Square-root remainder in generalized quantum equipartition

Problem

Let \(H\) be a \(d\)-dimensional Hilbert space. For each \(n\), let \(\mathcal A_n\subseteq\mathcal D(H^{\otimes n})\) and \(\mathcal B_n\subseteq\mathcal L(H^{\otimes n})_+\) be nonempty compact, convex, permutation-invariant sets, each closed under tensor products. For a positive-operator set \(\mathcal C\), define its positive polar by

\begin{equation} \mathcal C_+^\circ :=\{X\geq0:\operatorname{Tr}(XY)\leq1 \text{ for every }Y\in\mathcal C\}. \tag{1} \end{equation}

Assume that both families of polars from Eq. (1) satisfy

\begin{equation} (\mathcal A_m)_+^\circ\otimes(\mathcal A_n)_+^\circ \subseteq(\mathcal A_{m+n})_+^\circ, \qquad (\mathcal B_m)_+^\circ\otimes(\mathcal B_n)_+^\circ \subseteq(\mathcal B_{m+n})_+^\circ. \tag{2} \end{equation}

In addition to Eq. (2), assume that some constant \(C<\infty\) satisfies

\begin{equation} D_{\max}(\rho_n\|\sigma_n)\leq Cn, \qquad \log_2\operatorname{Tr}\sigma_n\leq Cn \tag{3} \end{equation}

for all \(\rho_n\in\mathcal A_n\) and \(\sigma_n\in\mathcal B_n\). The linear growth condition in Eq. (3) uses \(D_{\max}(\rho\|\sigma):=\inf\{\lambda:\rho\leq2^\lambda\sigma\}\). For positive operators \(\rho\) and \(\sigma\), define

\begin{equation} \begin{aligned} D(\rho\|\sigma) &:=\operatorname{Tr}[\rho(\log_2\rho-\log_2\sigma)],\\ D_H^\varepsilon(\rho\|\sigma) &:=-\log_2\inf_{\substack{0\leq Q\leq I\\ \operatorname{Tr}(Q\rho)\geq1-\varepsilon}} \operatorname{Tr}(Q\sigma), \end{aligned} \tag{4} \end{equation}

where \(D(\rho\|\sigma)=+\infty\) unless \(\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma\). Define the divergences between the two sets from Eq. (4) by

\begin{equation} \begin{aligned} D(\mathcal A_n\|\mathcal B_n) &:=\inf_{\rho_n\in\mathcal A_n,\,\sigma_n\in\mathcal B_n} D(\rho_n\|\sigma_n),\\ D_H^\varepsilon(\mathcal A_n\|\mathcal B_n) &:=\inf_{\rho_n\in\mathcal A_n,\,\sigma_n\in\mathcal B_n} D_H^\varepsilon(\rho_n\|\sigma_n), \end{aligned} \tag{5} \end{equation}

The two quantities in Eq. (5) determine the first- and finite-blocklength orders of interest. Set

\begin{equation} D^\infty(\mathcal A\|\mathcal B) :=\lim_{n\to\infty}\frac1nD(\mathcal A_n\|\mathcal B_n). \tag{6} \end{equation}

Does every fixed \(\varepsilon\in(0,1)\) admit a constant \(K_\varepsilon<\infty\) such that, for all sufficiently large \(n\),

\begin{equation} \left|D_H^\varepsilon(\mathcal A_n\|\mathcal B_n) -nD^\infty(\mathcal A\|\mathcal B)\right| \leq K_\varepsilon\sqrt n? \tag{7} \end{equation}

Source

Fang, Fawzi, and Fawzi establish an \(O(n^{2/3}\log n)\) generalized equipartition remainder and explicitly leave improvement to the \(O(\sqrt n)\) scale open [FFF26].

Progress

For two fixed i.i.d. states, quantum hypothesis testing has the second-order expansion

\begin{equation} D_H^\varepsilon(\rho^{\otimes n}\|\sigma^{\otimes n}) =nD(\rho\|\sigma) +\sqrt{nV(\rho\|\sigma)}\,\Phi^{-1}(\varepsilon)+O(\log n), \tag{8} \end{equation}

where \(V\) is the relative-entropy variance and \(\Phi\) is the standard normal distribution function [Li14].
Under the assumptions in the problem, the generalized quantum asymptotic equipartition property proves the first-order limit in Eq. (6) for every fixed error [FFF26].
The best general quantitative estimate currently established is

\begin{equation} D_H^\varepsilon(\mathcal A_n\|\mathcal B_n) -nD^\infty(\mathcal A\|\mathcal B) =O(n^{2/3}\log n) \tag{9} \end{equation}

at fixed \(\varepsilon\). The authors explicitly leave replacement of Eq. (9) by the square-root scale in Eq. (7) open [FFF26].

Comment

Equation (7) asks only for a universal order bound. Identifying a Gaussian coefficient analogous to the variance term in Eq. (8) would be a strictly stronger problem.

References

[Li14]: K. Li, “Second-Order Asymptotics for Quantum Hypothesis Testing,” The Annals of Statistics 42, 171–189 (2014).DOI arXiv
[FFF26]: K. Fang, H. Fawzi, and O. Fawzi, “Generalized Quantum Asymptotic Equipartition,” Communications in Mathematical Physics 407, 208 (2026).DOI arXiv

Square-root remainder in generalized quantum equipartition

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