On the notion of entanglement in Hilbert spaces

Холево, Александр Семенович; Shirokov, M. E.; Werner, R. F.

doi:10.1070/rm2005v060n02abeh000830

Cited by 54 publications

(102 citation statements)

References 4 publications

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“…It is known that the phase conjugating channel is entanglement breaking [36,37]. It is also known that every entanglement breaking channel has a description in terms of rank one Kraus operators [35].…”

Section: Entanglement Breaking Propertymentioning

confidence: 99%

“…And a theorem of Choi [30] gives a necessary and sufficient test for extremality of a channel in terms of Kraus operators. It is known that a channel is entanglement breaking if and only if it can be described in terms of a set of rank one Kraus operators [35][36][37]. Further, the work of [38] and [39,40] relate error correctability to the structure of the Kraus operators of a channel.…”

Section: Introductionmentioning

confidence: 99%

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Operator-sum representation for bosonic Gaussian channels

2011

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Operator-sum or Kraus representations for single-mode Bosonic Gaussian channels are developed, and several of their consequences explored. The fact that the two-mode metaplectic operators acting as unitary purification of these channels do not, in their canonical form, mix the position and momentum variables is exploited to present a procedure which applies uniformly to all families in the Holevo classification. In this procedure the Kraus operators of every quantum-limited Gaussian channel can be simply read off from the matrix elements of a corresponding metaplectic operator. Kraus operators are employed to bring out, in the Fock basis, the manner in which the antilinear, unphysical matrix transposition map when accompanied by injection of a threshold classical noise becomes a physical channel, denoted D(κ) in the Holevo classification. The matrix transposition channels D(κ), D(κ −1 ) turn out to be a dual pair in the sense that their Kraus operators are related by the adjoint operation. The amplifier channel with amplification factor κ and the beamsplitter channel with attenuation factor κ −1 turn out to be mutually dual in the same sense. The action of the quantum-limited attenuator and amplifier channels as simply scaling maps on suitable quasiprobabilities in phase space is examined in the Kraus picture. Consideration of cumulants is used to examine the issue of fixed points. The semigroup property of the amplifier and attenuator families leads in both cases to a Zeno-like effect arising as a consequence of interrupted evolution. In the cases of entanglement-breaking channels a description in terms of rank one Kraus operators is shown to emerge quite simply. In contradistinction, it is shown that there is not even one finite rank operator in the entire linear span of Kraus operators of the quantum-limited amplifier or attenuator families, an assertion far stronger than the statement that these are not entanglement breaking channels. A characterization of extremality in terms of Kraus operators, originally due to Choi, is employed to show that all quantum-limited Gaussian channels are extremal. The fact that every noisy Gaussian channel can be realised as product of a pair of quantum-limited channels is used to construct a discrete set of linearly independent Kraus operators for noisy Gaussian channels, including the classical noise channel, and these Kraus operators have a particularly simple structure.

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Section: Entanglement Breaking Propertymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Operator-sum representation for bosonic Gaussian channels

2011

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“…In this section we consider nontrivial class of entanglement-breaking channels generalizing the example considered in [5]. For channels of this class the χ-capacity and the minimal output entropy can be explicitly calculated.…”

Section: On a Class Of Channels With Finite χ-Capacitymentioning

confidence: 99%

“…where {|k } k∈Z\{0} be an orthonormal basis in H. This channel is entanglement-breaking [6], [5] and Φ {qn} (S(H)) = co(S …”

Section: General Propertiesmentioning

confidence: 99%

“…So, we may consider the class of CE-channels as an intermediate one and use technical advantages of dealing with such channels to approach general infinite dimensional channels and systems. For example, the channel of this type was used in [5] in considering properties of infinite dimensional entanglement-breaking channels and of the set of separable states.…”

Section: Introductionmentioning

confidence: 99%

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On Channels with Finite Holevo Capacity

Shirokov¹

2009

Theory Probab. Appl.

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Attainability and Lower Semi-continuity of the Relative Entropy of Entanglement and Variations on the Theme

Lami

Shirokov

2023

Ann. Henri Poincaré

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The relative entropy of entanglement $$E_R$$ E R is defined as the distance of a multipartite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a closest separable state, even in infinite dimensions; also, $$E_R$$ E R is everywhere lower semi-continuous. We use this to derive a dual variational expression for $$E_R$$ E R in terms of an external supremum instead of infimum. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multipartite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity—more generally, all relative entropy distances from the sets of states with non-negative $$\lambda $$ λ -quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states, and for this reason, the techniques we develop involve a bouquet of classical results from functional analysis. We complement our analysis by giving explicit and asymptotically tight continuity estimates for $$E_R$$ E R and closely related quantities in the presence of an energy constraint.

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On the notion of entanglement in Hilbert spaces

Cited by 54 publications

References 4 publications

Operator-sum representation for bosonic Gaussian channels

Operator-sum representation for bosonic Gaussian channels

On Channels with Finite Holevo Capacity

Attainability and Lower Semi-continuity of the Relative Entropy of Entanglement and Variations on the Theme

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