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.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:2610.28902Oct 22, 20260 citations

Logarithmic Sobolev Inequalities for Finite Dimensional Quantum Markov Chains

Cambyse Rouzé

Bolyai Society mathematical studiesJan 1, 20242 citations

Bounds on the capacity of Wiener phase noise channels

Luca Barletta

Jan 1, 20150 citations

Improved quantum hypercontractivity inequality for the qubit depolarizing channel

Salman Beigi

The hypercontractivity inequality for the qubit depolarizing channel Ψt states that ‖Ψt⊗n(X)‖p≤‖X‖q, provided that p ≥ q > 1 and t≥lnp−1q−1. In this paper, we present an improvement of this inequality. We first prove an improved quantum logarithmic-Sobolev inequality and then use the well-known equivalence of logarithmic-Sobolev inequalities and hypercontractivity inequalities to obtain our main result. As applications of these results, we present an asymptotically tight quantum Faber–Krahn inequality on the hypercube and a new quantum Schwartz–Zippel lemma.The hypercontractivity inequality for the qubit depolarizing channel Ψt states that ‖Ψt⊗n(X)‖p≤‖X‖q, provided that p ≥ q > 1 and t≥lnp−1q−1. In this paper, we present an improvement of this inequality. We first prove an improved quantum logarithmic-Sobolev inequality and then use the well-known equivalence of logarithmic-Sobolev inequalities and hypercontractivity inequalities to obtain our main result. As applications of these results, we present an asymptotically tight quantum Faber–Krahn inequality on the hypercube and a new quantum Schwartz–Zippel lemma.

arXiv:2105.00462Journal of Mathematical PhysicsDec 1, 20214 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 ofA 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 SurveysJan 1, 20222 citations

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-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.