arXiv:2609.29485

Local and Global Risk Bounds for Quantum Entropy Estimation under Projective-Design Measurements

Xinyu Song

quant-phmath.STSubmitted Aug 25, 2026• Announced Aug 26, 2026• 0 citations

Abstract

We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least $(d/n)\log^2\{n/(4d)\}$ when $d\ge C$ and $n\ge Cd$, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius $r$ around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by $\varepsilon<1$ from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most $d^3r^2/n+d^4/n^2+d^6r^6$. Lower bounds under the same frame control yield the local minimax rate $d^3r^2/n+d^4/n^2$ when $n\ge Cd^2$ and $r$ lies in an explicit matching range. For every fixed upper bound on $\varepsilon$, approximation changes only the constants, not the powers of $d,n,r$. At the critical radius $r=n^{-1/2}$, the local risk is asymptotically negligible relative to the global risk when $n\log^2(n/d)\gg d^3$. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.

Comments: 21 pages, 1 figure, no tables

Related papers

A Frame-Spread Lower Bound for Quantum Entropy Estimation under Fixed Rank-One Measurements

Xinyu Song

quant-pharXiv:2610.00270Statistics & Probability LettersAug 26, 20260 citations

Batched and Complete U-Statistics for Trace-Polynomial Estimation from Classical Shadows

Xinyu Song

quant-pharXiv:2608.22962Aug 25, 20260 citations

Minimax Quantum State Tomography with Periodic Clifford Measurements

Hongru Zhao

quant-pharXiv:2610.00210Sep 22, 20260 citations

Minimax Quantum State Tomography in a Known Trace-Distance Ball

Yonglong Li

quant-pharXiv:2610.05374Oct 6, 20260 citations

Breaking the Multiplicative Overhead in Quantum Entropy Estimation

Junxiang Huang, Chenyang Li, Lu-Fan Zhang +2 more

quant-pharXiv:2609.40179Oct 1, 20260 citations

A Unified Complexity Framework for Quantum Property Testing

Qisheng Wang

quant-pharXiv:2608.02600Aug 4, 20260 citations

arXiv:2609.29485

Local and Global Risk Bounds for Quantum Entropy Estimation under Projective-Design Measurements

Xinyu Song

quant-phmath.STSubmitted Aug 25, 2026• Announced Aug 26, 2026• 0 citations

arXiv HTML PDF

Abstract

We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least

(d/n)\log^2\{n/(4d)\}

when

d\ge C

and

n\ge Cd

, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius

r

around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by

\varepsilon<1

from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most

d^3r^2/n+d^4/n^2+d^6r^6

. Lower bounds under the same frame control yield the local minimax rate

d^3r^2/n+d^4/n^2

when

n\ge Cd^2

and

r

lies in an explicit matching range. For every fixed upper bound on

\varepsilon

, approximation changes only the constants, not the powers of

d,n,r

. At the critical radius

r=n^{-1/2}

, the local risk is asymptotically negligible relative to the global risk when

n\log^2(n/d)\gg d^3

. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.

Comments: 21 pages, 1 figure, no tables

Related papers

A Frame-Spread Lower Bound for Quantum Entropy Estimation under Fixed Rank-One Measurements

Xinyu Song

We study estimation of the von Neumann entropy of a $d$ -dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let $κ_ν\in[1,d+1]$ denote the normalized trace-zero frame spread; $κ_ν=1$ for exact complex projective $2$ -designs. For any fixed rank-one measurement and $n\le c_2d^2/κ_ν$ , the minimax mean-squared error is at least $c_1\log^2\{d^2/(κ_νn)\}$ . For bounded frame spread and every fixed $0<a<2$ , the minimax risk is of order $\log^2 d$ throughout $n\lesssim d^{2-a}$ ; hence fixed uniform accuracy requires $Ω(d^2)$ observations. Existing tomography bounds give bounded risk from $O(d^3\log^3 d)$ observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.We study estimation of the von Neumann entropy of a $d$-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let $κ_ν\in[1,d+1]$ denote the normalized trace-zero frame spread; $κ_ν=1$ for exact complex projective $2$-designs. For any fixed rank-one measurement and $n\le c_2d^2/κ_ν$, the minimax mean-squared error is at least $c_1\log^2\{d^2/(κ_νn)\}$. For bounded frame spread and every fixed $0<a<2$, the minimax risk is of order $\log^2 d$ throughout $n\lesssim d^{2-a}$; hence fixed uniform accuracy requires $Ω(d^2)$ observations. Existing tomography bounds give bounded risk from $O(d^3\log^3 d)$ observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.

