Tight uniform continuity bounds for the quantum conditional mutual information, for the Holevo quantity, and for capacities of quantum channels

Shirokov, M. E.

doi:10.1063/1.4987135

Cited by 53 publications

(81 citation statements)

References 45 publications

(190 reference statements)

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“…Quantitative continuity analysis of characteristics of quantum systems and channels is important for different tasks of quantum information theory. 1 This is confirmed by a number of works devoted to this question [1,2,3,4,14,20,26,34,37,39,40,43,48,50].…”

Section: Introductionmentioning

confidence: 84%

“…Asymptotically tight continuity bounds for the entropy and for the conditional entropy under the energy constraint have been obtained by Winter [50]. Asymptotically tight continuity bounds for the QCMI under the energy constraint on one of the subsystems has been obtained in [37] by using Winter's technique. The aim of this section is to show that our technique also gives asymptotically tight continuity bounds for these quantities without any claim of their superiority.…”

Section: Basic Examplesmentioning

confidence: 99%

“…It is an important characteristic related to the classical capacity of a quantum channel [16,46]. In analysis of continuity of the Holevo quantity we will use three measures of divergence between ensembles µ = {p i , ρ i } and ν = {q i , σ i } described in detail in [31,37].…”

Section: Close-to-tight Continuity Bound For the Output Holevo Quantimentioning

confidence: 99%

“…In [50] he obtained asymptotically tight continuity bounds for the von Neumann entropy and for the quantum conditional entropy under the energy constraint by using the two-step approach based on the AFW-method combined with finite-dimensional approximation of states with bounded energy. Winter's approach was used in [37] to obtain asymptotically tight continuity bounds for the quantum conditional mutual information under the energy constraint on one subsystem.…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 84%

Section: Basic Examplesmentioning

confidence: 99%

Section: Close-to-tight Continuity Bound For the Output Holevo Quantimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Shirokov

2020

Quantum Inf Process

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“…For this purpose we shall make explicit use of the the continuity argument of Refs. [27,28] which allows one to connect the capacities values of two channels via their relative distance measured in terms of the diamond norm metric [29,30]. While our derivation in many respects mimics the one presented by Bravyi et al, we stress that in order to account for all possible encoding strategy, we have explicitly to deal with the dimension of the ancillary memory element Q Alice can use in the process.…”

Section: Introductionmentioning

confidence: 99%

Quantum-capacity bounds in spin-network communication channels

2019

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Using the Lieb-Robinson inequality and the continuity property of the quantum capacities in terms of the diamond norm, we derive an upper bound on the values that these capacities can attain in spin-network communication models of arbitrary topology. Differently from previous results we make no assumptions on the encoding mechanisms that the sender of the messages adopts in loading information on the network.

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Convergence Rates for the Quantum Central Limit Theorem

Becker

Datta

Lami

et al. 2021

Commun. Math. Phys.

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Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state $$\rho $$ ρ with finite second moments, converges to the Gaussian state with the same first and second moments as $$\rho $$ ρ . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate $$\mathcal {O}\left( n^{-1/2}\right) $$ O n - 1 / 2 in the Hilbert–Schmidt norm whenever the third moments of $$\rho $$ ρ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities $$\lambda ^{1/n}$$ λ 1 / n fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate $$\mathcal {O}\Big (n^{-\frac{1}{2(m+1)}}\Big )$$ O ( n - 1 2 ( m + 1 ) ) . This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function $$\chi _\rho $$ χ ρ is uniformly bounded by some $$\eta _\rho <1$$ η ρ < 1 outside of any neighbourhood of the origin; also, $$\eta _\rho $$ η ρ can be made to depend only on the energy of the state $$\rho $$ ρ .

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Tight uniform continuity bounds for the quantum conditional mutual information, for the Holevo quantity, and for capacities of quantum channels

Abstract: We start with Fannes' type and Winter's type tight continuity bounds for the quantum conditional mutual information and their specifications for states of special types.

Cited by 53 publications

References 45 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

Quantum-capacity bounds in spin-network communication channels

Convergence Rates for the Quantum Central Limit Theorem

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