arXiv:2604.22627

The Exact Replica Threshold for Nonlinear Moments of Quantum States

Shuai Zeng

quant-phcs.CCcs.ITcs.LGphysics.comp-phSubmitted Apr 24, 2026• Announced Apr 27, 2026• 0 citations

Abstract

Joint measurements on multiple copies of a quantum state provide access to nonlinear observables such as $\operatorname{tr}(ρ^t)$, but whether replica number marks a sharp information-theoretic resource boundary has remained unclear. For every fixed order $t\ge 3$, existing protocols show that $\lceil t/2\rceil$ replicas already suffice for polynomial-sample estimation of $\operatorname{tr}(ρ^t)$, yet it has remained open whether one fewer replica must necessarily incur a sample-complexity barrier growing with the dimension. We prove that this is indeed the case in the sample/copy-access model with replica-limited joint measurements: any protocol restricted to $\lceil t/2\rceil-1$ replicas requires dimension-growing sample complexity, while $\lceil t/2\rceil$ replicas suffice by prior work. Thus the exact replica threshold for fixed-order pure moments is $\lceil t/2\rceil$. Equivalently, for fixed-order pure moments, one additional coherent replica is not merely useful but marks the exact threshold between polynomial-sample estimation and a dimension-growing regime in the replica-limited model. We further show that the same threshold law extends to a broad family of observable-weighted moments $\operatorname{tr}(Oρ^t)$, including Pauli observables and other observables with bounded operator norm and macroscopic trace norm. Coherent replica number therefore acts as a genuinely discrete resource for nonlinear quantum-state estimation.

Related papers

Optimal single-copy estimation of quantum state moments: why $\operatorname{Tr}(ρ^3)$ and $\operatorname{Tr}(ρ^4)$ are equally hard

Zhenhuan Liu

quant-pharXiv:2609.37781Sep 30, 20260 citations

Pauli-resolved virtual distillation

Si-Yuan Chen, Congcong Zheng, Kun Wang +1 more

quant-pharXiv:2609.24132Sep 22, 20260 citations

Random Projections for Multi-Copy Quantum Algorithms

Xiaoyu Liu, Jordi Tura, Johannes Knörzer

quant-pharXiv:2606.20238Jun 19, 20260 citations

Exponential Advantage from One More Replica in Estimating Nonlinear Properties of Quantum States

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Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements

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Power and Limits of Collective Local Measurements in Multicopy State Discrimination

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quant-pharXiv:2609.07233Sep 8, 20260 citations

arXiv:2604.22627

The Exact Replica Threshold for Nonlinear Moments of Quantum States

Shuai Zeng

quant-phcs.CCcs.ITcs.LGphysics.comp-phSubmitted Apr 24, 2026• Announced Apr 27, 2026• 0 citations

arXiv HTML PDF

Abstract

Joint measurements on multiple copies of a quantum state provide access to nonlinear observables such as

\operatorname{tr}(ρ^t)

, but whether replica number marks a sharp information-theoretic resource boundary has remained unclear. For every fixed order

t\ge 3

, existing protocols show that

\lceil t/2\rceil

replicas already suffice for polynomial-sample estimation of

\operatorname{tr}(ρ^t)

, yet it has remained open whether one fewer replica must necessarily incur a sample-complexity barrier growing with the dimension. We prove that this is indeed the case in the sample/copy-access model with replica-limited joint measurements: any protocol restricted to

\lceil t/2\rceil-1

replicas requires dimension-growing sample complexity, while

\lceil t/2\rceil

replicas suffice by prior work. Thus the exact replica threshold for fixed-order pure moments is

\lceil t/2\rceil

. Equivalently, for fixed-order pure moments, one additional coherent replica is not merely useful but marks the exact threshold between polynomial-sample estimation and a dimension-growing regime in the replica-limited model. We further show that the same threshold law extends to a broad family of observable-weighted moments

\operatorname{tr}(Oρ^t)

, including Pauli observables and other observables with bounded operator norm and macroscopic trace norm. Coherent replica number therefore acts as a genuinely discrete resource for nonlinear quantum-state estimation.