quant-pharXiv:2610.00270Statistics & Probability LettersAug 26, 20260 citations

Batched and Complete U-Statistics for Trace-Polynomial Estimation from Classical Shadows

Xinyu Song

We study estimation of the trace polynomial $\operatorname{tr} p(PρP)$ from global classical shadows, where $ρ$ is an unknown quantum state and $P$ is a fixed projector. Disjoint batching and complete U-statistics yield unbiased estimators of the same trace moments, but assign different sample-size factors to the degenerate terms in their Hoeffding decompositions. Under the global Clifford protocol, exact degree-two variance formulas show that, on a null projected block of rank $s$ , the quadratic degenerate term has order $s^2/N$ under batching and $s^2/N^2$ under complete symmetrization. For a logarithmic-degree polynomial used in entropy approximation, the quadratic coefficient raises the batched variance to at least order $s^2N\log^2N$ at the classical entropy cutoff. For complete U-statistics, we derive a cross-degree covariance identity and an exact variance decomposition for polynomial estimators. We also bound every Hoeffding order at a fixed degree and obtain a growing-dimensional risk bound for a small-spectrum entropy functional. The higher-order bounds retain a polynomial dependence on the ambient dimension and therefore do not cover logarithmically increasing degrees. Monte Carlo experiments confirm the degree-two formulas, and exact calculations illustrate the entropy risks.We study estimation of the trace polynomial $\operatorname{tr} p(PρP)$ from global classical shadows, where $ρ$ is an unknown quantum state and $P$ is a fixed projector. Disjoint batching and complete U-statistics yield unbiased estimators of the same trace moments, but assign different sample-size factors to the degenerate terms in their Hoeffding decompositions. Under the global Clifford protocol, exact degree-two variance formulas show that, on a null projected block of rank $s$, the quadratic degenerate term has order $s^2/N$ under batching and $s^2/N^2$ under complete symmetrization. For a logarithmic-degree polynomial used in entropy approximation, the quadratic coefficient raises the batched variance to at least order $s^2N\log^2N$ at the classical entropy cutoff. For complete U-statistics, we derive a cross-degree covariance identity and an exact variance decomposition for polynomial estimators. We also bound every Hoeffding order at a fixed degree and obtain a growing-dimensional risk bound for a small-spectrum entropy functional. The higher-order bounds retain a polynomial dependence on the ambient dimension and therefore do not cover logarithmically increasing degrees. Monte Carlo experiments confirm the degree-two formulas, and exact calculations illustrate the entropy risks.

quant-pharXiv:2608.22962Aug 25, 20260 citations

Minimax Quantum State Tomography with Periodic Clifford Measurements

Hongru Zhao

Quantum state tomography provides a foundation for characterizing state preparation and predicting measurement outcomes. We characterize the minimax expected trace norm risk of quantum state tomography over classes with polynomial spectral decay under randomized nonadaptive single copy measurements. Under an explicit sufficient block size condition for the two layer periodic Clifford ensemble, we establish statewise expected trace norm oracle inequalities for operator norm minimum distance and measurement weighted projected least squares, with bounds that depend on each state's actual spectral tail, while the estimators require no prior knowledge of the state's structural characteristics, including its rank, eigenbasis, or spectral structure. Over these polynomial spectral decay classes, both estimators attain the minimax expected trace norm rates up to constants. As a direct consequence, the same estimators attain the minimax rate over every fixed rank constrained class, while remaining adaptive to the unknown structural characteristics of the state. For admissible logarithmic block sizes, the periodic Clifford measurements attain these guarantees with logarithmic elementary gate depth. The main technical ingredient is a uniform bound on the fourth moment of projectors for the periodic Clifford ensemble. The bound is sharp in dimension and holds for arbitrary test states, including those entangled across blocks. Covariance concentration and positivity yield adaptive oracle bounds, while a shared rotation information argument on spectral tail packings establishes matching lower bounds. We also provide polynomial time reconstruction algorithms, numerical comparisons, and Lean formalizations of all labeled theoretical results.Quantum state tomography provides a foundation for characterizing state preparation and predicting measurement outcomes. We characterize the minimax expected trace norm risk of quantum state tomography over classes with polynomial spectral decay under randomized nonadaptive single copy measurements. Under an explicit sufficient block size condition for the two layer periodic Clifford ensemble, we establish statewise expected trace norm oracle inequalities for operator norm minimum distance and measurement weighted projected least squares, with bounds that depend on each state's actual spectral tail, while the estimators require no prior knowledge of the state's structural characteristics, including its rank, eigenbasis, or spectral structure. Over these polynomial spectral decay classes, both estimators attain the minimax expected trace norm rates up to constants. As a direct consequence, the same estimators attain the minimax rate over every fixed rank constrained class, while remaining adaptive to the unknown structural characteristics of the state. For admissible logarithmic block sizes, the periodic Clifford measurements attain these guarantees with logarithmic elementary gate depth. The main technical ingredient is a uniform bound on the fourth moment of projectors for the periodic Clifford ensemble. The bound is sharp in dimension and holds for arbitrary test states, including those entangled across blocks. Covariance concentration and positivity yield adaptive oracle bounds, while a shared rotation information argument on spectral tail packings establishes matching lower bounds. We also provide polynomial time reconstruction algorithms, numerical comparisons, and Lean formalizations of all labeled theoretical results.

