Advanced Alicki–Fannes–Winter method for energy-constrained quantum systems and its use

Shirokov, M. E.

doi:10.1007/s11128-020-2581-2

Cited by 15 publications

(56 citation statements)

References 67 publications

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“…Our first result is concerned with providing a tight version of Winter's bound in [4] on the difference of von Neumann entropies for two states, when the Hamiltonian imposing an energy constraint on the input states is the number operator. The bound obtained by Winter is asymptotically tight, see also [9,10,11].…”

Section: Introductionmentioning

confidence: 88%

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

Becker¹,

Datta²,

Jabbour³

2021

Preprint

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We prove a variety of new and refined uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality and the notion of maximal coupling of random variables. We then employ this classical result to derive the first tight energy-constrained continuity bound for the von Neumann entropy of states of infinite-dimensional quantum systems, when the Hamiltonian is the number operator, which is arguably the most relevant Hamiltonian in the study of infinitedimensional quantum systems in the context of quantum information theory. The above scheme works only for Shannon-and von Neumann entropies. Hence, to deal with more general entropies, e.g. α-Rényi and α-Tsallis entropies, with α ∈ (0, 1), for which continuity bounds are known only for finite-dimensional systems, we develop a novel approximation scheme which relies on recent results on operator Hölder continuous functions. This approach is, as we show, motivated by continuity bounds for α-Rényi and α-Tsallis entropies of random variables that follow from the Hölder continuity of the entropy functionals. We also provide bounds for α > 1. Finally, we settle an open problem on related approximation questions posed in the recent works by Shirokov [1, 2] on the so-called Finite-dimensional Approximation (FA) property.

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Section: Introductionmentioning

confidence: 88%

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

Becker¹,

Datta²,

Jabbour³

2021

Preprint

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show abstract

“…Our bounds, which we prove by further refining the techniques in Ref. [25,26], turn out to be asymptotically tight in many physically interesting cases, and imply that the aforementioned quantifiers, whose importance for the study of entanglement theory can hardly be overestimated, are uniformly continuous on energy-bounded sets of states.…”

Section: Introductionmentioning

confidence: 53%

“…And yet, such a highly discontinuous behaviour does not represent a problem physically, because it typically involves infinite-energy states. Throughout this section we will see how it is possible to restore a (slightly weaker) form of continuity for the relative entropy of entanglement and related quantities by looking only at the physically meaningful energy-constrained states [25,26,49,50].…”

Section: Tight Uniform Continuity Bounds For the Relative Entropy Of ...mentioning

confidence: 99%

“…Significantly more accurate continuity bounds for the function D F under an energy constraint can be obtained by using the methods proposed in Ref. [25,26]. To apply them, we shall also require that the function under examination be bounded by a multiple of the local entropy or the sum of local entropies.…”

Section: A Energy Constraintsmentioning

confidence: 99%

“…From now on, assume that H is a grounded Hamiltonian H on H. The methods of Ref. [25,26] need one more regularity assumption on H: informally, this captures the fact that its energy levels should not become too dense as the energy grows; formally, the requirement is that lim λ→0 + Tr e −λH λ = 1 .…”

Section: A Energy Constraintsmentioning

confidence: 99%

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Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme

Lami¹,

Shirokov²

2021

Preprint

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The relative entropy of entanglement ER is defined as the distance of a multi-partite 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 (unique) closest separable state, even in infinite dimension; also, ER is everywhere lower semi-continuous. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multi-partite 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 λ-quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states. We complement our findings by giving explicit and asymptotically tight continuity estimates for ER and closely related quantities in the presence of an energy constraint. CONTENTS I. Introduction II. Notation A. Topologies for quantum systems B. Relative entropy of resource III. Main results A. The statements B. Proofs C. On the continuity of the relative entropy of resource D. On the continuity of the minimiser IV. Applications A. Relative entropy of entanglement B. Relative entropy of multi-partite entanglement C. Relative entropy of NPT entanglement D. Relative entropy of non-classicality, Wigner non-positivity, and generalisations thereof E. Relative entropy of non-Gaussianity V. Tight uniform continuity bounds for the relative entropy of entanglement and generalisations thereof A. Energy constraints B. Uniform continuity on energy-constrained states: bipartite case C. Uniform continuity on energy-constrained states: multipartite case VI. Conclusions and outlook VII. Acknowledgements References A. On the proof of an inequality involving the multi-partite relative entropy of entanglement

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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.

show abstract

Advanced Alicki–Fannes–Winter method for energy-constrained quantum systems and its use

Cited by 15 publications

References 67 publications

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme

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

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