Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

One-mode quantum-limited Gaussian channels have Gaussian maximizers

Giacomo De Palma, Dario Trevisan, Vittorio Giovannetti

Published Oct 31, 2016• 5 citations
OpenAlex

Abstract

We prove that states saturate the p->q norms of the one-mode quantum-limited attenuator and amplifier. The proof starts from the majorization result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), and is based on a new logarithmic Sobolev inequality. Our result extends to noncommutative probability the seminal theorem Gaussian kernels have only maximizers (Lieb, Invent. Math. 102, 179 (1990)), stating that operators saturate the p->q norms of integral kernels. Our result also implies that the p->q norms of the thinning are saturated by geometric probability distributions. Moreover, the multimode extension of our result would imply the multiplicativity of the p->q norms of quantum-limited channels.

Related papers

Gaussian States Minimize the Output Entropy of the One-Mode Quantum Attenuator

Giacomo De Palma, Dario Trevisan, Vittorio Giovannetti

We prove that Gaussian thermal input states minimize the output von Neumann entropy of the one-mode Gaussian quantum-limited attenuator for fixed input entropy. The Gaussian quantum-limited attenuator models the attenuation of an electromagnetic signal in the quantum regime. The Shannon entropy of an attenuated real-valued classical signal is a simple function of the entropy of the original signal. A striking consequence of energy quantization is that the output von Neumann entropy of the quantum-limited attenuator is no more a function of the input entropy alone. The proof starts from the majorization result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), and is based on a new isoperimetric inequality. Our result implies that geometric input probability distributions minimize the output Shannon entropy of the thinning for fixed input entropy. Moreover, our result opens the way to the multimode generalization that permits to determine both the triple tradeoff region of the Gaussian quantum-limited attenuator and the classical capacity region of the Gaussian degraded quantum broadcast channel.

arXiv:1605.00441•IEEE Transactions on Information Theory•Nov 1, 2016•31 citations

Gaussian States Minimize the Output Entropy of One-Mode Quantum Gaussian Channels

Giacomo De Palma, Dario Trevisan, Vittorio Giovannetti

We prove the long-standing conjecture stating that Gaussian thermal input states minimize the output von Neumann entropy of one-mode phase-covariant quantum Gaussian channels among all the input states with a given entropy. Phase-covariant quantum Gaussian channels model the attenuation and the noise that affect any electromagnetic signal in the quantum regime. Our result is crucial to prove the converse theorems for both the triple trade-off region and the capacity region for broadcast communication of the Gaussian quantum-limited amplifier. Our result extends to the quantum regime the entropy power inequality that plays a key role in classical information theory. Our proof exploits a completely new technique based on the recent determination of the p→q norms of the quantum-limited amplifier [De Palma et al., arXiv:1610.09967]. This technique can be applied to any quantum channel.

arXiv:1610.09970•Physical Review Letters•Apr 21, 2017•31 citations

From Qubit Tensorization to Gaussian Optimizers in Bosonic Systems

Salman Beigi, Hami Mehrabi

We develop a method for deriving Gaussian optimality results for bosonic quantum systems from inequalities for qubit systems. Inspired by Gross's derivation of the Gaussian logarithmic Sobolev inequality from an inequality on the two-point space, we use qubits as quantum two-point spaces and lift bosonic Weyl operators to pseudo-Weyl operators on large qubit systems. By controlling weighted norms and relative entropy in the fluctuation limit, we establish two transfer theorems that convert tensorized qubit inequalities into inequalities for bosonic Gaussian channels. As a first application, we prove that the sharp $2$-logarithmic Sobolev inequality for the bosonic Ornstein--Uhlenbeck semigroup tensorizes without loss in the constant. We also establish a sharp output entropy inequality for tensor powers of a class of qubit channels. This includes a quantum Mrs.~Gerber's lemma determining the minimum output entropy of tensor powers of the qubit depolarizing channel at fixed input entropy. Applying our relative entropy transfer theorem, we obtain multimode constrained minimum output entropy bounds for Gaussian additive-noise channels and thermal attenuators in the heating range.