quant-pharXiv:2610.00210Sep 22, 20260 citations

Minimax Quantum State Tomography in a Known Trace-Distance Ball

Yonglong Li

We study quantum state tomography when the unknown state lies in a trace-distance ball of radius $R$ around a known reference state $ρ$ . For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write $Γ=d(\mathrm{Tr}\sqrtρ)^2+d^2R$ . For $d\ge2$ , $0<R\le1$ , and $n\ge1$ copies, the expected risk is $Θ(\min\{R,\sqrt{Γ/n}\})$ . At failure probability $0<δ\le1/3$ , the minimax confidence radius is $Θ\!\left(\min\left\{R,\sqrt{\frac{Γ+\log(1/δ)}{n}}\right\}\right).$ For $\varepsilon\le c_0R$ with a universal $c_0>0$ , the copy complexity is $Θ((Γ+\log(1/δ))/\varepsilon^2)$ . For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.We study quantum state tomography when the unknown state lies in a trace-distance ball of radius $R$ around a known reference state $ρ$. For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write $Γ=d(\mathrm{Tr}\sqrtρ)^2+d^2R$. For $d\ge2$, $0<R\le1$, and $n\ge1$ copies, the expected risk is $Θ(\min\{R,\sqrt{Γ/n}\})$. At failure probability $0<δ\le1/3$, the minimax confidence radius is \[ Θ\!\left(\min\left\{R,\sqrt{\frac{Γ+\log(1/δ)}{n}}\right\}\right). \] For $\varepsilon\le c_0R$ with a universal $c_0>0$, the copy complexity is $Θ((Γ+\log(1/δ))/\varepsilon^2)$. For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.

