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arXiv:1912.06129 · Communications in Mathematical Physics

Convergence Rates for the Quantum Central Limit Theorem

Simon Becker, Nilanjana Datta, Ludovico Lami, Cambyse Rouzé

Published Feb 15, 2021• 20 citations
arXivHTMLPDF

Abstract

Abstract Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state $$\rho $$ ρ with finite second moments, converges to the Gaussian state with the same first and second moments as $$\rho $$ ρ . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate $$\mathcal {O}\left( n^{-1/2}\right) $$ O n - 1 / 2 in the Hilbert–Schmidt norm whenever the third moments of $$\rho $$ ρ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities $$\lambda ^{1/n}$$ λ 1 / n fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate $$\mathcal {O}\Big (n^{-\frac{1}{2(m+1)}}\Big )$$ O ( n - 1 2 ( m + 1 ) ) . This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function $$\chi _\rho $$ χ ρ is uniformly bounded by some $$\eta _\rho <1$$ η ρ < 1 outside of any neighbourhood of the origin; also, $$\eta _\rho $$ η ρ can be made to depend only on the energy of the state $$\rho $$ ρ .

DOI
10.1007/s00220-021-03988-1

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Abstract Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state ρ\rho ρ ρ with finite second moments, converges to the Gaussian state with the same first and second moments as ρ\rho ρ ρ . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate O(n−1/2)\mathcal {O}\left( n^{-1/2}\right) O(n−1/2) O n - 1 / 2 in the Hilbert–Schmidt norm whenever the third moments of ρ\rho ρ ρ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities λ1/n\lambda ^{1/n}λ1/n λ 1 / n fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate O(n−12(m+1))\mathcal {O}\Big (n^{-\frac{1}{2(m+1)}}\Big )O(n−2(m+1)1​) O ( n - 1 2 ( m + 1 ) ) . This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function χρ\chi _\rho χρ​ χ ρ is uniformly bounded by some ηρ<1\eta _\rho <1ηρ​<1 η ρ < 1 outside of any neighbourhood of the origin; also, ηρ\eta _\rho ηρ​ η ρ can be made to depend only on the energy of the state ρ\rho ρ ρ .
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.
We prove that quantum many-body systems on a one-dimensional lattice locally relax to Gaussian states under non-equilibrium dynamics generated by a bosonic quadratic Hamiltonian. This is true for a large class of initial states—pure or mixed—which have to satisfy merely weak conditions concerning the decay of correlations. The considered setting is a proven instance of a situation where dynamically evolving closed quantum systems locally appear as if they had truly relaxed, to maximum entropy states for fixed second moments. This furthers the understanding of relaxation in suddenly quenched quantum many-body systems. The proof features a non-commutative central limit theorem for non-i.i.d. random variables, showing convergence to Gaussian characteristic functions, giving rise to trace-norm closeness. We briefly link our findings to the ideas of typicality and concentration of measure.
We introduce a quantum convolution and a conceptual framework to study states in discrete-variable (DV) quantum systems. All our results suggest that stabilizer states play a role in DV quantum systems similar to the role that Gaussian states play in continuous-variable systems. Hence we suggest the name ``discrete quantum Gaussians'' for stabilizer states. For example, we prove that the convolution of two stabilizer states is another stabilizer state, and that stabilizer states extremize both quantum entropy and Fisher information. We establish a ``maximal entropy principle,'' a ``second law of thermodynamics for quantum convolution,'' and a ``quantum central limit theorem.'' The latter is based on iterating the convolution of a zero-mean quantum state, which we prove converges to a stabilizer state. We bound the exponential rate of convergence of the quantum central limit theorem by the ``magic gap,'' defined in terms of the support of the characteristic function of the state. We elaborate on our general results with a discussion of some examples, as well as extending many of them to quantum channels.
The quantum central limit theorem derived by Cushen and Hudson provides the foundations for understanding how subsystems of large bosonic systems evolving unitarily do reach equilibrium. It finds important applications in the context of quantum interferometry, for example, with photons. A practical feature of current photonic experiments, however, is that photons carry their own internal degrees of freedom pertaining to, e.g., the polarization or spatiotemporal mode they occupy, which makes them partially distinguishable. The ensuing deviation from ideal indistinguishability is well known to have observable consequences, for example in relation with boson bunching, but an understanding of its role in bosonic equilibration phenomena is still missing. Here, we generalize the Cushen-Hudson quantum central limit theorem to encompass scenarios with partial distinguishability, implying an asymptotic convergence of the subsystem's reduced state towards a multimode Gaussian state defined over the internal degrees of freedom. While these asymptotic internal states may not be directly accessible, we show that particle number distributions carry important signatures of distinguishability, which may be used to diagnose experimental imperfections in large boson sampling experiments.
