Linear transformations which preserve trace and positive semidefiniteness of operators

Jamiołkowski, Andrzej

doi:10.1016/0034-4877(72)90011-0

Cited by 1,265 publications

(960 citation statements)

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“…We will in the following always understand Eq. (5) in this form and call L and the corresponding semigroup purely dissipative if H = 0 w.r.t. such a representation.…”

Section: Preliminariesmentioning

confidence: 99%

Dividing Quantum Channels

Wolf

Cirac

2008

Commun. Math. Phys.

288

344

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Abstract:We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of 'indivisible' channels which can not be written as non-trivial products of other channels and study the set of 'infinitesimal divisible' channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.

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“…We will in the following always understand Eq. (5) in this form and call L and the corresponding semigroup purely dissipative if H = 0 w.r.t. such a representation.…”

Section: Preliminariesmentioning

confidence: 99%

Dividing Quantum Channels

Wolf

Cirac

2008

Commun. Math. Phys.

288

344

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“…Thus, for such times (t ′ , t), the map E(t, t ′ ) in the form (14) ceases to take the form of a random unitary map. Indeed, using the Jamiolkowski isomorphism [43] it is straightforward to see that |D(t, t ′ )| < 1 or…”

Section: Random Unitary Representationsmentioning

confidence: 99%

System–environment correlations and non-Markovian dynamics

Pernice

Helm

Strunz

2012

J. Phys. B: At. Mol. Opt. Phys.

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We determine the total state dynamics of a dephasing open quantum system using the standard environment of harmonic oscillators. Of particular interest are random unitary approaches to the same reduced dynamics and system-environment correlations in the full model. Concentrating on a model with an at times negative dephasing rate, the issue of "non-Markovianity" will also be addressed. Crucially, given the quantum environment, the appearance of non-Markovian dynamics turns out to be accompanied by a loss of system-environment correlations. Depending on the initial purity of the qubit state, these system-environment correlations may be purely classical over the whole relevant time scale, or there may be intervals of genuine system-environment entanglement. In the latter case, we see no obvious relation between the build-up or decay of these quantum correlations and "Non-Markovianity".

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“…Here use is made of the isomorphism between positive maps and operators on H 1 ⊗H 2 (Jamiolkowski, 1972). Horodecki et al (1996a) note that a map Λ will be positive iff the associated operator S(Λ) is Hermitian and Tr(S(Λ)P ⊗ Q) ≥ 0, for all P , Q on H 1 , H 2 respectively.…”

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confidence: 99%

Quantum Information Theory and the Foundations of Quantum Mechanics

Timpson¹

2013

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of Thesis Submitted for the Degree of Doctor of PhilosophyThis thesis is a contribution to the debate on the implications of quantum information theory for the foundational problems of quantum mechanics. In Part I an attempt is made to shed some light on the nature of information and quantum information theory. It is emphasized that the everyday notion of information is to be firmly distinguished from the technical notions arising in information theory;however it is maintained that in both settings 'information' functions as an abstract noun, hence does not refer to a particular or substance. The popular claim 'Information is Physical' is assessed and it is argued that this proposition faces a destructive dilemma. Accordingly, the slogan may not be understood as an ontological claim, but at best, as a methodological one. A novel argument is provided against Dretske's (1981) attempt to base a semantic notion of information on ideas from information theory.The function of various measures of information content for quantum systems is explored and the applicability of the Shannon information in the quantum context maintained against the challenge of Brukner and Zeilinger (2001). The phenomenon of quantum teleportation is then explored as a case study serving to emphasize the value of recognising the logical status of 'information' as an abstract noun: it is argued that the conceptual puzzles often associated with this phenomenon result from the familiar error of hypostatizing an abstract noun.The approach of Deutsch and Hayden (2000) to the questions of locality and information flow in entangled quantum systems is assessed. It is suggested that the approach suffers from an equivocation between a conservative and an ontological reading; and the differing implications of each is examined. Some results are presented on the characterization of entanglement in the Deutsch-Hayden formalism.Part I closes with a discussion of some philosophical aspects of quantum computation.In particular, it is argued against Deutsch that the Church-Turing hypothesis is not underwritten by a physical principle, the Turing Principle. Some general morals are drawn concerning the nature of quantum information theory.In Part II, attention turns to the question of the implications of quantum information theory for our understanding of the meaning of the quantum formalism. Following some preliminary remarks, two particular information-theoretic approaches to the foundations of quantum mechanics are assessed in detail. It is argued that Zeilinger's (1999) Foundational Principle is unsuccessful as a foundational principle for quantum mechanics. The information-theoretic characterization theorem of Clifton, is assessed more favourably, but the generality of the approach is questioned and it is argued that the implications of the theorem for the traditional foundational problems in quantum mechanics remains obscure.

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Linear transformations which preserve trace and positive semidefiniteness of operators

Cited by 1,265 publications

References 1 publication

Dividing Quantum Channels

Dividing Quantum Channels

System–environment correlations and non-Markovian dynamics

Quantum Information Theory and the Foundations of Quantum Mechanics

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