Stochastic couplings for von neumann algebras

Cecchini, Carlo

doi:10.1007/bfb0083549

Cited by 7 publications

(10 citation statements)

References 3 publications

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“…Since its range is contained in the center of A 2 , (b) can be obtained by Ref. 4. The implication from (b) to (c) follows from: ω(a 1 a 2 ) = a + 1 Ω, a 2 Ω = a + 1 Ω, E 1 a 2 Ω = a + 1 Ω, ω 2 (a 2 )Ω = ω 1 (a 1 )ω 2 (a 2 ) .…”

Section: Independencementioning

confidence: 98%

Independence and Markovianity for States on Von Neumann Algebras

Cecchini

2007

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Several possible generalizations of the classical notion of markovianity are given for states defined on a von Neumann algebras generated on a triple of subalgebras. Their mutual relation is discussed in the particular case in which they mutually commute, and the generalization of the classical; time reversal theorem is proved. A structure theorem for a class of Markov chains is also proved.Proof. Our claim follows if we remark that ω(a 1 a {2,3} ) = J {2,3} λ {2,3},1 (a 1 )Ω, a {2,3} Ω and ω

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Section: Independencementioning

confidence: 98%

Independence and Markovianity for States on Von Neumann Algebras

Cecchini

2007

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…such that E * ρ 1 is a compound state of ρ 1 and Λ * ρ 1 . Several particular solutions of this problem have been proposed in [9], [10], [11], [21], [22], [23], however an explicit description of all the possible solutions to this problem is still missing.…”

Section: Certain Quantum Channels Are Naturally Associated To Classicmentioning

confidence: 99%

Compound Channels, Transition Expectations, and Liftings

Accardi¹,

Ohya

2008

Selected Papers of M Ohya

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In Section 1, we introduce the notion of lifting as a generalization of the notion of compound state introduced in [21], [22] and we show that this notion allows an unified approach to the problems of quantum measurement and of signal transmission through quantum channels. The dual of a linear lifting is a transition expectation in the sense of [3] and we characterize those transition expectations which arise from compound states in the sense of [22]. In Section 2, we characterize those liftings whose range is contained in the closed convex hull of product states and we prove that the corresponding quantum Markov chains [2] are uniquely determined by a classical generalization of both the quantum random walks of [4] and the locally diagonalizable states considered in [3]. In Section 4, as a first application of the above results, we prove that the attenuation (beam splitting) process for optical communication treated in [21] can be described in a simpler and more general way in terms of liftings and of transition expectations. The error probabilty of information transmission in the attenuation process is rederived from our new description. We also obtain some new results concerning the explicit computation of error probabilities in the squeezing case.

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“…In this section, we study QMS as defined in section II (see (2) and Theorem II.I). We will treat only the simplest case where .…”

Section: A Representation Theoremmentioning

confidence: 99%

“…For the moment, several candidates are put forward by different authors [1,2,3] and any proposal has specific advantages and disadvantages w.r.t, mathematical properties. To our opinion, the respective relevance of the different notions for the Quantum Processes which arise from physical problems, might serve as an additional useful argument in the discussion.…”

Section: Introductionmentioning

confidence: 99%

“…It is not clear however that including also non-invariant QMS would not yield a better (lower) value for e0.Due to Remark V.4, we only have to consider two possibilities for the representation ~r of SU(2) occurring in the covariance condition (25).1. The trivial representation: a(u) = 11 for all u E SU(2). In this case, (19) becomes:E~.=~ = E~ for all u E SU(2).…”

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confidence: 99%

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Working with Quantum Markov States and their classical analogues

Nachtergaele

1990

Quantum Probability and Applications V

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We investigate in detail the structure of the set of Quantum Markov States (QMS), as defined by Accardi and Frigerio. A detailed study of their classical analogues reveals that they contain a lot more than just the Markovian measures. Furthermore we prove a canonical representation theorem and this result is then used to analyze the consequences of group invariance for a QMS. The theory is applied in a variational approximation for the ground state of the antiferromagnetic tteisenberg model. We thus show that the QMS might provide an interesting new tool to investigate ground state problems for 1dimensional quantum spin systems.

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Stochastic couplings for von neumann algebras

Cited by 7 publications

References 3 publications

Independence and Markovianity for States on Von Neumann Algebras

Independence and Markovianity for States on Von Neumann Algebras

Compound Channels, Transition Expectations, and Liftings

Working with Quantum Markov States and their classical analogues

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