We prove the Quantum Voronovskaya--Damasclin Theorem, establishing the first complete asymptotic expansion for Quantum Neural Network Operators (QNNOs) when approximating arbitrary quantum channels. This result generalizes the classical Voronovskaya theorem for Bernstein polynomials to the non‑commutative, multi‑dimensional setting of quantum information theory. Within a rigorous functional analytic framework, we introduce quantum analogues of Sobolev and Hölder spaces Cm,γ(H)\mathcal{C}^{m,\gamma}(\mathcal{H})Cm,γ(H) based on Fréchet differentiability in the Liouville representation, and we measure approximation errors using the diamond (completely bounded trace) norm. Our expansion, Ψn(Φ)(ρ)=Φ(ρ)+∑j=1maj(Φ,ρ)nj+∑j=1⌊m/2⌋bj(Φ,ρ)nj+γ+∑j=1⌊m/3⌋cj(Φ,ρ)nj+2γ+O ⁣(n−(m+γ)(log⁡n)3m/2), \Psi_n(\Phi)(\rho) = \Phi(\rho) + \sum_{j=1}^{m} \frac{a_j(\Phi,\rho)}{n^j} + \sum_{j=1}^{\lfloor m/2\rfloor} \frac{b_j(\Phi,\rho)}{n^{j+\gamma}} + \sum_{j=1}^{\lfloor m/3\rfloor} \frac{c_j(\Phi,\rho)}{n^{j+2\gamma}} + O\!\left(n^{-(m+\gamma)}(\log n)^{3m/2}\right), Ψn​(Φ)(ρ)=Φ(ρ)+∑j=1m​njaj​(Φ,ρ)​+∑j=1⌊m/2⌋​nj+γbj​(Φ,ρ)​+∑j=1⌊m/3⌋​nj+2γcj​(Φ,ρ)​+O(n−(m+γ)(logn)3m/2),explicitly separates three distinct contributions: integer‑order polynomial terms involving Fréchet derivatives and even kernel moments; fractional corrections governed by Marchaud fractional derivatives that capture Hölder regularity of order γ\gammaγ; and intrinsically non‑commutative commutator terms that vanish in classical settings. The remainder estimate is sharp, with an explicit constant depending on mmm, γ\gammaγ, and the Hilbert space dimension ddd. As applications, we derive a quantum central limit theorem for QNNOs, construct optimal interpolation geodesics between channels using Kubo--Ando means, and develop a quantum Richardson extrapolation method that reveals fundamental acceleration limits imposed by fractional smoothness. These results establish a rigorous bridge between classical approximation theory, fractional calculus, and quantum machine learning, providing a powerful tool for the design and analysis of quantum neural networks.
Abstract When can a quantum optical source be replaced by a coherent state with controlled error in every subsequent experiment? We turn its Husimi phase-space entropy excess into an error bound for the entire quantum state. For every fixed Rényi order p>0p>0p>0 and every density operator on any finite number of modes, D(ρ)2≤Cpδp(ρ),D(ρ)=inf⁡α∥ρ−∣α⟩⟨α∣∥1,D(\rho)^2\le C_p\delta_p(\rho),\qquad D(\rho)=\inf_\alpha\|\rho-|\alpha\rangle\langle\alpha|\|_1,D(ρ)2≤Cp​δp​(ρ),D(ρ)=infα​∥ρ−∣α⟩⟨α∣∥1​, where δp\delta_pδp​ is the total Rényi–Wehrl deficit and CpC_pCp​ is independent of mode count, energy, rank and photon-number support. One coherent replacement then bounds every measurement-probability error after any quantum channel. We identify the source structure that determines the finer error near the coherent manifold: after optimising displacement, D≍a+bD\asymp a+bD≍a+b, where aaa is the population outside the reference vacuum and bbb its coherence with the remaining state. This distinguishes linear thermal response from square-root squeezing response and yields intermediate rates. The guarantee holds throughout the positive-order range, but its optimal coefficient necessarily diverges at small order: Cpopt∼4/[plog⁡(1/p)]C_p^{\rm opt}\sim4/[p\log(1/p)]Cpopt​∼4/[plog(1/p)] as p↓0p\downarrow0p↓0. A reduction of fractional Husimi moments to one active mode proves this sharp limit. The results specify both when entropy certifies a multimode coherent approximation and how its accuracy depends on the structure of the source. Version 1.1 — 9 October 2026 This revision extends the original mode-uniform Wehrl theorem to every positive Rényi order and develops its use as a coherent-state approximation guarantee. All positive orders and arbitrary density operators. The full trace-norm bound is uniform over every finite mode number, without a finite-energy assumption. Explicit sufficient coefficients include C1=64/log⁡2C_1=64/\log 2C1​=64/log2 and C2=16C_2=16C2​=16. Source-dependent accuracy. A local population–coherence law separates thermal, squeezed and intermediate responses. The same coherent reference controls subsequent quantum channels and measurement probabilities. Sharp small-order limit. The optimal global coefficient satisfies plog⁡(1/p)Cpopt→4p\log(1/p)C_p^{\rm opt}\to4plog(1/p)Cpopt​→4. The complete proof uses a one-active-mode reduction of fractional Husimi moments. Expanded presentation. Two vector figures explain source-dependent error, the role of total entropy deficit and the order-dependent bounds. The 19-page PDF includes the complete manuscript, proofs and updated references. This is an expanded revision of Uniform Wehrl stability across all bosonic mode numbers: total entropy deficit controls full trace-norm distance to coherent states, first published on 6 October 2026.