quant-pharXiv:2610.05374Oct 6, 20260 citations

Breaking the Multiplicative Overhead in Quantum Entropy Estimation

Junxiang Huang, Chenyang Li, Lu-Fan Zhang +2 more

The von Neumann, Tsallis, and Rényi entropies are fundamental measures of quantum information. A common estimation strategy multiplies the worst-case costs of spectral transformation and statistical readout, leaving gaps to query lower bounds. We reduce this overhead with multi-level algorithms that use local normalization and precision allocation, while variable-time estimation accounts for the probability of reaching expensive spectral tests. With controlled purified access and its inverse, we obtain bounds for additive error $\varepsilon$ , rank upper bound $R$ , and fixed order $α$ . Tsallis entropy estimation has near-optimal rank-independent query complexity $\widetilde O(\varepsilon^{-1/(α-1)})$ for $1<α<2$ , when the allowed dimension and rank accommodate the lower-bound instances, and $\widetilde O(1/\varepsilon)$ for $α\ge2$ , where $\widetilde O$ suppresses logarithmic factors. The first saves a factor $1/\varepsilon$ ; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from $\widetilde O(R/\varepsilon^2)$ to $\widetilde O(R/\varepsilon)$ and obtain $\widetilde O(R/\varepsilon)$ for noninteger Rényi orders $1<α<3$ , with further bounds at other orders. Our functional-estimation theorem replaces the global product by a sum of local costs weighted by function magnitudes and spectral masses. We further estimate fixed logarithmic moments and entropy variance using $\widetilde O(R/\varepsilon)$ queries, providing efficient access to fluctuation parameters that enter finite-blocklength quantum compression and pure-state entanglement conversion. More broadly, our framework extends beyond entropy to a broad class of density-matrix functionals, offering a systematic approach toward optimal query complexity in quantum spectral estimation.The von Neumann, Tsallis, and Rényi entropies are fundamental measures of quantum information. A common estimation strategy multiplies the worst-case costs of spectral transformation and statistical readout, leaving gaps to query lower bounds. We reduce this overhead with multi-level algorithms that use local normalization and precision allocation, while variable-time estimation accounts for the probability of reaching expensive spectral tests. With controlled purified access and its inverse, we obtain bounds for additive error $\varepsilon$, rank upper bound $R$, and fixed order $α$. Tsallis entropy estimation has near-optimal rank-independent query complexity $\widetilde O(\varepsilon^{-1/(α-1)})$ for $1<α<2$, when the allowed dimension and rank accommodate the lower-bound instances, and $\widetilde O(1/\varepsilon)$ for $α\ge2$, where $\widetilde O$ suppresses logarithmic factors. The first saves a factor $1/\varepsilon$; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from $\widetilde O(R/\varepsilon^2)$ to $\widetilde O(R/\varepsilon)$ and obtain $\widetilde O(R/\varepsilon)$ for noninteger Rényi orders $1<α<3$, with further bounds at other orders. Our functional-estimation theorem replaces the global product by a sum of local costs weighted by function magnitudes and spectral masses. We further estimate fixed logarithmic moments and entropy variance using $\widetilde O(R/\varepsilon)$ queries, providing efficient access to fluctuation parameters that enter finite-blocklength quantum compression and pure-state entanglement conversion. More broadly, our framework extends beyond entropy to a broad class of density-matrix functionals, offering a systematic approach toward optimal query complexity in quantum spectral estimation.

quant-pharXiv:2609.40179Oct 1, 20260 citations

A Unified Complexity Framework for Quantum Property Testing

Qisheng Wang

We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form $\mathcal{L}_φ(ρ) = \operatorname{tr}(φ(dρ))/d$ , where $ρ$ is an unknown $d$ -dimensional quantum state and $φ$ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating $\mathcal{L}_φ(ρ)$ from properties of $φ$ , combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields (nearly) tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). In particular, combined with known upper bounds, we show that the sample complexity of von Neumann entropy estimation is $Θ\!\left(\frac{d^2}{\varepsilon\log^2(d)\max\{1, \varepsilon\log^2(d)\}} + \frac{\log^2(d)}{\varepsilon^2}\right),$ and the sample complexity of spectrum estimation is $Θ\!\left(\frac{d^2}{\varepsilon^2\log^2(d)\max\{1, \varepsilon^2\log^2(d)\}}\right).$ In summary, our results resolve several open problems and establish the optimality of 37 quantum algorithms since 2015, up to constant or polylogarithmic factors.We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form $\mathcal{L}_φ(ρ) = \operatorname{tr}(φ(dρ))/d$, where $ρ$ is an unknown $d$-dimensional quantum state and $φ$ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating $\mathcal{L}_φ(ρ)$ from properties of $φ$, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields (nearly) tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). In particular, combined with known upper bounds, we show that the sample complexity of von Neumann entropy estimation is \[ Θ\!\left(\frac{d^2}{\varepsilon\log^2(d)\max\{1, \varepsilon\log^2(d)\}} + \frac{\log^2(d)}{\varepsilon^2}\right), \] and the sample complexity of spectrum estimation is \[ Θ\!\left(\frac{d^2}{\varepsilon^2\log^2(d)\max\{1, \varepsilon^2\log^2(d)\}}\right). \] In summary, our results resolve several open problems and establish the optimality of 37 quantum algorithms since 2015, up to constant or polylogarithmic factors.

quant-pharXiv:2608.02600Aug 4, 20260 citations

We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least

(d/n)\log^2\{n/(4d)\}

when

d\ge C

and

n\ge Cd

, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius

r

around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by

\varepsilon<1

from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most

d^3r^2/n+d^4/n^2+d^6r^6

. Lower bounds under the same frame control yield the local minimax rate

d^3r^2/n+d^4/n^2

when

n\ge Cd^2

and

r

lies in an explicit matching range. For every fixed upper bound on

\varepsilon

, approximation changes only the constants, not the powers of

d,n,r

. At the critical radius

r=n^{-1/2}

, the local risk is asymptotically negligible relative to the global risk when

n\log^2(n/d)\gg d^3

We study estimation of the von Neumann entropy of a

d

-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let

κ_ν\in[1,d+1]

denote the normalized trace-zero frame spread;

κ_ν=1

for exact complex projective

2

-designs. For any fixed rank-one measurement and

n\le c_2d^2/κ_ν

, the minimax mean-squared error is at least

c_1\log^2\{d^2/(κ_νn)\}

. For bounded frame spread and every fixed

0<a<2

, the minimax risk is of order

\log^2 d

throughout

n\lesssim d^{2-a}

; hence fixed uniform accuracy requires

Ω(d^2)

observations. Existing tomography bounds give bounded risk from

O(d^3\log^3 d)

observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.

