Infinite-Copy Certification: Quantum Verification

• Infinite-Copy Certification is a research area that explores optimal quantum state certification using numerous copies via diverse measurement models.
• It employs methods such as collective measurements and device-independent protocols to achieve instance-optimal and asymptotic scaling laws.
• Key challenges include aligning finite-sample scaling with infinite tensor-product structures, ensuring robust error handling, and overcoming operational limitations.

As used across recent literature, Infinite-Copy Certification does not denote a single standardized problem. It refers to a cluster of closely related questions in which access to many copies of a quantum resource is the decisive asymptotic variable. In the most direct formulation, one is given a known hypothesis state $\sigma$, many copies of an unknown state $\rho$, and a measurement model, and asks for the optimal copy complexity of distinguishing $\rho=\sigma$ from $\|\rho-\sigma\|_1\ge \epsilon$. In that setting, the sharp instance-wise law with unrestricted collective measurements is, up to logarithmic factors and mild spectral trimming, $\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$ (O'Donnell et al., 8 Jul 2025). In adjacent literatures, the same phrase is used more broadly for asymptotic-surrogate questions: certifying arbitrarily large finite tensor products, certifying many leftover copies after testing a subset, or approaching ideal projective verification as system size grows (Šupić et al., 2019, Gočanin et al., 2021, Cha et al., 8 Jun 2026). A separate operator-algebraic line gives a literal infinite-copy theorem: exact embezzlement of a pure target state forces the catalyst to contain infinitely many mutually commuting local copies of that target (Liu, 5 Sep 2025).

1. Distinct formulations of the problem

The phrase is best understood as an umbrella for several models rather than as the name of one theorem. The main distinctions are the measurement primitive, the object being certified, and whether “infinite-copy” means a finite-sample scaling law, an asymptotic surrogate, or an actual infinite tensor-product structure.

Formulation	Representative statement	Representative paper
Known-state certification from many copies	Distinguish $H_0:\rho=\sigma$ from $H_1:\|\rho-\sigma\|_1\ge\epsilon$ with optimal copy complexity	(O'Donnell et al., 8 Jul 2025)
Device-independent many-copy certification	Test a subset of independent copies and certify the unmeasured remainder via extractability	(Gočanin et al., 2021)
Unbounded finite tensor-product self-testing	Lift a single-copy self-test to certify $\ket{\psi'}^{\otimes n}$ for arbitrary finite $n$	(Šupić et al., 2019)
Near-projective asymptotic verification	Restricted single-copy measurements approach ideal projector behavior as system size grows	(Cha et al., 8 Jun 2026)
Literal infinite-copy containment	Exact embezzlement implies internal containment of $g^{\otimes\infty}$	(Liu, 5 Sep 2025)

The finite-sample formulation is the most direct descendant of ordinary quantum state certification. There the resource is the number $\rho$0 of copies of $\rho$1, and the strongest model allows fully entangled measurements across all $\rho$2 copies. In weaker models, the same state-certification problem changes substantially because the accessible statistics change. That distinction is central to the subject: several papers show that copy complexity is not determined by the hypothesis test alone, but by the joint choice of test and measurement primitive (O'Donnell et al., 8 Jul 2025, Liu et al., 2024, Chen et al., 2022).

A second family of works uses “infinite-copy” in a looser asymptotic sense. Those papers do not prove statements about an actual infinite tensor product or a single verifier acting on infinitely many copies at once. Rather, they identify regimes in which either the number of certifiable copies is arbitrary but finite, or the leading coefficient of copy complexity converges to that of an ideal projective test as another parameter tends to infinity (Šupić et al., 2019, Cha et al., 8 Jun 2026).

2. Instance-optimal many-copy certification with collective measurements

The most explicit answer to the finite-dimensional many-copy problem is given for the task

$\rho$3

where $\rho$4 is known, the tester receives $\rho$5 copies of $\rho$6, and measurements may be fully entangled across all copies. Success probability is required to be at least $\rho$7, and standard amplification raises this to $\rho$8 at multiplicative cost $\rho$9 (O'Donnell et al., 8 Jul 2025).