Related papers

Optimal single-copy estimation of quantum state moments: why $\operatorname{Tr}(ρ^3)$ and $\operatorname{Tr}(ρ^4)$ are equally hardOptimal single-copy estimation of quantum state moments: why $\operatorname{Tr}(ρ^3)$ and $\operatorname{Tr}(ρ^4)$ are equally hard

Zhenhuan Liu

Estimating nonlinear properties of an unknown quantum state with restrictive experimental accessibility is a fundamental problem in quantum learning. We study the sample complexity of estimating the state moments $\operatorname{Tr}(ρ^t)$ , allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity $Θ(\sqrt d)$ at constant additive error, the optimal dimension dependence for higher moments has remained unclear. We resolve this question for every fixed integer $t\ge 2$ : for any sufficiently small constant additive error, the optimal sample complexity is $Θ\!\left(d^{\frac{\lceil\log_2 t\rceil}{1+\lceil\log_2 t\rceil}}\right)$ with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer $h\geq1$ , all moment orders $2^{h-1}<t\leq2^h$ share the same exponent. In particular, the third and fourth moments both require $Θ(d^{2/3})$ copies. Our results also apply to multistate overlaps, weighted and partial-transpose moments, fixed-rank spectrum estimation, and distributed estimation of nonlinear quantities. We also establish an $Ω(d^{1-1/t})$ lower bound for arbitrary nonadaptive single-copy protocols, matching the known upper bound and demonstrating an advantage from adaptivity for every $t\geq4$ . Our upper bound uses adaptive state filtering to reduce higher-order moment estimation to lower-order moment estimation in a smaller state space. For the lower bound, we directly compare moment-matched ensembles through a smooth interpolation, bypassing the maximally mixed state as an intermediate hypothesis and yielding matching bounds beyond $\sqrt d$ .Estimating nonlinear properties of an unknown quantum state with restrictive experimental accessibility is a fundamental problem in quantum learning. We study the sample complexity of estimating the state moments $\operatorname{Tr}(ρ^t)$, allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity $Θ(\sqrt d)$ at constant additive error, the optimal dimension dependence for higher moments has remained unclear. We resolve this question for every fixed integer $t\ge 2$: for any sufficiently small constant additive error, the optimal sample complexity is \[ Θ\!\left(d^{\frac{\lceil\log_2 t\rceil}{1+\lceil\log_2 t\rceil}}\right) \] with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer $h\geq1$, all moment orders $2^{h-1}<t\leq2^h$ share the same exponent. In particular, the third and fourth moments both require $Θ(d^{2/3})$ copies. Our results also apply to multistate overlaps, weighted and partial-transpose moments, fixed-rank spectrum estimation, and distributed estimation of nonlinear quantities. We also establish an $Ω(d^{1-1/t})$ lower bound for arbitrary nonadaptive single-copy protocols, matching the known upper bound and demonstrating an advantage from adaptivity for every $t\geq4$. Our upper bound uses adaptive state filtering to reduce higher-order moment estimation to lower-order moment estimation in a smaller state space. For the lower bound, we directly compare moment-matched ensembles through a smooth interpolation, bypassing the maximally mixed state as an intermediate hypothesis and yielding matching bounds beyond $\sqrt d$.

