Convergence Rates for Quantum Evolution and Entropic Continuity Bounds in Infinite Dimensions

Becker, Simon; Datta, Nilanjana

doi:10.1007/s00220-019-03594-2

Cited by 27 publications

(41 citation statements)

References 52 publications

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“…This quantity plays important role in quantum information theory [15,46], its nonnegativity is a basic result well known as strong subadditivity of von Neumann entropy [29]. If system Z is trivial then (6) coincides with (5). In infinite dimensions formula (6) may contain the uncertainty "∞ − ∞".…”

Section: Basic Notationsmentioning

confidence: 99%

“…while condition (11) is equivalent to F H A (E) = o(E) as E → +∞. It is essential that condition (14) holds for the Hamiltonians of many real quantum systems [5,39]. 4 The functionF…”

Section: The Set Of Quantum States With Bounded Energymentioning

confidence: 99%

“…Note: The condition of part B of Proposition 1 is valid for the Hamiltonians of many real quantum systems [5].…”

Section: The Set Of Quantum States With Bounded Energymentioning

confidence: 99%

“…where a i and a * i are the annihilation and creation operators of the i-th mode [16]. Note that this Hamiltonian satisfies condition (20) with a = 1 + 1/ℓ [5,6].…”

Section: The Case When a Is The ℓ-Mode Quantum Oscillatormentioning

confidence: 99%

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

Shirokov

2020

Quantum Inf Process

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We describe an universal method for quantitative continuity analysis of entropic characteristics of energy-constrained quantum systems and channels. It gives asymptotically tight continuity bounds for basic characteristics of quantum systems of wide class (including multi-mode quantum oscillators) and channels between such systems under the energy constraint.The main application of the proposed method is the advanced version of the uniform finite-dimensional approximation theorem for basic capacities of energyconstrained quantum channels.5 Advanced version of the uniform finite-dimensional approximation theorem for capacities of energy-constrained channels 25

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Section: Basic Notationsmentioning

confidence: 99%

“…while condition (11) is equivalent to F H A (E) = o(E) as E → +∞. It is essential that condition (14) holds for the Hamiltonians of many real quantum systems [5,39]. 4 The functionF…”

Section: The Set Of Quantum States With Bounded Energymentioning

confidence: 99%

“…Note: The condition of part B of Proposition 1 is valid for the Hamiltonians of many real quantum systems [5].…”

Section: The Set Of Quantum States With Bounded Energymentioning

confidence: 99%

“…where a i and a * i are the annihilation and creation operators of the i-th mode [16]. Note that this Hamiltonian satisfies condition (20) with a = 1 + 1/ℓ [5,6].…”

Section: The Case When a Is The ℓ-Mode Quantum Oscillatormentioning

confidence: 99%

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

Shirokov

2020

Quantum Inf Process

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“…In the next section, we show that the conditional output state (1) for the approximate operation (4) does not converge uniformly to the result of the ideal photon subtraction (3) even in the case of energy-constrained states. 29,30].…”

Section: Approximate Photon Subtraction and Photon Additionmentioning

confidence: 99%

On Quantum Operations of Photon Subtraction and Photon Addition

Filippov

2019

Lobachevskii J Math

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The conventional photon subtraction and photon addition transformations, ̺ → ta̺a † and ̺ → ta † ̺a, are not valid quantum operations for any constant t > 0 since these transformations are not trace nonincreasing. For a fixed density operator ̺ there exist fair quantum operations, N− and N+, whose conditional output states approximate the normalized outputs of former transformations with an arbitrary accuracy. However, the uniform convergence for some classes of density operators ̺ has remained essentially unknown. Here we show that, in the case of photon addition operation, the uniform convergence takes place for the energy-second-moment-constrained states such that tr[̺H 2 ] ≤ E2 < ∞, H = a † a. In the case of photon subtraction, the uniform convergence takes place for the energy-second-moment-constrained states with nonvanishing energy, i.e., the states ̺ such that tr[̺H] ≥ E1 > 0 and tr[̺H 2 ] ≤ E2 < ∞. We prove that these conditions cannot be relaxed and generalize the results to the cases of multiple photon subtraction and addition.

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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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Convergence Rates for Quantum Evolution and Entropic Continuity Bounds in Infinite Dimensions

Cited by 27 publications

References 52 publications

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

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

On Quantum Operations of Photon Subtraction and Photon Addition

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

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