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arXiv:2610.28902

From Qubit Tensorization to Gaussian Optimizers in Bosonic Systems

Salman Beigi, Hami Mehrabi

quant-phSubmitted Oct 20, 2026• Announced Oct 22, 2026• 0 citations
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Abstract

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.

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66 pages

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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.
The main topic of this thesis is the proof of two fundamental entropic inequalities for quantum Gaussian channels. Quantum Gaussian channels model the propagation of electromagnetic waves through optical fibers and free space in the quantum regime, and provide the mathematical model to determine the maximum rates achievable by quantum communication devices for communication and quantum key distribution. The first inequality proven in this thesis is the quantum Entropy Power Inequality, which provides a lower bound to the output entropy of the beam-splitter or of the squeezing in terms of the entropies of the two inputs. The second inequality states that Gaussian states minimize the output entropy of the one-mode quantum Gaussian attenuator among all the input states with a given entropy, and was a longstanding conjecture in quantum communication theory. The generalization of this inequality to the multimode attenuator would determine its triple trade-off region and its capacity region for broadcast communication to two receivers. The thesis contains further results in Gaussian quantum information, quantum statistical mechanics and relativistic quantum information. The most important of these results concerns the Eigenstate Thermalization Hypothesis (ETH). The ETH is an assumption in quantum statistical mechanics stating that the eigenstates of the Hamiltonian of a system+bath compound look as thermal states if we can access only the system. We prove that the ETH must hold if the system thermalizes for any initial product state of the system+bath compound with a well-defined temperature.
We prove the multimode conditional quantum Entropy Power Inequality for bosonic quantum systems. This inequality determines the minimum conditional von Neumann entropy of the output of the most general linear mixing of bosonic quantum modes among all the input states of the modes with given conditional entropies. Bosonic quantum systems constitute the mathematical model for the electromagnetic radiation in the quantum regime, which provides the most promising platform for quantum communication and quantum key distribution. We apply our multimode conditional quantum Entropy Power Inequality to determine new lower bounds to the squashed entanglement of a large family of bosonic quantum Gaussian states. The squashed entanglement is one of the main entanglement measures in quantum communication theory, providing the best known upper bound to the distillable key. Exploiting this result, we determine a new lower bound to the squashed entanglement of the multimode bosonic Gaussian channels that are extreme,i.e., that cannot be decomposed as a non-trivial convex combination of quantum channels. The squashed entanglement of a quantum channel provides an upper bound to its secret-key capacity,i.e., the capacity to generate a secret key shared between the sender and the receiver. Lower bounds to the squashed entanglement are notoriously hard to prove. Our results contribute to break this barrier and will stimulate further research in the field of quantum communication with bosonic quantum systems.
This dissertation deals with entropic inequalities for bosonic channels in quantum information theory. Such inequalities have applications in foundational questions of quantum communication and can be used to make statements about the convergence of quantum dynamical semigroups. We prove, among other results, an inequality for the output entropy of a quantum channel which models additive classical noise and apply this inequality to derive bounds on the classical capacity of a general family of non-Gaussian quantum channels.
Abstract We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of the relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or L2L_2L2​ L 2 -estimates. As an application, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive tight bounds for local generators.
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.
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