quant-pharXiv:2609.37781Sep 30, 20260 citations

Pauli-resolved virtual distillation

Si-Yuan Chen, Congcong Zheng, Kun Wang +1 more

Learning the full Pauli profile of the virtually distilled quantum state $ρ^m/\text{tr}(ρ^m)$ has so far required exponentially many copies of $ρ$ . We show that all $4^n$ squared Pauli moments $[\text{tr}(Pρ^m)]^2$ can be learned to additive error $\varepsilon$ with confidence $1-δ$ from one $2m$ -replica measurement setting using $O(m[n+\log(1/δ)]/\varepsilon^2)$ copies. This is an exponential speedup in the system size $n$ over previous protocols. For $m = 2$ , we propose the coherent Bell difference sampling circuit that realizes this measurement on current devices. The speedup originates from paired replicas that cancel the anticommutation signs of Pauli operators, collapsing the incompatibility of the Pauli family. We certify this collapse by introducing the quantum Bernstein norm, a computable incompatibility measure for nonlinear functionals. A further information-theoretic $m$ -replica protocol recovers the signed moments $\text{tr}(Pρ^m)$ using $O(m[n+\log(1/δ)]/\varepsilon^4)$ copies. For fixed $m$ in the stated lower-bound regime, it attains the optimal replica number, since any protocol with fewer replicas requires exponentially many copies.Learning the full Pauli profile of the virtually distilled quantum state $ρ^m/\text{tr}(ρ^m)$ has so far required exponentially many copies of $ρ$. We show that all $4^n$ squared Pauli moments $[\text{tr}(Pρ^m)]^2$ can be learned to additive error $\varepsilon$ with confidence $1-δ$ from one $2m$-replica measurement setting using $O(m[n+\log(1/δ)]/\varepsilon^2)$ copies. This is an exponential speedup in the system size $n$ over previous protocols. For $m = 2$, we propose the coherent Bell difference sampling circuit that realizes this measurement on current devices. The speedup originates from paired replicas that cancel the anticommutation signs of Pauli operators, collapsing the incompatibility of the Pauli family. We certify this collapse by introducing the quantum Bernstein norm, a computable incompatibility measure for nonlinear functionals. A further information-theoretic $m$-replica protocol recovers the signed moments $\text{tr}(Pρ^m)$ using $O(m[n+\log(1/δ)]/\varepsilon^4)$ copies. For fixed $m$ in the stated lower-bound regime, it attains the optimal replica number, since any protocol with fewer replicas requires exponentially many copies.

quant-pharXiv:2609.24132Sep 22, 20260 citations

Random Projections for Multi-Copy Quantum Algorithms

Xiaoyu Liu, Jordi Tura, Johannes Knörzer

Estimating nonlinear properties of quantum states is a central task in quantum information science. Multivariate traces, $\mathrm{tr}(ρ_1 \cdots ρ_K)$ , and nonlinear observables such as $\mathrm{tr}(ρ^K)$ , for integer $K$ , can be accessed through collective measurements on multiple state copies, but standard protocols based on swap tests require coherent operations on the full Hilbert space and become experimentally unfeasible for large systems. In this work, we introduce a framework for multi-copy measurements based on random projections onto lower-dimensional subspaces prior to the collective measurement, which is then performed only on the reduced Hilbert space. This procedure yields a tunable tradeoff between coherent quantum resources and statistical sampling overhead, allowing the amount of coherent processing to be matched to the capabilities of the underlying hardware. We derive explicit formulas relating the Haar-averaged projected moments to multivariate traces of the original states and analyze the sampling overhead induced by the projection procedure. Specifically, after compressing an $n$ -qubit state to a reduced $q$ -qubit subspace, estimating $\mathrm{tr}(ρ^K)$ requires approximately $O(2^{(n-q)(K-1)})$ copies of $ρ$ , with each qubit projected out increasing the sampling cost by a factor of $2^{K-1}$ . Our results establish how coherent multi-copy operations can be traded for additional state copies, enabling multi-copy quantum protocols to be optimized for the available hardware resources.Estimating nonlinear properties of quantum states is a central task in quantum information science. Multivariate traces, $\mathrm{tr}(ρ_1 \cdots ρ_K)$, and nonlinear observables such as $\mathrm{tr}(ρ^K)$, for integer $K$, can be accessed through collective measurements on multiple state copies, but standard protocols based on swap tests require coherent operations on the full Hilbert space and become experimentally unfeasible for large systems. In this work, we introduce a framework for multi-copy measurements based on random projections onto lower-dimensional subspaces prior to the collective measurement, which is then performed only on the reduced Hilbert space. This procedure yields a tunable tradeoff between coherent quantum resources and statistical sampling overhead, allowing the amount of coherent processing to be matched to the capabilities of the underlying hardware. We derive explicit formulas relating the Haar-averaged projected moments to multivariate traces of the original states and analyze the sampling overhead induced by the projection procedure. Specifically, after compressing an $n$-qubit state to a reduced $q$-qubit subspace, estimating $\mathrm{tr}(ρ^K)$ requires approximately $O(2^{(n-q)(K-1)})$ copies of $ρ$, with each qubit projected out increasing the sampling cost by a factor of $2^{K-1}$. Our results establish how coherent multi-copy operations can be traded for additional state copies, enabling multi-copy quantum protocols to be optimized for the available hardware resources.

