Kraus representation of destruction of states for one qudit

Abstract

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 ...