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

arXiv:2510.04077

The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology

Sviatoslav V. Dzhenzher

Published Oct 5, 2025• 1 citation
arXivHTMLPDF

Abstract

We consider random linear continuous operators $Ω\to \mathcal{L}(\mathcal{H}, \mathcal{H})$ on a Hilbert space $\mathcal{H}$. For example, such random operators may be random quantum channels. The Central Limit Theorem is known for the sums of i.i.d. random operators. Instead of the sum, there may be considered the composition of random exponents $e^{A_i/n}$. We obtain the Central Limit Theorem in the Weak Operator Topology for centralized and normalized random exponents of i.i.d. linear continuous operators on a Hilbert space.

DOI
10.48550/arxiv.2510.04077

Related papers

The Law of Large Numbers for Discrete Generalized Quantum Channels

Sviatoslav V. Dzhenzher, Vsevolod Zhanovich Sakbaev

We consider random linear operators $$\Omega\to\mathcal{L}(\mathcal{T}_{p},\mathcal{T}_{p})$$ acting in a $$p$$ th Schatten class $$\mathcal{T}_{p}$$ in a separable Hilbert space $$\mathcal{H}$$ for some $$1\leqslant p<\infty$$ . Such a superoperator is called pre-channel since it is an extension of a quantum channel to a wider class of operators without requirements of trace-preserving and positivity. Instead of sum of i.i.d. variables there may be considered the composition of random semigroups $$e^{A_{i}t/n}$$ in the Banach space $$\mathcal{T}_{p}$$ . The law of large numbers is known in the case $$p=2$$ in the form of the usual law of large numbers for random operators in a Hilbert space. We obtain the law of large numbers for the case $$1\leqslant p\leqslant 2$$ .

arXiv:2504.10033•Lobachevskii Journal of Mathematics•Jun 1, 2025•1 citation

Random Quantization of Hamiltonian Systems

John Edward Gough, Yu. N. Orlov, Vsevolod Zhanovich Sakbaev +1 more

Abstract A quantization of a Hamiltonian system is an ambiguous procedure. Accordingly, we introduce the notion of random quantization, related random variables with values in the set of self-adjoint operators, and random processes with values in the group of unitary operators. The procedures for the averaging of random unitary groups and averaging of random self-adjoint operators are defined. The generalized weak convergence of a sequence of measures and the corresponding generalized convergence in distribution of a sequence of random variables are introduced. The generalized convergence in distribution for some sequences of compositions of random mappings is obtained. In the case of a sequence of compositions of shifts by independent random vectors of Euclidean space, the obtained convergence coincides with the statement of the central limit theorem for a sum of independent random vectors. The results are applied to the dynamics of quantum systems arising in random quantization of a Hamiltonian system.

Doklady Mathematics•May 1, 2021•19 citations

Central Limit Theorems for Open Quantum Random Walks ∗

Stéphane Attal, Nadine Guillotin‐Plantard, Christophe Sabot +2 more

Open Quantum Random Walks are the exact quantum generalization of Markov chains on finite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on nets, their distribution seems to always converges to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on Z^d we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z^d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.

•Jan 1, 2012•0 citations

Banach and counting measures, and dynamics of singular quantum states generated by averaging of operator random walks

E. A. Dzhenzher, S. V. Dzhenzher, V. Zh. Sakbaev

In this paper the random channels and their compositions in the space of quantum states are studied. For compositions of i.i.d. random unitary channels, the limit behaviour of probability distributions is described. The sufficient condition for convergence in probability is obtained. The generalized convergence in distribution w.r.t. weak operator topology is obtained. The analysis of transmission of pure and normal states to the set of singular states is done. The dynamics of quantum states is described in terms of the evolution of the values of quadratic forms of operators from the algebra that implements the representation of canonical commutation relations.

quant-pharXiv:2603.25151•Lobachevskii Journal of Mathematics•Mar 27, 2026•0 citations

Limit theorems for the evolution of quantum pure states

Sviatoslav V. Dzhenzher, V. Zh. Sakbaev

Limit theorems of strong law of large numbers and central limit theorem types are obtained for the compositions of independent identically distributed random unitary channels.