quant-pharXiv:2606.20238Jun 19, 20260 citations

Exponential Advantage from One More Replica in Estimating Nonlinear Properties of Quantum States

Qi Ye, Zhenhuan Liu, Dong-Ling Deng

Inferring nonlinear features of quantum states is fundamentally important across quantum information science, but remains challenging due to the intrinsic linearity of quantum mechanics. It is widely recognized that quantum memory and coherent operations help avoid exponential sample complexity, by mapping nonlinear properties onto linear observables over multiple copies of the target state. In this work, we prove that such a conversion is not only sufficient but also necessary. Specifically, we prove that the estimation of $\mathrm{tr}(ρ^{k} O)$ for a broad class of observables $O$ is exponentially hard for any protocol restricted to $(k-1)$ -replica joint measurements, whereas access to one additional replica reduces the complexity to a constant. These results establish, for the first time, an exponential separation between $(k-1)$ - and $k$ -replica protocols for any integer $k>2$ , thereby introducing a fine-grained hierarchy of replica-based quantum advantage and resolving an open question in the literature. The technical core is a general indistinguishability principle showing that any ensemble constructed from large Haar random states via tensor products and mixtures is hard to distinguish from its average. Leveraging this principle, we further prove that $k$ -replica joint measurements are also necessary for distinguishing rank- $k$ density matrices from rank- $(k-1)$ ones. Overall, our work delineates sharp boundaries on the power of joint measurements, highlighting resource-complexity trade-offs in quantum learning theory and deepening the understanding of quantum mechanics' intrinsic linearity.Inferring nonlinear features of quantum states is fundamentally important across quantum information science, but remains challenging due to the intrinsic linearity of quantum mechanics. It is widely recognized that quantum memory and coherent operations help avoid exponential sample complexity, by mapping nonlinear properties onto linear observables over multiple copies of the target state. In this work, we prove that such a conversion is not only sufficient but also necessary. Specifically, we prove that the estimation of $\mathrm{tr}(ρ^{k} O)$ for a broad class of observables $O$ is exponentially hard for any protocol restricted to $(k-1)$-replica joint measurements, whereas access to one additional replica reduces the complexity to a constant. These results establish, for the first time, an exponential separation between $(k-1)$- and $k$-replica protocols for any integer $k>2$, thereby introducing a fine-grained hierarchy of replica-based quantum advantage and resolving an open question in the literature. The technical core is a general indistinguishability principle showing that any ensemble constructed from large Haar random states via tensor products and mixtures is hard to distinguish from its average. Leveraging this principle, we further prove that $k$-replica joint measurements are also necessary for distinguishing rank-$k$ density matrices from rank-$(k-1)$ ones. Overall, our work delineates sharp boundaries on the power of joint measurements, highlighting resource-complexity trade-offs in quantum learning theory and deepening the understanding of quantum mechanics' intrinsic linearity.

arXiv:2509.24000Sep 28, 20250 citations

Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements

Ashwin Nayak, Xingyu Zhou

We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples. For sufficiently small $\varepsilon$ , estimating an unknown state on $\mathbb{C}^d$ of rank at most $r$ to trace norm error $\varepsilon$ with constant success probability requires, and is achievable with, $Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)$ samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\sqrt{t}$ . Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples. For sufficiently small $\varepsilon$, estimating an unknown state on $\mathbb{C}^d$ of rank at most $r$ to trace norm error $\varepsilon$ with constant success probability requires, and is achievable with, $$Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)$$ samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\sqrt{t}$. Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.

quant-pharXiv:2609.10514Sep 10, 20260 citations

Power and Limits of Collective Local Measurements in Multicopy State Discrimination

