Optimal convergence rates in trace distance and relative entropy for the quantum central limit theorem

Beigi, Salman; Goodarzi, Milad M.; Mehrabi, Hami

Optimal convergence rates in trace distance and relative entropy for the quantum central limit theorem

Oct 29, 2024

40 pages

e-Print:

2410.21998 [quant-ph]

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Abstract: (arXiv)

A quantum analogue of the Central Limit Theorem (CLT), first introduced by Cushen and Hudson (1971), states that the

n

-fold convolution

\rho^{\boxplus n}

of an

m

-mode quantum state

\rho

with zero first moments and finite second moments converges weakly, as

n

increases, to a Gaussian state

\rho_G

with the same first and second moments. 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}

and

\rho_G

decays at the optimal rate of

\mathcal{O}(n^{-1/2})

, consistent with known convergence rates. Furthermore, for states with finite fourth-order moments (plus a small correction

\delta

for

m>1

), we prove that the relative entropy between

\rho^{\boxplus n}

and

\rho_G

decays at the optimal rate of

\mathcal{O}(n^{-1})

. Both of these rates are proven to be optimal, even when assuming the finiteness of all moments of

\rho

. These results relax previous assumptions on higher-order moments, yielding convergence rates that match the best known results in the classical setting. 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. As a by-product of this, an upper bound on the relative entropy of non-Gaussianity is derived, which is of independent interest.

Note:

40 pages

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[1]

On an inequality of Lieb and Thirring

H. Araki

- Lett.Math.Phys. 19 (1990) 167-170
•
DOI:
- 10.1007/BF01045887

[2]

A central limit theorem in many-body quantum dynamics

G.B. Arous
,
K. Kirkpatrick
,
B. Schlein

- Commun.Math.Phys. 321 (2013) 371-417

[3]

Convergence rates for the quantum central limit theorem

S. Becker
,
N. Datta
,
L. Lami
,
C. Rouz

- Commun.Math.Phys. 383 (2021) 223-279
•
DOI:
- 10.1007/s00220-021-03988-1

[4]

Towards Optimal Convergence Rates for the Quantum Central Limit Theorem

e-Print:
- 2310.09812

[5]

The accuracy of the Gaussian approximation to the sum of independent variates

A.C. Berry

- Trans.Am.Math.Soc. 49 (1941) 122-136
•
DOI:
- 10.1090/S0002-9947-1941-0003498-3

[6]

Matrix analysis. Number 169 in Graduate texts in mathematics

R. Bhatia

•
DOI:
- 10.1007/978-1-4612-0653-8

[7]

Normal approximation and asymptotic expansions

R.N. Bhattacharya
,
R.R. Rao

•
DOI:
- 10.1137/1.9780898719895

[8]

Optimal trace-distance bounds for free-fermionic states: Testing and improved tomography

e-Print:
- 2409.17953

[9]

Rate of convergence and edgeworth-type expansion in the entropic central limit theorem. The Annals of Probability, pages 2479-2512

S.G. Bobkov
,
G.P. Chistyakov
,
F. Götze

•
DOI:
- 10.1214/12-AOP780

[10]

Berry-esseen bounds in the entropic central limit theorem

S.G. Bobkov
,
G.P. Chistyakov
,
F. Götze

- Probab.Theor.Related Fields 159 (2014) 435-478
•
DOI:
- 10.1007/s00440-013-0510-3

[11]

Reversible Framework for Quantum Resource Theories

- Phys.Rev.Lett. 115 (2015) 7, 070503
•
DOI:
- 10.1103/PhysRevLett.115.070503

[12]

Continuous-variable entanglement distillation and noncommutative central limit theorems. Physical Review A--Atomic

E.T. Campbell
,
M.G. Genoni
,
J. Eisert

- Physics 87 (2013) 042330

[13]

A quantum central limit theorem for non-equilibrium systems: exact local relaxation of correlated states

M. Cramer
,
J. Eisert

- New J.Phys. 12 (2010) 055020

[14]

A quantum-mechanical central limit theorem

C.D. Cushen
,
R.L. Hudson

- J.Appl.Probab. 8 (1971) 454-469
•
DOI:
- 10.2307/3212170

[15]

On the central limit theorem for unsharp quantum random variables

A. Dimi
,
B. Daki

- New J.Phys. 20 (2018) 063051

[16]

Probability: Theory and Examples. Cambridge University Press, Cambridge, 5th edition

R. Durrett

•
DOI:
- 10.1017/9781108591034

[17]

An introduction to probability theory and its applications, volume 2, volume 81. John

W. Feller

[18]

Quantifying the non-gaussian character of a quantum state by quantum relative entropy. Physical Review A--Atomic

M.G. Genoni
,
M.G. Paris
,
K. Banaszek

- Physics 78 (2008) 060303
•
DOI:
- 10.1103/PhysRevA.78.060303

[19]

An algebraic version of the central limit theorem. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 42(2):129-134

N. Giri
,
W. von Waldenfels

[20]

Non-commutative central limits

D. Goderis
,
A. Verbeure
,
P. Vets

- Probab.Theor.Related Fields 82 (1989) 527-544

[21]

Central limit theorem for mixing quantum systems and the ccr-algebra of fluctuations

D. Goderis
,
P. Vets

- Commun.Math.Phys. 122 (1989) 249-265

[22]

Quantum information

M. Hayashi

[23]

Quantum estimation and the quantum central limit theorem. American Mathematical Society Translations Series, 2(277):95

M. Hayashi

[24]

On the superradiant phase transition for molecules in a quantized radiation field: the dicke maser model

- Annals Phys. 76 (1973) 2, 360-404
•
DOI:
- 10.1016/0003-4916(73)90039-0

[25]

Phase transitions in reservoir-driven open systems with applications to lasers and superconductors. In Condensed Matter Physics and Exactly Soluble Models: Selecta of Elliott H

K. Hepp
,
E.H. Lieb

•
DOI:
- 10.1007/978-3-662-06390-3_13

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