arXiv:2601.01845•Jan 5, 2026•0 citations

Central Limit Theorem for Bosonic Quantum Channels

Hami Mehrabi, Ludovico Lami, Mark M. Wilde

In this paper, we develop an extension of the Central Limit Theorem (CLT) to the setting of bosonic quantum channels. This extension provides a deeper understanding of Gaussian bosonic channels as extremal objects. Using our CLT for bosonic quantum channels, we recover both the classical CLT and the CLT for bosonic quantum states, thereby offering a unified perspective that connects classical probability theory with continuous-variable quantum systems. Moreover, using our result, we can provide necessary uncertainty relations that every physical (possibly non-Gaussian) bosonic quantum channel must satisfy. As another application of our limit theorems, we derive tight lower bounds on the energy-constrained quantum capacity of linear bosonic channels by relating it to the capacity of their associated Gaussian bosonic channels, further reinforcing the role of Gaussian channels as extremal.

quant-pharXiv:2605.16782•May 19, 2026•0 citations
We consider random linear continuous operators Ω→L(H,H)Ω\to \mathcal{L}(\mathcal{H}, \mathcal{H})Ω→L(H,H) on a Hilbert space H\mathcal{H}H. For example, such random operators may be random quantum channels. The Central Limit Theorem is known for the sums of i.i.d. random operators. Instead of the sum, there may be considered the composition of random exponents eAi/ne^{A_i/n}eAi​/n. We obtain the Central Limit Theorem in the Weak Operator Topology for centralized and normalized random exponents of i.i.d. linear continuous operators on a Hilbert space.
We consider random linear operators Ω→L(Tp,Tp)\Omega\to\mathcal{L}(\mathcal{T}_{p},\mathcal{T}_{p})Ω→L(Tp​,Tp​) acting in a ppp th Schatten class Tp\mathcal{T}_{p}Tp​ in a separable Hilbert space H\mathcal{H}H for some 1⩽p<∞1\leqslant p<\infty1⩽p<∞ . Such a superoperator is called pre-channel since it is an extension of a quantum channel to a wider class of operators without requirements of trace-preserving and positivity. Instead of sum of i.i.d. variables there may be considered the composition of random semigroups eAit/ne^{A_{i}t/n}eAi​t/n in the Banach space Tp\mathcal{T}_{p}Tp​ . The law of large numbers is known in the case p=2p=2p=2 in the form of the usual law of large numbers for random operators in a Hilbert space. We obtain the law of large numbers for the case 1⩽p⩽21\leqslant p\leqslant 21⩽p⩽2 .
Abstract A quantization of a Hamiltonian system is an ambiguous procedure. Accordingly, we introduce the notion of random quantization, related random variables with values in the set of self-adjoint operators, and random processes with values in the group of unitary operators. The procedures for the averaging of random unitary groups and averaging of random self-adjoint operators are defined. The generalized weak convergence of a sequence of measures and the corresponding generalized convergence in distribution of a sequence of random variables are introduced. The generalized convergence in distribution for some sequences of compositions of random mappings is obtained. In the case of a sequence of compositions of shifts by independent random vectors of Euclidean space, the obtained convergence coincides with the statement of the central limit theorem for a sum of independent random vectors. The results are applied to the dynamics of quantum systems arising in random quantization of a Hamiltonian system.
Open Quantum Random Walks are the exact quantum generalization of Markov chains on finite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on nets, their distribution seems to always converges to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on Z^d we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z^d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.
In this paper the random channels and their compositions in the space of quantum states are studied. For compositions of i.i.d. random unitary channels, the limit behaviour of probability distributions is described. The sufficient condition for convergence in probability is obtained. The generalized convergence in distribution w.r.t. weak operator topology is obtained. The analysis of transmission of pure and normal states to the set of singular states is done. The dynamics of quantum states is described in terms of the evolution of the values of quadratic forms of operators from the algebra that implements the representation of canonical commutation relations.
In this paper, we develop an extension of the Central Limit Theorem (CLT) to the setting of bosonic quantum channels. This extension provides a deeper understanding of Gaussian bosonic channels as extremal objects. Using our CLT for bosonic quantum channels, we recover both the classical CLT and the CLT for bosonic quantum states, thereby offering a unified perspective that connects classical probability theory with continuous-variable quantum systems. Moreover, using our result, we can provide necessary uncertainty relations that every physical (possibly non-Gaussian) bosonic quantum channel must satisfy. As another application of our limit theorems, we derive tight lower bounds on the energy-constrained quantum capacity of linear bosonic channels by relating it to the capacity of their associated Gaussian bosonic channels, further reinforcing the role of Gaussian channels as extremal.