Mao-Sheng Li, Yan-Ling Wang, Zhu-Jun Zheng

More than two decades ago, Bennett \emph{et al.} [Phys. Rev. A \textbf{59}, 1070 (1999)] asked whether perfect local discrimination of orthogonal quantum states can require more than two copies. This question was subsequently answered for adaptive protocols that process the copies separately [Phys. Rev. Lett. \textbf{126}, 210505 (2021)], but remained open when each laboratory is allowed to process its local copies collectively. Here we resolve this stronger setting and identify collective access across repeated local inputs as a distinct resource. For every fixed odd-prime local dimension, there exist complete maximal-stabilizer eigenbases whose copy complexity under individual-copy separable measurements diverges with system size. Under collective processing this behavior changes sharply: every maximal-stabilizer eigenbasis in odd-prime local dimension is perfectly decoded by one-round collective LOCC using at most three copies. Collective processing, however, does not remove multicopy hardness in general. For every fixed local dimension $d\ge2$ , we prove the existence, within an explicit phase family, of complete bases whose copy complexity remains unbounded even under collective separable measurements, with a square root of the number of subsystems as lower-bound scale. Thus sample number, spatial measurement power, and coherent access across repeated local inputs are distinct resources in distributed quantum measurement.More than two decades ago, Bennett \emph{et al.} [Phys. Rev. A \textbf{59}, 1070 (1999)] asked whether perfect local discrimination of orthogonal quantum states can require more than two copies. This question was subsequently answered for adaptive protocols that process the copies separately [Phys. Rev. Lett. \textbf{126}, 210505 (2021)], but remained open when each laboratory is allowed to process its local copies collectively. Here we resolve this stronger setting and identify collective access across repeated local inputs as a distinct resource. For every fixed odd-prime local dimension, there exist complete maximal-stabilizer eigenbases whose copy complexity under individual-copy separable measurements diverges with system size. Under collective processing this behavior changes sharply: every maximal-stabilizer eigenbasis in odd-prime local dimension is perfectly decoded by one-round collective LOCC using at most three copies. Collective processing, however, does not remove multicopy hardness in general. For every fixed local dimension $d\ge2$, we prove the existence, within an explicit phase family, of complete bases whose copy complexity remains unbounded even under collective separable measurements, with a square root of the number of subsystems as lower-bound scale. Thus sample number, spatial measurement power, and coherent access across repeated local inputs are distinct resources in distributed quantum measurement.

quant-pharXiv:2609.07233Sep 8, 20260 citations

Joint measurements on multiple copies of a quantum state provide access to nonlinear observables such as

\operatorname{tr}(ρ^t)

, but whether replica number marks a sharp information-theoretic resource boundary has remained unclear. For every fixed order

t\ge 3

, existing protocols show that

\lceil t/2\rceil

replicas already suffice for polynomial-sample estimation of

\operatorname{tr}(ρ^t)

\lceil t/2\rceil-1

replicas requires dimension-growing sample complexity, while

\lceil t/2\rceil

replicas suffice by prior work. Thus the exact replica threshold for fixed-order pure moments is

\lceil t/2\rceil

\operatorname{tr}(Oρ^t)

Optimal single-copy estimation of quantum state moments: why

\operatorname{Tr}(ρ^3)

and

\operatorname{Tr}(ρ^4)

are equally hard

\operatorname{Tr}(ρ^t)

, allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity

Θ(\sqrt d)

at constant additive error, the optimal dimension dependence for higher moments has remained unclear. We resolve this question for every fixed integer

t\ge 2

: for any sufficiently small constant additive error, the optimal sample complexity is

Θ\!\left(d^{\frac{\lceil\log_2 t\rceil}{1+\lceil\log_2 t\rceil}}\right)

with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer

h\geq1

, all moment orders

2^{h-1}<t\leq2^h

share the same exponent. In particular, the third and fourth moments both require

Θ(d^{2/3})

copies. Our results also apply to multistate overlaps, weighted and partial-transpose moments, fixed-rank spectrum estimation, and distributed estimation of nonlinear quantities. We also establish an

Ω(d^{1-1/t})

lower bound for arbitrary nonadaptive single-copy protocols, matching the known upper bound and demonstrating an advantage from adaptivity for every

t\geq4

. Our upper bound uses adaptive state filtering to reduce higher-order moment estimation to lower-order moment estimation in a smaller state space. For the lower bound, we directly compare moment-matched ensembles through a smooth interpolation, bypassing the maximally mixed state as an intermediate hypothesis and yielding matching bounds beyond