The central theorem is nearly instance-optimal. For suitable truncations $\rho=\sigma$0 and $\rho=\sigma$1, obtained by zeroing out small spectral mass and renormalizing, the copy complexity satisfies

$\rho=\sigma$2

where the $\rho=\sigma$3 and $\rho=\sigma$4 hide polylogarithmic factors in $\rho=\sigma$5 (O'Donnell et al., 8 Jul 2025). Using

$\rho=\sigma$6

this becomes

$\rho=\sigma$7

up to logarithms and mild spectral trimming. Spectrally, if $\rho=\sigma$8 has eigenvalues $\rho=\sigma$9, then

$\|\rho-\sigma\|_1\ge \epsilon$0

The instance difficulty is therefore governed by the square of the $\|\rho-\sigma\|_1\ge \epsilon$1-type mass of the spectrum: spectrally flatter states are harder; more concentrated spectra are easier (O'Donnell et al., 8 Jul 2025).

The special cases are especially informative. If $\|\rho-\sigma\|_1\ge \epsilon$2 is pure, then $\|\rho-\sigma\|_1\ge \epsilon$3, so $\|\rho-\sigma\|_1\ge \epsilon$4 and the theorem gives $\|\rho-\sigma\|_1\ge \epsilon$5. If $\|\rho-\sigma\|_1\ge \epsilon$6, then $\|\rho-\sigma\|_1\ge \epsilon$7, giving $\|\rho-\sigma\|_1\ge \epsilon$8. More generally, if $\|\rho-\sigma\|_1\ge \epsilon$9 is close to maximally mixed on an $\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$0-dimensional support, then $\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$1, and the copy complexity is $\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$2 (O'Donnell et al., 8 Jul 2025).

This result refines the older worst-case picture. Earlier robust certification algorithms already showed that certification is easier than tomography: for known $\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$3, one can certify in fidelity with $\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$4 copies and in trace distance with $\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$5 copies, both optimal up to constants; tomography requires on the order of $\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$6 copies in general (Bădescu et al., 2017). The newer instance-wise theorem identifies the correct state-dependent parameter when collective measurements are unrestricted (O'Donnell et al., 8 Jul 2025).

The lower-bound technique is itself a major part of the subject. The same work introduces a quantum analogue of the Ingster–Suslina method, based on the quantum $\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$7-divergence and a mixture argument for $\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$8. For mixedness testing, this yields a particularly simple proof that

$\widetilde{\Theta}(d\,F(\sigma,I/d)/\epsilon^2)$9

and hence $H_0:\rho=\sigma$0 (O'Donnell et al., 8 Jul 2025). The “infinite-copy” content of this line is therefore finite-sample but sharp: it identifies the asymptotically correct scaling law in $H_0:\rho=\sigma$1 and $H_0:\rho=\sigma$2 for each fixed hypothesis state.

3. Restricted measurements, shared randomness, and copy-number hierarchies

When the tester is not allowed fully collective measurements, the notion of infinite-copy certification changes from a single scaling law to a hierarchy of measurement models. For incoherent or unentangled measurements, one copy is measured at a time, possibly adaptively, and the copy complexity can increase polynomially.

For mixedness testing under incoherent measurements, the exact copy complexity is

$H_0:\rho=\sigma$3

and adaptivity does not help. The same work shows that the earlier instance-optimal bounds for non-adaptive certification of general $H_0:\rho=\sigma$4 remain valid, up to polylogarithmic factors, even for arbitrary adaptive incoherent protocols (Chen et al., 2022). This gives a sharp separation between collective and one-copy-at-a-time access: the worst-case rate is $H_0:\rho=\sigma$5 with collective measurements, but $H_0:\rho=\sigma$6 with incoherent measurements (Chen et al., 2022).

A related refinement concerns the role of shared randomness. For unentangled certification, deterministic fixed measurement schemes require

$H_0:\rho=\sigma$7

copies, whereas shared-randomness-assisted unentangled certification requires only

$H_0:\rho=\sigma$8

and unrestricted entangled measurements achieve $H_0:\rho=\sigma$9 (Liu et al., 2024). The paper formulates this separation through the spectrum of the average Lüders channel associated with the one-copy measurements. This indicates that, even in the asymptotic large-copy regime, “infinite-copy certification” is not a property of the hypothesis test alone; it is a property of the hypothesis test together with the admissible copy-wise interaction model (Liu et al., 2024).

