Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup

Carbone, Raffaella; Sasso, Emanuela

doi:10.1007/s00440-007-0073-2

Cited by 20 publications

(31 citation statements)

References 27 publications

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“…For any 1 < q < p and any 0 < λ < 1 the p → q norm of E λ is infinite and is asymptotically achieved by thermal Gaussian states with infinite temperature: 6) and the claim follows. One-mode quatum Gaussian channels have Gaussian maximizers 21…”

Section: )mentioning

confidence: 85%

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The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have GaussianMaximizers

Palma

Trevisan

Giovannetti

2018

Ann. Henri Poincaré

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We determine the p → q norms of the Gaussian one-mode quantum-limited attenuator and amplifier and prove that they are achieved by Gaussian states, extending to noncommutative probability the seminal theorem "Gaussian kernels have only Gaussian maximizers" (Lieb, Invent. Math. 102, 179 (1990)). The quantum-limited attenuator and amplifier are the building blocks of quantum Gaussian channels, which play a key role in quantum communication theory since they model in the quantum regime the attenuation and the noise affecting any electromagnetic signal. Our result is crucial to prove the longstanding conjecture stating that Gaussian input states minimize the output entropy of one-mode phase-covariant quantum Gaussian channels for fixed input entropy. Our proof technique is based on a new noncommutative logarithmic Sobolev inequality, and it can be used to determine the p → q norms of any quantum semigroup.Mathematics Subject Classification (2010). 46B28; 46N50; 81P45; 81V80; 94A15.

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Section: )mentioning

confidence: 85%

“…Proof. IfX has rank 1, we have from Lemma 3.8 6) and the claim follows since S a X = 0. IfX has rank at least 2, then S a X > 0 and the claim follows from Theorem 3.1 with 0 < z < 1 chosen such that S a (ω z ) = S a X .…”

Section: )mentioning

confidence: 88%

The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have GaussianMaximizers

Palma

Trevisan

Giovannetti

2018

Ann. Henri Poincaré

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“…We remark that it would significantly strengthen known results about the qOU semigroup. Specifically, Carbone et al [26] established that the qOU semigroup is hypercontractive. In [26,Proposition 4.2], the following inequality was shown for the Log-Sobolev-2 constant 1 α 2 of L µ,λ :…”

Section: Application To the Ornstein-uhlenbeck Semigroupmentioning

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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“…for any integrable, continuously differentiable function f , which has an integrable partial derivative ∂ x j f , where z j denotes the j th component of the vector z ∈ Z. Secondly, for any square integrable function h on R 2n , we have that 5) which is the classical Plancherel identity. We also employ the well-known polarization identity:…”

Section: Proofs Of Theorem and Theoremmentioning

confidence: 99%

Contractivity properties of a quantum diffusion semigroup

Datta

Pautrat

Rouzé

2017

Journal of Mathematical Physics

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We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of a state with positive Wigner function, evolving under the action of the semigroup, decrease at least inverse polynomially in time, while its entropy increases at least logarithmically in time.

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Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup

Cited by 20 publications

References 27 publications

The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have GaussianMaximizers

The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have GaussianMaximizers

Geometric inequalities from phase space translations

Contractivity properties of a quantum diffusion semigroup

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