A continuity theorem for Stinespring's dilation

Kretschmann, Dennis; Schlingemann, Dirk; Werner, R. F.

doi:10.1016/j.jfa.2008.07.023

Cited by 60 publications

(119 citation statements)

References 20 publications

Supporting

Mentioning

116

Contrasting

Unclassified

Order By: Relevance

“…Working with modules has several advantages. The results we get are of course same as that of [13], when the range algebra is a von Neumann algebra or an injective C * -algebra. However, we show that one may not even get a metric (triangle inequality may fail) when the range algebra is a general C * -algebra.…”

Section: Introductionsupporting

confidence: 72%

“…Thus there exists a one-one correspondence between the GNS-constructions {(E, x 1 ), (E, x 2 )} and the Stinespring representations ϕ 2 ) coincides with the definition given in [13]. In particular, if B = B(C) = C, then β(ϕ 1 , ϕ 2 ) is the Bures distance given in [7].…”

Section: Definitionmentioning

confidence: 74%

“…The following proposition says that β(ϕ 1 , ϕ 2 ) coincide with the alternative definition, given in [13], of Bures distance for CP-maps between arbitrary C * -algebras. Subsequently will not be needing this definition and we present it here for the sake of completeness.…”

Section: Definitionmentioning

confidence: 80%

“…The proofs are similar, though we have also taken some ideas from [7]. We also give several examples and answer a question of [13] in the negative.…”

Section: Bures Distance: Von Neumann Algebrasmentioning

confidence: 86%

“…In this Section, motivated by the work of [13], we show that Bures distance can be computed using intertwiners between two (minimal) GNS-constructions of CP-maps. Suppose E is a common representation module for ϕ i and…”

Section: Intertwiners and Computation Of Bures Distancementioning

confidence: 99%

See 4 more Smart Citations

Bures Distance for Completely Positive Maps

Bhat

Sumesh

2013

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

View full text Add to dashboard Cite

D. Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between C * -algebras by D. Kretschmann, D. Schlingemann and R. F. Werner. We present a Hilbert C * -module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples. We also prove a rigidity theorem, showing that representation modules of completely positive maps which are close to the identity map contain a copy of the original algebra.

show abstract

Section: Introductionsupporting

confidence: 72%

Section: Definitionmentioning

confidence: 74%

Section: Definitionmentioning

confidence: 80%

“…The proofs are similar, though we have also taken some ideas from [7]. We also give several examples and answer a question of [13] in the negative.…”

Section: Bures Distance: Von Neumann Algebrasmentioning

confidence: 86%

Section: Intertwiners and Computation Of Bures Distancementioning

confidence: 99%

See 3 more Smart Citations

Bures Distance for Completely Positive Maps

Bhat

Sumesh

2013

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

View full text Add to dashboard Cite

show abstract

Irreversibility of Entanglement Loss

Buscemi

2008

Theory of Quantum Computation, Communication, and Cryptography

View full text Add to dashboard Cite

Abstract. The action of a channel on a quantum system, when non trivial, always causes deterioration of initial quantum resources, understood as the entanglement initially shared by the input system with some reference purifying it. One effective way to measure such a deterioration is by measuring the loss of coherent information, namely the difference between the initial coherent information and the final one: such a difference is "small", if and only if the action of the channel can be "almost perfectly" corrected with probability one. In this work, we generalise this result to different entanglement loss functions, notably including the entanglement of formation loss, and prove that many inequivalent entanglement measures lead to equivalent conditions for approximate quantum error correction. In doing this, we show how different measures of bipartite entanglement give rise to corresponding distance-like functions between quantum channels, and we investigate how these induced distances are related to the cb-norm.

show abstract

State Convertibility in the von Neumann Algebra Framework

Crann

Kribs

Levene

et al. 2020

Commun. Math. Phys.

View full text Add to dashboard Cite

We establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi-finite von Neumann algebras. Namely, we establish a generalisation to this setting of Nielsen's theorem on the convertibility of quantum states under local operations and classical communication (LOCC) schemes. Along the way, we introduce an appropriate generalisation of LOCC operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von Neumann algebras. As an application of our result in the setting of II1-factors, we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of Vidal, thus yielding a new way to quantify entanglement in that context. Building on previous work in the infinite-dimensional setting, we show that trace vectors play the role of maximally entangled states for general II1-factors. Examples are drawn from infinite spin chains, quasi-free representations of the CAR, and discretised versions of the CCR.

show abstract

A continuity theorem for Stinespring's dilation

Cited by 60 publications

References 20 publications

Bures Distance for Completely Positive Maps

Bures Distance for Completely Positive Maps

Irreversibility of Entanglement Loss

State Convertibility in the von Neumann Algebra Framework

Contact Info

Product

Resources

About