We study estimation of the trace polynomial

\operatorname{tr} p(PρP)

from global classical shadows, where

ρ

is an unknown quantum state and

P

is a fixed projector. Disjoint batching and complete U-statistics yield unbiased estimators of the same trace moments, but assign different sample-size factors to the degenerate terms in their Hoeffding decompositions. Under the global Clifford protocol, exact degree-two variance formulas show that, on a null projected block of rank

s

, the quadratic degenerate term has order

s^2/N

under batching and

s^2/N^2

under complete symmetrization. For a logarithmic-degree polynomial used in entropy approximation, the quadratic coefficient raises the batched variance to at least order

s^2N\log^2N

at the classical entropy cutoff. For complete U-statistics, we derive a cross-degree covariance identity and an exact variance decomposition for polynomial estimators. We also bound every Hoeffding order at a fixed degree and obtain a growing-dimensional risk bound for a small-spectrum entropy functional. The higher-order bounds retain a polynomial dependence on the ambient dimension and therefore do not cover logarithmically increasing degrees. Monte Carlo experiments confirm the degree-two formulas, and exact calculations illustrate the entropy risks.

We study quantum state tomography when the unknown state lies in a trace-distance ball of radius

R

around a known reference state

ρ

. For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write

Γ=d(\mathrm{Tr}\sqrtρ)^2+d^2R

. For

d\ge2

0<R\le1

, and

n\ge1

copies, the expected risk is

Θ(\min\{R,\sqrt{Γ/n}\})

. At failure probability

0<δ\le1/3

, the minimax confidence radius is

Θ\!\left(\min\left\{R,\sqrt{\frac{Γ+\log(1/δ)}{n}}\right\}\right).

For

\varepsilon\le c_0R

with a universal

c_0>0

, the copy complexity is

Θ((Γ+\log(1/δ))/\varepsilon^2)

. For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.

\varepsilon

, rank upper bound

R

, and fixed order

α

. Tsallis entropy estimation has near-optimal rank-independent query complexity

\widetilde O(\varepsilon^{-1/(α-1)})

for

1<α<2

, when the allowed dimension and rank accommodate the lower-bound instances, and

\widetilde O(1/\varepsilon)

for

α\ge2

, where

\widetilde O

suppresses logarithmic factors. The first saves a factor

1/\varepsilon

; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from

\widetilde O(R/\varepsilon^2)

\widetilde O(R/\varepsilon)

and obtain

\widetilde O(R/\varepsilon)

for noninteger Rényi orders

1<α<3

, with further bounds at other orders. Our functional-estimation theorem replaces the global product by a sum of local costs weighted by function magnitudes and spectral masses. We further estimate fixed logarithmic moments and entropy variance using

\widetilde O(R/\varepsilon)

queries, providing efficient access to fluctuation parameters that enter finite-blocklength quantum compression and pure-state entanglement conversion. More broadly, our framework extends beyond entropy to a broad class of density-matrix functionals, offering a systematic approach toward optimal query complexity in quantum spectral estimation.

We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form

\mathcal{L}_φ(ρ) = \operatorname{tr}(φ(dρ))/d

, where

ρ

is an unknown

d

-dimensional quantum state and

φ

is a given function. A master theorem is established that derives sample complexity lower bounds for estimating

\mathcal{L}_φ(ρ)

from properties of

φ

, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields (nearly) tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). In particular, combined with known upper bounds, we show that the sample complexity of von Neumann entropy estimation is

Θ\!\left(\frac{d^2}{\varepsilon\log^2(d)\max\{1, \varepsilon\log^2(d)\}} + \frac{\log^2(d)}{\varepsilon^2}\right),

and the sample complexity of spectrum estimation is

Θ\!\left(\frac{d^2}{\varepsilon^2\log^2(d)\max\{1, \varepsilon^2\log^2(d)\}}\right).

In summary, our results resolve several open problems and establish the optimality of 37 quantum algorithms since 2015, up to constant or polylogarithmic factors.