quant-pharXiv:2610.28902•Oct 22, 2026•0 citations

Logarithmic Sobolev inequality and Hypothesis of Quantum Gaussian Maximizers

Alexander Semenovich Holevo

A long-standing problem in quantum Shannon theory is the classical capacity of bosonic Gaussian channels of various kinds [6].The Hypothesis of Gaussian Maximizers (HGM) states that the full capacity of such channels is attained at Gaussian encodings.A breakthrough was made in the papers [1] and [2], where HGM was proved for the important class of multimode gauge co-or contra-variant Gaussian channels [4].In [8] and [5] the solution was extended to a much broader class of channels satisfying a certain 'threshold condition'.At the same time the HGM remains open for a large variety of Gaussian channels lying beyond the scope of the threshold condition [7].In this note we sketch a novel approach to such problems, based on principles of convex programming, and illustrate it using the characteristic case of an approximate position measurement with energy constraint, underlying noisy Gaussian homodyning in quantum optics.Rather remarkably, for this particular model the method reduces the solution of the optimization problem to a generalization of the celebrated log-Sobolev inequality.Let S be the convex set of all density operators in a separable Hilbert space H of the quantum system and X and Y be standard measurable spaces.Consider a measurement channel M : ρ → p ρ (y) = Tr ρm(y), where ρ ∈ S and m(y) is a uniformly bounded positive operator-valued function of y ∈ Y such that m(y) µ(dy) = I (I is the identity operator on H and µ is a measure).The encoding E = {π(dx), ρ(x)} is a probability measure π(dx) with a measurable family of

Russian Mathematical Surveys•Jan 1, 2022•2 citations

Gaussian optimizers for entropic inequalities in quantum information

Giacomo De Palma, Dario Trevisan, Vittorio Giovannetti +1 more

We survey the state of the art for the proof of the quantum Gaussian optimizer conjectures of quantum information theory. These fundamental conjectures state that quantum Gaussian input states are the solution to several optimization problems involving quantum Gaussian channels. These problems are the quantum counterpart of three fundamental results of functional analysis and probability: the Entropy Power Inequality, the sharp Young’s inequality for convolutions, and the theorem “Gaussian kernels have only Gaussian maximizers.” Quantum Gaussian channels play a key role in quantum communication theory: they are the quantum counterpart of Gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime. The quantum Gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space. The restriction of the quantum-limited Gaussian attenuator to input states diagonal in the Fock basis coincides with the thinning, which is the analog of the rescaling for positive integer random variables. Quantum Gaussian channels provide then a bridge between functional analysis and discrete probability.

arXiv:1803.02360•Journal of Mathematical Physics•Aug 1, 2018•28 citations

Multimode Gaussian optimizers for the Wehrl entropy and quantum Gaussian channels

Giacomo De Palma, Dario Trevisan, Vittorio Giovannetti

We prove in the multimode scenario a fundamental relation between the Wehrl and the von Neumann entropy, stating that the minimum Wehrl entropy among all the quantum states with a given von Neumann entropy is achieved by thermal Gaussian states. We also prove that thermal Gaussian input states minimize the output von Neumann entropy of multimode quantum Gaussian attenuators, amplifiers and phase-contravariant channels among all the input states diagonal in some product basis and with a given entropy. This result constitutes a major step towards the proof of the same property for generic input states, which is still an open conjecture. This conjecture is necessary to determine the maximum communication rates for the triple trade-off coding and broadcast communication with the Gaussian quantum-limited attenuator. Finally, we prove that the tensor product of n identical geometric input probability distributions minimizes the output Shannon entropy of the n-mode thinning among all the input probability distributions with a given entropy.

