Destruction of states in quantum mechanics

Caban, Paweł; Rembieliński, Jakub; Smoliński, Kordian A.; Walczak, Zbigniew

doi:10.1088/0305-4470/35/14/308

Cited by 5 publications

(5 citation statements)

References 15 publications

(9 reference statements)

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“…Therefore, in our opinion, we need an approach enabling a description of the system that can be in two-particle states as well as the one-particle and even zero-particle states, which can arise during the time evolution (decay) of the initial system. It seems to us that the mentioned issues can be resolved by an assumption that the decaying particle can be found in a particle state as well as in the state of the absence of the particle, i.e., in the vacuum state * Electronic address: P.Caban@merlin.fic.uni.lodz.pl † Electronic address: J.Rembielinski@merlin.fic.uni.lodz.pl ‡ Electronic address: K.A.Smolinski@merlin.fic.uni.lodz.pl § Electronic address: Z.Walczak@merlin.fic.uni.lodz.pl (it is not a vacuum in the sense used in quantum field theory, but rather in a sense used in [9]).…”

Section: Introductionmentioning

confidence: 99%

Unstable particles as open quantum systems

et al. 2005

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We present the probability preserving description of the decaying particle within the framework of quantum mechanics of open systems taking into account the superselection rule prohibiting the superposition of the particle and vacuum. In our approach the evolution of the system is given by a family of completely positive trace preserving maps forming one-parameter dynamical semigroup. We give the Kraus representation for the general evolution of such systems which allows one to write the evolution for systems with two or more particles. Moreover, we show that the decay of the particle can be regarded as a Markov process by finding explicitly the master equation in the Lindblad form. We also show that there are remarkable restrictions on the possible strength of decoherence.Comment: 11 pp, 2 figs (published version

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Section: Introductionmentioning

confidence: 99%

Unstable particles as open quantum systems

et al. 2005

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“…Despite the fact that the state |0 is usually called vacuum, it is not the vacuum in the sense of quantum field theory, but rather in the sense used in [39], i.e., it is an absence of a particle. This equation leads to the probability density of finding the particle evolving according to the Geiger-Nutall exponential law.…”

Section: Schrödinger Picturementioning

confidence: 99%

Many Unstable Particles from an Open Quantum System's Perspective

Smoliński

2014

Open Syst. Inf. Dyn.

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We postulate a master equation, written in the language of creation and annihilation operators, as a candidate for unambiguous quantum mechanical description of unstable particles. We have found Kraus representation for the evolution driven by this master equation and study its properties. Both Schrödinger and Heisenberg picture of the system evolution are presented. We show that the resulting time evolution leads to exponential decay law. Moreover, we analyse mixing of particle flavours and we show that it can lead to flavour oscillation phenomenon.

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“…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.…”

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confidence: 91%

“…The 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.…”

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confidence: 91%

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Kraus representation of destruction of states for one qudit

2003

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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 ...

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Destruction of states in quantum mechanics

Cited by 5 publications

References 15 publications

Unstable particles as open quantum systems

Unstable particles as open quantum systems

Many Unstable Particles from an Open Quantum System's Perspective

Kraus representation of destruction of states for one qudit

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