Therefore, the Hilbert space H of the system under consideration is the direct sum H 1 ⊕ H 0 , and the states are mixtures of the elements from End(H 1 ) and End(H 0 ).We now briefly review the destruction of states procedure introduced in Ref. [3]. LetΛ be an arbitrary observable on H 1 with the spectrum Λ and Ω be a subset of Λ. We denote the subspace spanned by all the eigenvectors corresponding to the eigenvalues from the subset Ω by H Ω ⊂ H 1 and the projector onto this subspace by Π Ω . If the qudit state is an element of End(H Ω ) then the qudit is destroyed, otherwise it is not. First, we measure with no selection the ob-Then, after the destruction we get either the vacuum with the probability tr(ρΠ Ω ) or the one-qudit state with the probability tr(ρΠ ⊥ Ω ). Therefore, the density operator is reduced to Π ⊥ Ω ρΠ ⊥ Ω + tr(ρΠ Ω )|vac ...