arXiv:1705.00499•May 1, 2017•3 citations
We prove that states saturate the p->q norms of the one-mode quantum-limited attenuator and amplifier. The proof starts from the majorization result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), and is based on a new logarithmic Sobolev inequality. Our result extends to noncommutative probability the seminal theorem Gaussian kernels have only maximizers (Lieb, Invent. Math. 102, 179 (1990)), stating that operators saturate the p->q norms of integral kernels. Our result also implies that the p->q norms of the thinning are saturated by geometric probability distributions. Moreover, the multimode extension of our result would imply the multiplicativity of the p->q norms of quantum-limited channels.
We prove that Gaussian thermal input states minimize the output von Neumann entropy of the one-mode Gaussian quantum-limited attenuator for fixed input entropy. The Gaussian quantum-limited attenuator models the attenuation of an electromagnetic signal in the quantum regime. The Shannon entropy of an attenuated real-valued classical signal is a simple function of the entropy of the original signal. A striking consequence of energy quantization is that the output von Neumann entropy of the quantum-limited attenuator is no more a function of the input entropy alone. The proof starts from the majorization result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), and is based on a new isoperimetric inequality. Our result implies that geometric input probability distributions minimize the output Shannon entropy of the thinning for fixed input entropy. Moreover, our result opens the way to the multimode generalization that permits to determine both the triple tradeoff region of the Gaussian quantum-limited attenuator and the classical capacity region of the Gaussian degraded quantum broadcast channel.
We prove the long-standing conjecture stating that Gaussian thermal input states minimize the output von Neumann entropy of one-mode phase-covariant quantum Gaussian channels among all the input states with a given entropy. Phase-covariant quantum Gaussian channels model the attenuation and the noise that affect any electromagnetic signal in the quantum regime. Our result is crucial to prove the converse theorems for both the triple trade-off region and the capacity region for broadcast communication of the Gaussian quantum-limited amplifier. Our result extends to the quantum regime the entropy power inequality that plays a key role in classical information theory. Our proof exploits a completely new technique based on the recent determination of the p→q norms of the quantum-limited amplifier [De Palma et al., arXiv:1610.09967]. This technique can be applied to any quantum channel.
We develop a method for deriving Gaussian optimality results for bosonic quantum systems from inequalities for qubit systems. Inspired by Gross's derivation of the Gaussian logarithmic Sobolev inequality from an inequality on the two-point space, we use qubits as quantum two-point spaces and lift bosonic Weyl operators to pseudo-Weyl operators on large qubit systems. By controlling weighted norms and relative entropy in the fluctuation limit, we establish two transfer theorems that convert tensorized qubit inequalities into inequalities for bosonic Gaussian channels. As a first application, we prove that the sharp 222-logarithmic Sobolev inequality for the bosonic Ornstein--Uhlenbeck semigroup tensorizes without loss in the constant. We also establish a sharp output entropy inequality for tensor powers of a class of qubit channels. This includes a quantum Mrs.~Gerber's lemma determining the minimum output entropy of tensor powers of the qubit depolarizing channel at fixed input entropy. Applying our relative entropy transfer theorem, we obtain multimode constrained minimum output entropy bounds for Gaussian additive-noise channels and thermal attenuators in the heating range.
A long-standing problem in quantum Shannon theory is the classical capacity of bosonic Gaussian channels of various kinds [6].The Hypothesis of Gaussian Maximizers (HGM) states that the full capacity of such channels is attained at Gaussian encodings.A breakthrough was made in the papers [1] and [2], where HGM was proved for the important class of multimode gauge co-or contra-variant Gaussian channels [4].In [8] and [5] the solution was extended to a much broader class of channels satisfying a certain 'threshold condition'.At the same time the HGM remains open for a large variety of Gaussian channels lying beyond the scope of the threshold condition [7].In this note we sketch a novel approach to such problems, based on principles of convex programming, and illustrate it using the characteristic case of an approximate position measurement with energy constraint, underlying noisy Gaussian homodyning in quantum optics.Rather remarkably, for this particular model the method reduces the solution of the optimization problem to a generalization of the celebrated log-Sobolev inequality.Let S be the convex set of all density operators in a separable Hilbert space H of the quantum system and X and Y be standard measurable spaces.Consider a measurement channel M : ρ → p ρ (y) = Tr ρm(y), where ρ ∈ S and m(y) is a uniformly bounded positive operator-valued function of y ∈ Y such that m(y) µ(dy) = I (I is the identity operator on H and µ is a measure).The encoding E = {π(dx), ρ(x)} is a probability measure π(dx) with a measurable family of
We survey the state of the art for the proof of the quantum Gaussian optimizer conjectures of quantum information theory. These fundamental conjectures state that quantum Gaussian input states are the solution to several optimization problems involving quantum Gaussian channels. These problems are the quantum counterpart of three fundamental results of functional analysis and probability: the Entropy Power Inequality, the sharp Young’s inequality for convolutions, and the theorem “Gaussian kernels have only Gaussian maximizers.” Quantum Gaussian channels play a key role in quantum communication theory: they are the quantum counterpart of Gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime. The quantum Gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space. The restriction of the quantum-limited Gaussian attenuator to input states diagonal in the Fock basis coincides with the thinning, which is the analog of the rescaling for positive integer random variables. Quantum Gaussian channels provide then a bridge between functional analysis and discrete probability.
We prove in the multimode scenario a fundamental relation between the Wehrl and the von Neumann entropy, stating that the minimum Wehrl entropy among all the quantum states with a given von Neumann entropy is achieved by thermal Gaussian states. We also prove that thermal Gaussian input states minimize the output von Neumann entropy of multimode quantum Gaussian attenuators, amplifiers and phase-contravariant channels among all the input states diagonal in some product basis and with a given entropy. This result constitutes a major step towards the proof of the same property for generic input states, which is still an open conjecture. This conjecture is necessary to determine the maximum communication rates for the triple trade-off coding and broadcast communication with the Gaussian quantum-limited attenuator. Finally, we prove that the tensor product of n identical geometric input probability distributions minimizes the output Shannon entropy of the n-mode thinning among all the input probability distributions with a given entropy.