Sample complexity of quantum resource testing via one-shot quantum blurring

Dmitry Grinko; Ludovico Lami

REVIEW 2 major objections 7 minor 35 references

Sample complexity of quantum resource testing via one-shot quantum blurring

T0 review · 2 major / 7 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Finite-copy bounds turn the generalised quantum Stein lemma into sample-complexity guarantees for resource testing.

desk verdict Solid one-shot upgrade of the GQSL that actually settles Fang–Hayashi and gives the first usable sample-complexity scaling; the long lemma chain holds up. read the letter →

arxiv 2607.24712 v1 pith:264JMBKF submitted 2026-07-27 quant-ph cs.ITmath-phmath.ITmath.MP

Dmitry Grinko , Ludovico Lami This is my paper

classification quant-phcs.ITmath-phmath.ITmath.MP MSC 81P4581P1894A17 PACS 03.67.-a03.67.Mn03.65.Ud

keywords quantumresourcetestinggeneralisedSteinlemmasamplecomplexityone-shotblurringregularisedrelativeentropyRényidivergencecontinuityentanglementmagic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper supplies the first rigorous finite-n estimates for quantum resource testing: discriminating many copies of a known resourceful state from every free (resourceless) state. Earlier work only controlled the asymptotic decay rate of the false-negative error; here that rate is made quantitative, with explicit correction terms that vanish as the number of copies grows. The resulting sample-complexity formula says that, for any fixed false-positive tolerance, the number of copies needed to drive the false-negative probability below δ scales as log(1/δ) divided by the regularised relative entropy of the resource. The same bounds settle a limit-commutation question for regularised Rényi divergences and, via the Brandão–Plenio correspondence, give concrete copy counts for distillation under asymptotically resource-non-generating operations. A sympathetic reader cares because the estimates turn an abstract asymptotic theorem into a practical recipe for how many copies an experiment or a distillation protocol actually needs.

What carries the argument

One-shot quantum blurring: a controlled replacement of a carefully chosen number of tensor factors by a free full-rank state, combined with a discrete polynomial approximant to the Kronecker delta and a variational Tikhonov regularisation that converts a large purification overlap into an operator inequality on the blurred state.

What would settle it

Exhibit a concrete Brandão–Plenio free-state family and a state ρ for which, inside the stated window on ε and n, every measurement with type-I error at most ε has type-II error larger than 2^{-D(ρ^{⊗n}∥F_n)} times the claimed penalty factor.

Watch

Extended reading notes

Core claim

Under the Brandão–Plenio axioms there exist constants such that, for all sufficiently large n and all type-I thresholds in an explicit window 2^{-n/240}⩽ε⩽ε₀, the hypothesis-testing relative entropy satisfies (1/n)D_H^ε(ρ^{⊗n}∥F_n)≥(1/n)D(ρ^{⊗n}∥F_n) minus an explicit penalty of order √( (1/n)log(1/ε) ) plus √ε. Consequently the sample complexity of asymmetric resource testing is Θ(log(1/δ)/D^∞(ρ∥F)) as δ o0, and the regularised Rényi relative entropies of the resource are continuous from below at α=1.

Load-bearing premise

The quantitative finite-n guarantee is proved only inside a restricted window on the type-I error and block length; outside that window the bound is not established, and the sample-complexity claim is recovered by choosing a smaller fixed tolerance that still lies inside the window.

Editorial extensions

If this is right

For fixed false-positive tolerance, O(log(1/δ)/D^∞(ρ∥F)) copies suffice to drive the false-negative probability below any δ o0.
Regularised Petz and sandwiched Rényi relative entropies of any Brandão–Plenio resource converge to the ordinary regularised relative entropy as α o1 from below.
Distilling k ebits under non-entangling operations requires only O(k/E(ρ)) copies of a bipartite state whose regularised relative entropy of entanglement is E(ρ).
The same finite-n estimates apply verbatim to magic-state testing and to any other resource theory obeying the five Brandão–Plenio axioms.
Exponential decay of both error types is simultaneously achievable with type-II exponent arbitrarily close to the Stein exponent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The explicit constants (blurring width ~√(s n), discrete-delta degree, universal κ≈0.19) can be numerically optimised for small fixed dimension, potentially yielding tighter laboratory sample-size prescriptions than the universal worst-case bounds.
Because the argument never uses more than the Brandão–Plenio axioms, any future resource theory that satisfies those axioms automatically inherits the same sample-complexity formula without further proof.
The one-shot blurring-plus-Tikhonov toolkit is likely reusable for other composite hypothesis-testing problems whose free sets are only known to be convex, permutation-invariant and full-rank.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real news is finite-n control, not another asymptotic restatement. They turn Lami’s blurring into a genuine one-shot tool (Tikhonov + discrete Chebyshev/Hahn delta + product-vector Kraus on the symmetric subspace) and get an explicit achievability bound: −log β_ε(ρ⊗n∥Fn) ≥ D(ρ⊗n∥Fn) − A n√ε − B √(n log 1/ε) inside a concrete window on ε and n. From that they close the α→1− / n→∞ commutation for regularised Rényi resource divergences and prove N_{ε,δ} = Θ(log(1/δ)/D∞) as δ→0. That is new, and it is what people actually need for entanglement/magic certification budgets.

The SI is complete and honest. Lemmas S4–S10 feed Theorem S1 cleanly; matching upper bounds via binary data processing lock the Stein and sample-complexity statements; they even include a weaker MSR-style bound and say so. Brandão–Plenio axioms and a full-rank free state are used as external structure, not smuggled conclusions. Self-citations to the asymptotic GQSL are tools with stated hypotheses.

Soft spots are real but proportionate. The quantitative bound only lives in a restricted window ((d²+2)² ≤ log(1/ε) ≤ n/240, hand-tuned κ and 240). They recover fixed-ε sample complexity by picking a smaller ε̃ inside the window, which is legitimate, but the constants are conservative and the upper-bound prefactor does depend on ρ via that choice (their “independent of ρ” claim is slightly overstated). A stress-check of the coefficient-norm step (S42) finds a loose intermediate inequality; the slack in (S43)–(S44) is huge, so the denominator condition and Theorem S1 still stand. The chain is long and not machine-checked—moderate, not high, confidence is fair.

This is for people who work on resource theories, finite-blocklength QI, or distillation/certification resource estimates. It deserves a serious referee. I would bring it to reading group and cite the sample-complexity and limit-commutation statements.

Referee Report

2 major / 7 minor

Summary. The paper studies asymmetric quantum resource testing: discriminating ρ^{⊗n} from an adversarially chosen free state σ_n ∈ F_n, for free-state families satisfying the Brandão–Plenio axioms. Its central result (Theorem 1 / Theorem S1) is the first one-shot achievability form of the generalised quantum Stein's lemma: −log β_ε(ρ^{⊗n}‖F_n) ≥ D(ρ^{⊗n}‖F_n) − An√ε − B√(n log(1/ε)), with explicit constants, in the regime (d²+2)² ≤ log(1/ε) ≤ n/240. From this the authors derive (i) the affirmative solution of the limit-commutation problem lim_{α→1⁻} D_α^∞(ρ‖F) = D^∞(ρ‖F), resolving the open problem of Fang–Hayashi [13] (Theorem 2 / S2), and (ii) the sample complexity N_{ε,δ}(ρ‖F) = Θ(log(1/δ)/D^∞(ρ‖F)) as δ→0 at fixed type-I threshold (Theorem 3 / S3), with matching converse from binary data processing. The technical engine is a finite-n version of Lami's quantum blurring, built from Tikhonov-regularised Kraus approximants on the symmetric subspace, a discrete Chebyshev delta approximant (Lemma S8), and Hahn-polynomial singular-value control (Lemma S9). The SI contains a complete, checkable proof chain (Lemmas S4–S10 → Theorem S1 → Theorems S2, S3) and an alternative, quantitatively weaker derivation from Mazzola–Sutter–Renner techniques (Section S.IV).

Significance. If the results hold — and my checking indicates they do, modulo the local (S42) repair — this is a significant contribution to finite-resource quantum information theory. It provides the first one-shot achievability formulation of the generalised quantum Stein's lemma with fully explicit constants (K_{d,τ}, κ ≈ 0.1926, the 240-regime), converts it into a quantitative sample-complexity statement with matching upper and lower bounds, and resolves the limit-commutation problem left open by Fang–Hayashi, unifying the error-exponent and Stein frameworks. The one-shot blurring toolkit (Tikhonov regularisation in variational form, discrete Chebyshev delta approximants, Hahn-polynomial singular-vector control) is genuinely new and likely reusable elsewhere. The proof chain is shipped completely and verifiably in the SI, and the authors include an honest quantitative comparison showing their bound beats what MSR-type techniques yield (s^{1/3} loss). The main caveats are the restricted ε-window, which limits the immediate practical bite of the finite-n guarantee, and the δ→0 nature of the Θ-statement.

major comments (2)

[S.II.A, proof of Theorem S1] Proof of Theorem S1, Eq. (S42): the middle-line estimate log(k+1) + (d²−2)log(k+d²−2) ≤ 2√(sn) does not follow from the displayed substitutions (S41). Inserting k+1, k+d²−2 ≤ 17√(sn) and using the cited log y ≤ √y bound gives a left side of order √17(d²−2)(sn)^{1/4}, which already at the boundary sn=(d²+2)², d=2 evaluates to ≈20 against a right side of 12, and scales like d³ versus d². The final line of (S42) appears repairable — a sharper binomial/cruder direct bound gives the term as ≲5.5(d²−2) ≤ 5.5√(sn), so the constant 8 becomes O(10) — and the quadratic-in-(sn) negative term in (S43) dwarfs any such linear-in-√(sn) constant, so no downstream statement should change. Nevertheless this line feeds the coefficient-norm control underlying the denominator condition of Lemma S7, on which Theorems S1–S3 all rest, so the derivation must be corrected explicitly rather than left as written.
[III, Theorem 1] Theorem 1 (main text) states the one-shot bound for all thresholds 2^{−n/240} ≤ ε ≤ ε0 without telling the reader that consistency with Theorem S1 forces ε0 ≤ 2^{−(d²+2)²}. The proved guarantee therefore only covers type-I thresholds exponentially small in d⁴; for a fixed, operationally natural ε and moderate local dimension d, Theorem 1 is vacuous, and Theorem 3's fixed-ε sample complexity is recovered only by shrinking to an auxiliary ε̃ inside the window ((S78)–(S82)). The window (d²+2)² ≤ log(1/ε) ≤ n/240 should be stated in Theorem 1 itself, and the Discussion's practical framing (certification of entanglement sources, magic-state factories) should be calibrated accordingly — ideally with a short worked regime example (what n, d the bound covers at a given ε). As written, a reader of the main text alone would overestimate the scope of the finite-n guarantee.

minor comments (7)

[S.II.B, Theorem S3] Theorem S3 states that the Θ constants 'only depend on the underlying resource theory and on ε, and not on the state ρ'. However ε̃ in (S78) is chosen using D^∞(ρ‖F), so the additive threshold (S79) — and hence the onset of the Θ regime in δ — is state-dependent. The claim is true for the asymptotic leading constants (4/D^∞ and (1−ε)/(2D^∞)); please make the quantifiers precise.
[S.II.A, proof of Theorem S2] In the proof of Theorem S2, the sentence 'We next identify the limiting hypothesis-testing exponent. Fix 0 < ε ≤ 1/2, fix σ_n ∈ F_n, and let T satisfy Tr ρ^{⊗n}T ≥ 1−ε. Binary data processing for the relative entropy gives' is immediately repeated verbatim before Eq. (S63); delete the duplicated fragment.
[Supplementary Information, title] The Supplementary Information is titled 'Sample complexity of entanglement testing via one-shot quantum blurring', whereas the paper concerns general quantum resource testing; please align the SI title with the manuscript title.
[S.II, dependency diagram] The proof-dependency diagram at the start of Section S.II contains a broken cross-reference ('Thms. S3 and ??').
[Throughout] Several typos: 'goverened' (§III), 'a is a penalty term' (§III), 'an thus' (§I), 'approriate' (§I), 'disscussions' (Acknowledgements), 'obtacle' (§IV), and 'even it was true' (twice in §II.B; should be 'even if it was true').
[Abstract / §I] The abstract and introduction's 'first rigorous finite-n bounds' should be qualified in light of the authors' own Note added and Section S.IV, where estimates of the same type (with a worse s^{1/3} correction) are derived from the techniques of [20]; a sentence in the introduction clarifying what is new (the one-shot blurring method and the optimal √s profile) would make the priority claim precise.
[S.III, Figure S2] Figure S2(b): the caption notes the polynomial exceeds the band by ≈33× on [1,L]; since the guarantee (S110) applies only at integer points, a half-sentence in the caption explaining that this overshoot between enforced zeros is expected would help readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: finite-n bounds are derived from Brandão–Plenio axioms plus new one-shot blurring lemmas, not assumed by definition or fit.

full rationale

This is a self-contained mathematical achievability paper. The target quantities (one-shot D_H^ε vs D(ρ^{⊗n}∥F_n), sample complexity N_{ε,δ}, and lim_{α→1−} D_α^∞ = D^∞) are not inputs to the argument. D^∞ and the Brandão–Plenio axioms are prior external structure; Theorems S1–S3 are proved from them via an explicit construction (blurring + Tikhonov/Kraus approximants + discrete Chebyshev delta, Lemmas S5–S10). Self-citations to Lami’s asymptotic GQSL and standard continuity/de Finetti lemmas supply tools with stated hypotheses; they do not restate or force the finite-n correction terms. There is no data fitting, no uniqueness theorem imported to forbid alternatives, and no renaming of a known empirical pattern. The restricted window on ε and n is an explicit hypothesis of the proof, not a circular premise. Honest finding: derivation chain is independent of the claimed conclusions.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central claims rest on the standard Brandão–Plenio free-state axioms, finite-dimensional quantum hypothesis-testing divergences, and several analytic approximation lemmas constructed in the paper. No data-fitted constants. The main modeling commitment is that free sets obey BP axioms (including a full-rank free state); the main proof commitments are the restricted (ε,n) window and the explicit polynomial/blurring parameter choices.

free parameters (2)

Universal Chebyshev constant κ = (15−2π)/(32√2) ≈ 0.1926 = ≈0.1926
Chosen in Lemma S8 via α=1/16, β=1−α−(π/2)√α to keep deg q_r ≤ r; not fitted to data, but a hand-selected sufficient constant that enters all quantitative rates.
Blurring/regime constants (240 in k=⌊√(240 s n)⌋, s≤1/240, K_{d,τ}=16(2 log d + log(1/c_τ)+1)) = 240; 16; s≤1/240
Sufficient numerical slack so that coefficient-norm and denominator conditions in Lemmas S7/S10 hold; determine the explicit finite-n window, not physical fits.

assumptions (5)

domain assumption Brandão–Plenio axioms (S13)–(S17): F_n closed convex; full-rank free τ≥c_τ 1/d; partial-trace stability; tensor stability; permutation invariance.
Assumed throughout; needed for free-set preservation under blurring and for finiteness of D^∞.
standard math Finite-dimensional Hilbert space and standard properties of Umegaki/Petz/sandwiched/hypothesis-testing relative entropies (data processing, Fuchs–van de Graaf, Fekete subadditivity).
Background QI/math used in §§S.I–S.II.
standard math Existence of symmetric purifications with fidelity overlap (Lemma S4 / Brandão–Plenio Lemma III.4).
Imported lemma used to lift smoothed states before blurring.
standard math One-shot comparison D_H^ε ≥ D_max^{√(1−ε)} + log 1/(1−ε) and additive-defect substate construction (Regula–Lami–Datta).
Cited smooth-entropy tools converting operator inequalities into hypothesis-testing bounds.
ad hoc to paper Restricted operating regime (d²+2)² ≤ log(1/ε) ≤ n/240 for the one-shot theorem.
Proof-only constraint enabling the polynomial degree and blurring width choices; applications for fixed ε reduce into this regime by choosing smaller ε̃.

invented entities (1)

One-shot quantum blurring with Tikhonov-regularised Kraus approximants on the symmetric subspace
purpose: Convert a fidelity overlap with a symmetric purification into a quantitative operator inequality controlling Tr(·)_+ against free blurred states at finite n.
Refines Lami’s asymptotic blurring; the finite-n control via discrete delta approximants and variational Tikhonov form is the paper’s core technical device.

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Cite this review

Pith. "Pith review of Sample complexity of quantum resource testing via one-shot quantum blurring." pith.science (2026). https://pith.science/paper/264JMBKF

@misc{pith2026260724712,
  author       = {Pith},
  title        = {Pith review of: Sample complexity of quantum resource testing via one-shot quantum blurring},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/264JMBKF}},
  note         = {Machine review of arXiv:2607.24712}
}

abstract

Quantum resource testing is a fundamental primitive of quantum information processing, profoundly connected to resource manipulation. Its goal is to discriminate $n$ copies of a given resourceful state $\rho$ from all free (i.e., resourceless) states; key instances for applications are entanglement testing and quantum magic testing. The asymptotic characterisation relies on the recently proven generalised quantum Stein's lemma, which establishes the rate of decay of the false negative error probability for a fixed false positive error probability. This result, however, is intrinsically asymptotic and thus can provide no finite-resource guarantees, which makes its practical implications unclear. Here, we establish the first rigorous finite-$n$ bounds on quantum resource testing and hence quantum resource manipulation, providing explicit estimates on the number of copies needed to achieve a prescribed performance. As notable consequences, we obtain (a) the convergence of the regularised R\'enyi relative entropies of a resource, which settles the important open problem from [Fang/Hayashi, IEEE ToIT 72:6, 2026]; and (b) the first sample-complexity bound for asymmetric resource testing: for any fixed false positive error probability, a false negative error probability of at most $\delta$ can be achieved with $n=O\left(\frac{\log(1/\delta)}{D^\infty(\rho\|F)}\right)$ copies of $\rho$, in the limit where $\delta \to 0$.

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Works this paper leans on

35 extracted references · 2 linked inside Pith

[13]

ThegeneralizedquantumStein’slemmaandthesecondlaw of quantum resource theories.Nature Physics, 21(12):1988–1993, 2025

MasahitoHayashiandHayataYamasaki. ThegeneralizedquantumStein’slemmaandthesecondlaw of quantum resource theories.Nature Physics, 21(12):1988–1993, 2025. 2, 5, 6, 7

1988
[1]

For every𝑛, the setF𝑛 is closed and convex
[2]

There exists a full-rank state𝜏∈F1, so that𝜏≥𝑐 𝜏 1 𝐴 𝑑 for some constant𝑐𝜏 >0, where1 𝐴/𝑑is the maximally mixed state on the𝑑-dimensional system𝐴
[3]

Free states are stable under the operation of discarding subsystems: whenever𝜎∈F𝑛+1 is free, its marginal on the first𝑛copies is also free, i.e.Tr𝑛+1 𝜎∈F 𝑛, whereTr𝑛+1 denotes the trace over the last copy
[4]

it belongs toF𝑛+𝑚

Free states can be combined by tensoring: if𝜎∈F𝑛 and𝜎′∈F𝑚, then𝜎⊗𝜎 ′ is also free, i.e. it belongs toF𝑛+𝑚
[5]

Beyond IID in Information Theory 14

The sets of free states are invariant under relabelling of the copies. More precisely, for every 𝜎∈F 𝑛 and every permutation𝜋∈𝑆 𝑛, one has𝑈𝜋𝜎𝑈† 𝜋∈F 𝑛, where𝑈𝜋 is the unitary that permutes the copies of the system𝐴. Entanglement theory [14] and the resource theory of ‘quantum magic’ [15] are two major examplesofquantumresourcetheories. Theformer, forinstan...

2026
[6]

Universal quantum computation with ideal Clifford gates and noisy ancillas.Physical Review A, 71:022316, 2005

Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas.Physical Review A, 71:022316, 2005. 1

2005
[7]

Quantum resource theories.Reviews of Modern Physics, 91:025001,

Eric Chitambar and Gilad Gour. Quantum resource theories.Reviews of Modern Physics, 91:025001,

Show all 35 references

[8]

Fernando G. S. L. Brandão and Martin B. Plenio. Entanglement theory and the second law of thermo- dynamics.Nature Physics, 4:873–877, 2008. 2, 5

2008
[9]

Fernando G. S. L. Brandão and Martin B. Plenio. A reversible theory of entanglement and its relation to the second law.Communications in Mathematical Physics, 295(3):829–851, 2010. 2, 3, 5

2010
[10]

The tangled state of quantum hypothesis testing.Nature Physics, 20:172–175,

MarioBerta,FernandoG.S.L.Brandão,GiladGour,LudovicoLami,MartinB.Plenio,BartoszRegula, and Marco Tomamichel. The tangled state of quantum hypothesis testing.Nature Physics, 20:172–175,
[11]

Plenio, Michael A

Vlatko Vedral, Martin B. Plenio, Michael A. Rippin, and Peter L. Knight. Quantifying entanglement. Physical Review Letters, 78:2275–2279, 1997. 2

1997
[12]

On a gap in the proof of the generalised quantum Stein’s lemma and its consequences for the reversibility of quantum resources.Quantum, 7:1103, 2023

MarioBerta,FernandoG.S.L.Brandão,GiladGour,LudovicoLami,MartinB.Plenio,BartoszRegula, and Marco Tomamichel. On a gap in the proof of the generalised quantum Stein’s lemma and its consequences for the reversibility of quantum resources.Quantum, 7:1103, 2023. 2 9

2023
[14]

AsolutionofthegeneralizedquantumStein’slemma.IEEETransactionsinInformation Theory, 71(6):4454–4484, 2025

LudovicoLami. AsolutionofthegeneralizedquantumStein’slemma.IEEETransactionsinInformation Theory, 71(6):4454–4484, 2025. 2, 3, 5, 6, 7, 8, 22, 35

2025
[15]

RenatoRenner.Securityofquantumkeydistribution.PhDthesis,ETHZurich,2005.PreprintarXiv:quant- ph/0512258. 2, 34

2005
[16]

Springer, 2015

Marco Tomamichel.Quantum Information Processing with Finite Resources: Mathematical Foundations. Springer, 2015

2015
[17]

Wilde.Principles of Quantum Communication Theory: A Modern Approach

Sumeet Khatri and Mark M. Wilde.Principles of Quantum Communication Theory: A Modern Approach
[18]

Error exponents of quantum state discrimination with composite correlated hypotheses.IEEE Transactions on Information Theory, 72(6):4140–4165, 2026

Kun Fang and Masahito Hayashi. Error exponents of quantum state discrimination with composite correlated hypotheses.IEEE Transactions on Information Theory, 72(6):4140–4165, 2026. 2, 3, 5, 6, 7

2026
[19]

Reinhard F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden- variable model.Physical Review A, 40:4277–4281, 1989. 3

1989
[20]

The resource theory of stabilizer quantum computation.New Journal of Physics, 16(1):013009, 2014

Victor Veitch, Seyed Ali Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. The resource theory of stabilizer quantum computation.New Journal of Physics, 16(1):013009, 2014. 3

2014
[21]

Conditional expectation in an operator algebra, IV (entropy and information)

Hisaharu Umegaki. Conditional expectation in an operator algebra, IV (entropy and information). K¯odai Mathematical Seminar Reports, 14(2):59–85, 1962. 4, 10

1962
[22]

The proper formula for relative entropy and its asymptotics in quantum probability.Communications in Mathematical Physics, 143:99–114, 1991

Fumio Hiai and Dénes Petz. The proper formula for relative entropy and its asymptotics in quantum probability.Communications in Mathematical Physics, 143:99–114, 1991. 4, 10

1991
[23]

AgeneralizationofquantumStein’slemma.Commu- nications in Mathematical Physics, 295:791–828, 2010

FernandoG.S.L.BrandãoandMartinB.Plenio. AgeneralizationofquantumStein’slemma.Commu- nications in Mathematical Physics, 295:791–828, 2010. 5, 13, 22, 34

2010
[24]

Quasi-entropies for finite quantum systems.Reports on Mathematical Physics, 23(1):57–65,

Denes Petz. Quasi-entropies for finite quantum systems.Reports on Mathematical Physics, 23(1):57–65,
[25]

Robust generalized quantum Stein’s lemma, 2026

Giulia Mazzola, David Sutter, and Renato Renner. Robust generalized quantum Stein’s lemma, 2026. 8, 32, 34

2026
[26]

Thequantumcapacityofchannelswitharbitrarilycorrelated noise.IEEE Transactions on Information Theory, 56(3):1447–1460, 2010

FrancescoBuscemiandNilanjanaDatta. Thequantumcapacityofchannelswitharbitrarilycorrelated noise.IEEE Transactions on Information Theory, 56(3):1447–1460, 2010. 11

2010
[27]

Generalized quantum asymptotic equipartition.Preprint arXiv:2411.04035, 2025

Kun Fang, Hamza Fawzi, and Omar Fawzi. Generalized quantum asymptotic equipartition.Preprint arXiv:2411.04035, 2025. 12

2025 arXiv
[28]

Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganz- zahligen Koeffizienten.Mathematische Zeitschrift, 17(1):228–249, 1923

Michael Fekete. Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganz- zahligen Koeffizienten.Mathematische Zeitschrift, 17(1):228–249, 1923. 13

1923
[29]

Aram W. Harrow. The church of the symmetric subspace.Preprint arXiv:1308.6595, 2013. 14

2013 arXiv
[30]

16, 18, 33

BartoszRegula,LudovicoLami,andNilanjanaDatta.Tightrelationsandequivalencesbetweensmooth relative entropies, 2025. 16, 18, 33

2025
[31]

AndreasWinter.Tightuniformcontinuityboundsforquantumentropies: conditionalentropy,relative entropy distance and energy constraints.Communications in Mathematical Physics, 347:291–313, 2016. 18, 35

2016
[32]

Lesky, and René F

Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw.Hypergeometric Orthogonal Polynomials and Their𝑞-Analogues. Springer Monographs in Mathematics. Springer, 2010. 29

2010
[33]

Symmetry of large physical systems implies independence of subsystems.Nature Physics, 3(9):645–649, 2007

Renato Renner. Symmetry of large physical systems implies independence of subsystems.Nature Physics, 3(9):645–649, 2007. 34

2007
[34]

Donald and Michał Horodecki

Matthew J. Donald and Michał Horodecki. Continuity of relative entropy of entanglement.Physics Letters A, 264(4):257–260, 1999. 35

1999
[35]

PhD thesis, University of Cambridge, 2006

Matthias Christandl.The Structure of Bipartite Quantum States: Insights from Group Theory and Cryptog- raphy. PhD thesis, University of Cambridge, 2006. 35 10 Supplementary Information: Sample complexity of entanglement testing via one-shot quantum blurring CONTENTS S.I. The s...

2006

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.

[1] [13]

ThegeneralizedquantumStein’slemmaandthesecondlaw of quantum resource theories.Nature Physics, 21(12):1988–1993, 2025

MasahitoHayashiandHayataYamasaki. ThegeneralizedquantumStein’slemmaandthesecondlaw of quantum resource theories.Nature Physics, 21(12):1988–1993, 2025. 2, 5, 6, 7

1988

[2] [1]

For every𝑛, the setF𝑛 is closed and convex

[3] [2]

There exists a full-rank state𝜏∈F1, so that𝜏≥𝑐 𝜏 1 𝐴 𝑑 for some constant𝑐𝜏 >0, where1 𝐴/𝑑is the maximally mixed state on the𝑑-dimensional system𝐴

[4] [3]

Free states are stable under the operation of discarding subsystems: whenever𝜎∈F𝑛+1 is free, its marginal on the first𝑛copies is also free, i.e.Tr𝑛+1 𝜎∈F 𝑛, whereTr𝑛+1 denotes the trace over the last copy

[5] [4]

it belongs toF𝑛+𝑚

Free states can be combined by tensoring: if𝜎∈F𝑛 and𝜎′∈F𝑚, then𝜎⊗𝜎 ′ is also free, i.e. it belongs toF𝑛+𝑚

[6] [5]

Beyond IID in Information Theory 14

The sets of free states are invariant under relabelling of the copies. More precisely, for every 𝜎∈F 𝑛 and every permutation𝜋∈𝑆 𝑛, one has𝑈𝜋𝜎𝑈† 𝜋∈F 𝑛, where𝑈𝜋 is the unitary that permutes the copies of the system𝐴. Entanglement theory [14] and the resource theory of ‘quantum magic’ [15] are two major examplesofquantumresourcetheories. Theformer, forinstan...

2026

[7] [6]

Universal quantum computation with ideal Clifford gates and noisy ancillas.Physical Review A, 71:022316, 2005

Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas.Physical Review A, 71:022316, 2005. 1

2005

[8] [7]

Quantum resource theories.Reviews of Modern Physics, 91:025001,

Eric Chitambar and Gilad Gour. Quantum resource theories.Reviews of Modern Physics, 91:025001,