Contractivity properties of a quantum diffusion semigroup

We prove the quantum conditional entropy power inequality for quantum additive noise channels. This inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory. We also show that this conditional entropy power inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of Gaussian input states. We apply the conditional entropy power inequality to find an array of information-theoretic inequalities for conditional entropies which are the analogs of inequalities which have already been established in the unconditioned setting. Furthermore, we give a simple proof of the convergence rate of the quantum Ornstein-Uhlenbeck semigroup based on entropy power inequalities. arXiv:1803.00470v2 [quant-ph]

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“…The quantum conditional Fisher information isoperimetric inequality stated in (14) can be restated as…”

Section: Concavity Of the Quantum Conditional Entropy Power Along Thementioning

confidence: 99%

“…In the context of mixing times of semigroups, the authors in [14] have used this convolution extensively and proved various properties which are related to the discussion of the entropy power inequality.…”

Section: Introductionmentioning

confidence: 99%

The conditional entropy power inequality for quantum additive noise channels

Palma

Huber

2018

Journal of Mathematical Physics

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“…After posting our paper to the arxiv, we were made aware of concurrent related work by Rouzé, Datta, and Pautrat. Their paper has now been made available [42].…”

Section: Remarkmentioning

confidence: 99%

Geometric inequalities from phase space translations

Huber

König

Vershynina

2017

Journal of Mathematical Physics

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We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results from a certain convolution operation: the latter maps a classical probability distribution on phase space and a quantum state to a quantum state. We show that this inequality also gives rise to several related inequalities whose counterparts are well-known in the classical setting: in particular, it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality, and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup. As an application, we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck semigroup, and argue that it implies fast convergence towards the fixed point for a large class of initial states.for the convolution (3) and for λ = 1/2, where S(ρ X ) = −tr(ρ X log ρ X ) denotes the von Neumann entropy. Subsequent work [14] managed to lift the restriction on λ, and generalized this result to more general (Gaussian) unitaries in place of U λ . A related inequality of the form , 1] was also shown in [13], generalizing classical results [15] (see also [16] for a discussion of the relationship between the two). A generalization to conditional entropies was proposed in [17], and an application to channel capacities was discussed in [18].A key tool in establishing these results is the quantum Fisher information J(ρ), defined for a state ρ as the divergence-based Fisher information of the family ρ (θ) θ obtained by displacing ρ along a phase space direction (see Section 3 for a precise definition). It was shown in [13] for λ = 1/2 and in [14] for general λ ∈ [0, 1] that this quantity satisfies the Fisher information inequality [3]

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“…They satisfy a number of convenient properties with respect to displacements in phase space as well as a data processing inequality. For more background we refer to [9,21,22]. For a specific kind of quantum channels, the so-called single-mode phase-insensitive 1 Gaussian channels, the classical capacity has recently been found by Giovannetti et al [4,25].…”

Section: Bosonic Noise Channelsmentioning

confidence: 99%

Coherent state coding approaches the capacity of non-Gaussian bosonic channels

Huber¹,

König²

2018

J. Phys. A: Math. Theor.

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The additivity problem asks if the use of entanglement can boost the information-carrying capacity of a given channel beyond what is achievable by coding with simple product states only. This has recently been shown not to be the case for phase-insensitive one-mode Gaussian channels, but remains unresolved in general. Here we consider two general classes of bosonic noise channels, which include phase-insensitive Gaussian channels as special cases: these are attenuators with general, potentially non-Gaussian environment states and classical noise channels with general probabilistic noise. We show that additivity violations, if existent, are rather minor for all these channels: the maximal gain in classical capacity is bounded by a constant independent of the input energy. Our proof shows that coding by simple classical modulation of coherent states is close to optimal.

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Contractivity properties of a quantum diffusion semigroup

Cited by 11 publications

References 47 publications

The conditional entropy power inequality for quantum additive noise channels

The conditional entropy power inequality for quantum additive noise channels

Geometric inequalities from phase space translations

Coherent state coding approaches the capacity of non-Gaussian bosonic channels

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