A different obstacle arises in robust certification of pure states by few-body measurements. Earlier single-qubit protocols could certify only $H_1:\|\rho-\sigma\|_1\ge\epsilon$0-infidelity neighborhoods of the target, and Appendix A of one paper shows that for almost all Haar-random targets there exists a nearly orthogonal $H_1:\|\rho-\sigma\|_1\ge\epsilon$1 accepted with probability at least $H_1:\|\rho-\sigma\|_1\ge\epsilon$2. This means that infinite repetition does not repair the test: estimating the acceptance probability perfectly still does not certify constant closeness (Coladangelo et al., 12 Feb 2026). The same paper resolves this, for all but an $H_1:\|\rho-\sigma\|_1\ge\epsilon$3 fraction of pure targets, by combining measurements in the $H_1:\|\rho-\sigma\|_1\ge\epsilon$4 and $H_1:\|\rho-\sigma\|_1\ge\epsilon$5 bases. With one $H_1:\|\rho-\sigma\|_1\ge\epsilon$6-qubit measurement and single-qubit measurements on the rest, the one-shot soundness is

$H_1:\|\rho-\sigma\|_1\ge\epsilon$7

and repetition gives optimal constant-in-$H_1:\|\rho-\sigma\|_1\ge\epsilon$8 copy complexity

$H_1:\|\rho-\sigma\|_1\ge\epsilon$9

for constant robustness (Coladangelo et al., 12 Feb 2026). This is an important conceptual point: an infinite supply of copies is useless if the one-shot test is non-robust.

At an even more structural level, the number of copies that may be coherently processed in each measurement round forms an infinite hierarchy. For every prime $\ket{\psi'}^{\otimes n}$0, there are explicit learning tasks of degree $\ket{\psi'}^{\otimes n}$1: they are exponentially hard with $\ket{\psi'}^{\otimes n}$2-copy measurements but efficiently solvable with $\ket{\psi'}^{\otimes n}$3-copy measurements. Analogous finite-degree tasks also exist for all square-free integers $\ket{\psi'}^{\otimes n}$4 (Nöller et al., 9 Oct 2025). A plausible implication is that copy number is not merely a quantitative resource; it induces distinct qualitative phases of certifiability.

4. Device-independent many-copy certification

In the device-independent setting, infinite-copy certification typically means certifying many copies or many leftover copies from Bell data without trusting the measurement devices. Here the central figure of merit is not direct fidelity but extractability or self-testing equivalence.

A finite-sample device-independent certification theorem is available for independent, not necessarily identical, copies. A source emits $\ket{\psi'}^{\otimes n}$5 independent states; each copy is selected for testing independently with probability $\ket{\psi'}^{\otimes n}$6; the tested subset is scored in a nonlocal game arising from a robust self-test; and the unmeasured remainder is certified. If the tested subset achieves empirical score $\ket{\psi'}^{\otimes n}$7, then

$\ket{\psi'}^{\otimes n}$8

where $\ket{\psi'}^{\otimes n}$9 is the Kullback–Leibler divergence, and the leftover average extractability obeys

$n$0

Because the failure probability decays exponentially in $n$1, the theorem has a clear asymptotic reading: for an independent source, testing a fixed fraction of a large batch certifies the rest with arbitrarily high confidence (Gočanin et al., 2021). The same paper is explicit that full non-IID certification of more than one leftover copy remains open.

Another line addresses arbitrary finite tensor powers directly. A single-copy self-test based on a Bell expression $n$2, under rank-one measurement assumptions and positivity conditions, can be lifted to a constant-input protocol that self-tests

$n$3

for every finite $n$4, with the same number of parties and the same number of measurement choices. The construction uses nonlinear Bell expressions built from conditional Bell values $n$5, and it certifies the tensor-product measurement structure as well as the state (Šupić et al., 2019). This is best described as certification of arbitrarily many finite copies, not of a literal infinite tensor product; the paper is explicit on that point.

Many-copy device-independent randomness certification gives a third asymptotic variant. A family of $n$6-settings Bell inequalities optimized for $n$7 copies of maximally entangled two-qubit states certifies global min-entropy

$n$8

which tends to $n$9 bits as $g^{\otimes\infty}$0 (Mahato et al., 2022). This is an asymptotic trend rather than a literal infinite-copy theorem: the certified randomness per Bell round is capped by the binary output alphabet, but the example shows how Bell inequalities can be tailored so that certification genuinely forces multi-copy use of the resource.

