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REVIEW 5 major objections 5 minor 39 references

Ensuring superior learning outcomes and data security for authorized learner

T0 review · 5 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that when the authorized learner's measured noise falls below a critical threshold in a quantum label-encoding protocol, no eavesdropper can be guaranteed equally good PAC learning, and the condition can be checked from…

desk verdict Central theorem unproven: the key tradeoff is imported from QKD, the proof has a sign error, and the δ-comparison does not follow; the learning-probability framing and experiments are suggestive but not validating. read the letter →

arxiv 2501.00754 v1 pith:EKP7QFEE submitted 2025-01-01 stat.ML cs.CRquant-ph

Jeongho Bang , Wooyeong Song , Kyujin Shin , Yong-Su Kim This is my paper · ORCID
classification stat.MLcs.CRquant-ph MSC 68Q3268T0581P68
keywords PAClearningquantumlabelencodingprobabilitydatasecurityeavesdroppernoisysamplecomplexityCNNimageclassificationkeydistribution
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 sets out to prove that an authorized learner can be guaranteed a better learning outcome than any eavesdropper when training labels are encoded in qubits. It works in the probably-approximately-correct (PAC) framework and introduces learning probability as a measurable stand-in for PAC confidence. The central result is a threshold condition: if the authorized learner measures a label noise $\eta_A$ below a critical value $\eta^\star$, then no eavesdropper can be guaranteed to match the authorized learner's accuracy and confidence simultaneously. What makes this practical is that the guarantee is certified using only quantities the authorized learner can measure about their own dataset, namely its size $|\Xi_{Q,A}|$ and noise $\eta_A$. The authors validate the prediction with CNN image classification, where decreasing $\eta_A$ widens the gap in learning probability between authorized and eavesdropping learners.

What carries the argument

The load-bearing object is quantum label encoding, a classical-quantum hybrid encoding in which each label $c(x)$ is stored as a qubit state $|c(x)\rangle$ rather than a classical bit. Around it, the paper builds a transmission protocol in which the authorized learner sends random states $|0\rangle$, $|1\rangle$, $|\pm\rangle$ to a data center; the data center returns the label-encoded state for $k=0,1$ or an unchanged $\pm$ test state, and the authorized learner uses the $\pm$ states to estimate their own noise $\eta_A$ via Eq. (19). The learning-probability function $P_L(|\Theta|,\epsilon)$ links the confidence $1-\delta$ in PAC learning to a directly measurable cumulative distribution over training-data consumption. The threshold $\eta^\star$ is imported from quantum key distribution security analyses, where information gain and disturbance trade off, and the proof combines this with the noisy sample-complexity bound $M_{b,\eta} = \frac{2}{\epsilon^2(1-2\eta)^2}\ln(2|H|/\delta)$.

What would settle it

Simulate the proposed protocol with an eavesdropper employing the optimal collective attack and directly measure both $\eta_A$ and $\eta_E$; the central claim collapses if any parameter regime shows $\eta_A < \eta^\star$ yet $\gamma_E|\Xi_{Q,E}| \ge \gamma_A|\Xi_{Q,A}|$, or if CNN trials reveal an eavesdropper consistently matching the authorized learner's accuracy and confidence in that regime.

Watch

Extended reading notes

Core claim

On the paper's own terms, