A quantum analogue of the Central Limit Theorem (CLT) for bosonic system, first introduced by Cushen and Hudson (1971), states that the n-fold convolution <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\rho ^{\boxplus n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ρ</mi> <mrow> <mo>⊞</mo> <mi>n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> of an m-mode quantum state <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>—with zero first moments and finite second moments—converges weakly, as n increases, to a Gaussian state <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho _G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> with the same first and second moments as those of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>, called its Gaussification. Recently, this result has been extended with estimates of the convergence rate in various distance measures. In this paper, we establish optimal rates of convergence in both the trace distance and quantum relative entropy. Specifically, we show that for a centered m-mode quantum state with finite third-order moments, the trace distance between <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\rho ^{\boxplus n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ρ</mi> <mrow> <mo>⊞</mo> <mi>n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\rho _G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> decays at the optimal rate of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {O}(n^{-1/2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, for states with finite fourth-order moments (order <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$4+\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo>+</mo> <mi>δ</mi> </mrow> </math></EquationSource> </InlineEquation> for an arbitrary small <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\delta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$m>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>), we prove that the relative entropy between <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\rho ^{\boxplus n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ρ</mi> <mrow> <mo>⊞</mo> <mi>n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\rho _G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> decays at the optimal rate of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {O}(n^{-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Both of these rates are proven to be optimal, even when assuming the finiteness of all moments of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>. These results relax previous assumptions on higher-order moments, yielding convergence rates that match the best known results in the classical setting. By giving explicit examples we also show that our moment assumptions are essentially minimal. We show that for any <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\theta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, there exists a quantum state <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> with finite moments of order less than <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$3-\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>-</mo> <mi>θ</mi> </mrow> </math></EquationSource> </InlineEquation>, such that the convergence rate of <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\rho ^{\boxplus n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ρ</mi> <mrow> <mo>⊞</mo> <mi>n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\rho _G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> in trace distance is not <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\mathcal {O}(n^{-1/2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Similarly, we show that for any <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\theta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, there exists a quantum state <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> with finite moments of order less than <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$4-\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo>-</mo> <mi>θ</mi> </mrow> </math></EquationSource> </InlineEquation>, such that the relative entropy between <InlineEquation ID="IEq24"> <EquationSource Format="TEX">$\rho ^{\boxplus n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ρ</mi> <mrow> <mo>⊞</mo> <mi>n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq25"> <EquationSource Format="TEX">$\rho _G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> does not decay at the rate <InlineEquation ID="IEq26"> <EquationSource Format="TEX">$\mathcal {O}(n^{-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Our proofs draw on techniques from the classical literature, including Edgeworth-type expansions of quantum characteristic functions, adapted to the quantum context. A key technical step in the proof of our entropic CLT is establishing an upper bound on the relative entropy distance between a general quantum state and its Gaussification, which is of independent interest.

Optimal Convergence Rates in Trace Distance and Relative Entropy for the Quantum Central Limit Theorem

期刊论文
发表日期：

2026年7月9日

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Salman Beigi,
Milad M. Goodarzi,
Hami Mehrabi

摘要

A quantum analogue of the Central Limit Theorem (CLT) for bosonic system, first introduced by Cushen and Hudson (1971), states that the n-fold convolution $\rho ^{\boxplus n}$ $ρ^{⊞ n}$ of an m-mode quantum state $\rho $ $ρ$ —with zero first moments and finite second moments—converges weakly, as n increases, to a Gaussian state $\rho _G$ $ρ_{G}$ with the same first and second moments as those of $\rho $ $ρ$ , called its Gaussification. Recently, this result has been extended with estimates of the convergence rate in various distance measures. In this paper, we establish optimal rates of convergence in both the trace distance and quantum relative entropy. Specifically, we show that for a centered m-mode quantum state with finite third-order moments, the trace distance between $\rho ^{\boxplus n}$ $ρ^{⊞ n}$ and $\rho _G$ $ρ_{G}$ decays at the optimal rate of $\mathcal {O}(n^{-1/2})$ $O (n^{- 1 / 2})$ . Furthermore, for states with finite fourth-order moments (order $4+\delta $ $4 + δ$ for an arbitrary small $\delta >0$ $δ > 0$ if $m>1$ $m > 1$ ), we prove that the relative entropy between $\rho ^{\boxplus n}$ $ρ^{⊞ n}$ and $\rho _G$ $ρ_{G}$ decays at the optimal rate of $\mathcal {O}(n^{-1})$ $O (n^{- 1})$ . Both of these rates are proven to be optimal, even when assuming the finiteness of all moments of $\rho $ $ρ$ . These results relax previous assumptions on higher-order moments, yielding convergence rates that match the best known results in the classical setting. By giving explicit examples we also show that our moment assumptions are essentially minimal. We show that for any $\theta >0$ $θ > 0$ , there exists a quantum state $\rho $ $ρ$ with finite moments of order less than $3-\theta $ $3 - θ$ , such that the convergence rate of $\rho ^{\boxplus n}$ $ρ^{⊞ n}$ to $\rho _G$ $ρ_{G}$ in trace distance is not $\mathcal {O}(n^{-1/2})$ $O (n^{- 1 / 2})$ . Similarly, we show that for any $\theta >0$ $θ > 0$ , there exists a quantum state $\rho $ $ρ$ with finite moments of order less than $4-\theta $ $4 - θ$ , such that the relative entropy between $\rho ^{\boxplus n}$ $ρ^{⊞ n}$ to $\rho _G$ $ρ_{G}$ does not decay at the rate $\mathcal {O}(n^{-1})$ $O (n^{- 1})$ . Our proofs draw on techniques from the classical literature, including Edgeworth-type expansions of quantum characteristic functions, adapted to the quantum context. A key technical step in the proof of our entropic CLT is establishing an upper bound on the relative entropy distance between a general quantum state and its Gaussification, which is of independent interest.