\sqrt d

Learning the full Pauli profile of the virtually distilled quantum state

ρ^m/\text{tr}(ρ^m)

has so far required exponentially many copies of

ρ

. We show that all

4^n

squared Pauli moments

[\text{tr}(Pρ^m)]^2

can be learned to additive error

\varepsilon

with confidence

1-δ

from one

2m

-replica measurement setting using

O(m[n+\log(1/δ)]/\varepsilon^2)

copies. This is an exponential speedup in the system size

n

over previous protocols. For

m = 2

, we propose the coherent Bell difference sampling circuit that realizes this measurement on current devices. The speedup originates from paired replicas that cancel the anticommutation signs of Pauli operators, collapsing the incompatibility of the Pauli family. We certify this collapse by introducing the quantum Bernstein norm, a computable incompatibility measure for nonlinear functionals. A further information-theoretic

m

-replica protocol recovers the signed moments

\text{tr}(Pρ^m)

using

O(m[n+\log(1/δ)]/\varepsilon^4)

copies. For fixed

m

in the stated lower-bound regime, it attains the optimal replica number, since any protocol with fewer replicas requires exponentially many copies.

Estimating nonlinear properties of quantum states is a central task in quantum information science. Multivariate traces,

\mathrm{tr}(ρ_1 \cdots ρ_K)

, and nonlinear observables such as

\mathrm{tr}(ρ^K)

, for integer

K

, can be accessed through collective measurements on multiple state copies, but standard protocols based on swap tests require coherent operations on the full Hilbert space and become experimentally unfeasible for large systems. In this work, we introduce a framework for multi-copy measurements based on random projections onto lower-dimensional subspaces prior to the collective measurement, which is then performed only on the reduced Hilbert space. This procedure yields a tunable tradeoff between coherent quantum resources and statistical sampling overhead, allowing the amount of coherent processing to be matched to the capabilities of the underlying hardware. We derive explicit formulas relating the Haar-averaged projected moments to multivariate traces of the original states and analyze the sampling overhead induced by the projection procedure. Specifically, after compressing an

n

-qubit state to a reduced

q

-qubit subspace, estimating

\mathrm{tr}(ρ^K)

requires approximately

O(2^{(n-q)(K-1)})

copies of

ρ

, with each qubit projected out increasing the sampling cost by a factor of

2^{K-1}

. Our results establish how coherent multi-copy operations can be traded for additional state copies, enabling multi-copy quantum protocols to be optimized for the available hardware resources.

\mathrm{tr}(ρ^{k} O)

for a broad class of observables

O

is exponentially hard for any protocol restricted to

(k-1)

-replica joint measurements, whereas access to one additional replica reduces the complexity to a constant. These results establish, for the first time, an exponential separation between

(k-1)

- and

k

-replica protocols for any integer

k>2

, thereby introducing a fine-grained hierarchy of replica-based quantum advantage and resolving an open question in the literature. The technical core is a general indistinguishability principle showing that any ensemble constructed from large Haar random states via tensor products and mixtures is hard to distinguish from its average. Leveraging this principle, we further prove that

k

-replica joint measurements are also necessary for distinguishing rank-

k

density matrices from rank-

(k-1)

ones. Overall, our work delineates sharp boundaries on the power of joint measurements, highlighting resource-complexity trade-offs in quantum learning theory and deepening the understanding of quantum mechanics' intrinsic linearity.

We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most

t

samples. For sufficiently small

\varepsilon

, estimating an unknown state on

\mathbb{C}^d

of rank at most

r

to trace norm error

\varepsilon

with constant success probability requires, and is achievable with,

Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)

samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most

t

samples improve the complexity of algorithms making single-sample measurements by at most a factor

\sqrt{t}

. Further, measuring order

r^2

samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on

t

samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.

d\ge2

, we prove the existence, within an explicit phase family, of complete bases whose copy complexity remains unbounded even under collective separable measurements, with a square root of the number of subsystems as lower-bound scale. Thus sample number, spatial measurement power, and coherent access across repeated local inputs are distinct resources in distributed quantum measurement.