5. Asymptotic-surrogate verification beyond collective-copy models

Some of the sharpest “infinite-copy” statements do not concern joint measurements on many copies at all. Instead they analyze how close a restricted verification primitive can get to an ideal projector as another system parameter grows.

For the $g^{\otimes\infty}$1-qubit GHZ state, Bell-Matching Certification uses only disjoint two-qubit Bell-basis measurements, plus one single-qubit $g^{\otimes\infty}$2-basis measurement when $g^{\otimes\infty}$3 is odd. The verification operator has exact second eigenvalue

$g^{\otimes\infty}$4

so the spectral gap is

$g^{\otimes\infty}$5

For $g^{\otimes\infty}$6 independent tests, the required number of rounds is

$g^{\otimes\infty}$7

and asymptotically

$g^{\otimes\infty}$8

Since $g^{\otimes\infty}$9, BM-Cert becomes asymptotically projective and asymptotically copy-optimal as $\rho$00 within its restricted measurement model (Cha et al., 8 Jun 2026). This is explicitly presented as an asymptotic surrogate for an infinite-copy certification question, not as a theorem about collective-copy measurements.

Continuous-variable graph-state certification provides another asymptotic surrogate, now in a non-i.i.d. many-register setting with realistic noise. The protocol tests $\rho$01 nullifiers on randomly selected registers, accepts if sufficiently many soft nullifier tests pass, and keeps $\rho$02 unmeasured registers. Its soundness uses Serfling’s bound rather than any i.i.d. assumption. Conditionally on acceptance, the retained $\rho$03-register state satisfies

$\rho$04

where $\rho$05 is a Gaussian-smeared overlap with the ideal graph state (Descamps et al., 2024). The key asymptotic point is that the Serfling error decays exponentially in $\rho$06, while completeness remains meaningful because the model explicitly incorporates finite squeezing and finite-precision quadrature measurement. In this setting, the infinite-copy interpretation is not an ideal limit of unphysical states; it is the limit in which a realistic noisy protocol becomes arbitrarily reliable.

6. Literal infinite-copy containment and unresolved boundaries

The most literal realization of infinite-copy certification appears in the operator-algebraic study of exact embezzlement. Let

$\rho$07

where $\rho$08 is a catalyst state, $\rho$09 is a pure entangled target state, and $\rho$10 is built from local -isomorphisms. Then exact embezzlement of $\rho$11 is possible only if $\rho$12 **locally contains infinitely many copies of $\rho$13* (Liu, 5 Sep 2025). The copies are realized by recursively iterating $\rho$14 and $\rho$15, and they are mutually commuting. The same paper proves an equivalence: $\rho$16 For fixed pure $\rho$17, exact embezzlement is therefore equivalent to local infinite-copy containment. As a consequence, any universal exact embezzler must generate a Type III$\rho$18 von Neumann factor (Liu, 5 Sep 2025). This is not an asymptotic sample-complexity statement; it is a structural theorem about actual infinite tensor-product content.

The coexistence of these distinct uses of the phrase defines the current boundaries of the subject. In finite-dimensional state certification with collective measurements, the main remaining issue is exact instance-optimality: removing logarithmic losses and reconciling the lower- and upper-bound truncation rules (O'Donnell et al., 8 Jul 2025). In device-independent certification, certifying more than one leftover copy in the fully non-IID setting remains open (Gočanin et al., 2021). In lifted many-copy self-testing, robustness beyond exact statistics is largely unresolved (Šupić et al., 2019). In few-qubit certification of pure states, constant robustness using only single-qubit measurements remains open (Coladangelo et al., 12 Feb 2026). In the operator-algebraic line, an approximate or noise-robust analogue of infinite-copy containment is also open (Liu, 5 Sep 2025).

Taken together, these works suggest a precise encyclopedia-level conclusion. Infinite-Copy Certification is not a single theorem but a research area organized around one recurring question: how the ability to access, test, or internally represent arbitrarily many copies changes what can be certified. In the strongest finite-sample model, the answer is essentially complete and instance-wise: the right parameter is $\rho$19 (O'Donnell et al., 8 Jul 2025). In restricted and device-independent models, the answer becomes model-dependent and often asymptotic rather than literal. In the operator-algebraic setting, the phrase regains its most literal meaning: exact operational capability certifies actual infinite-copy internal structure (Liu, 5 Sep 2025).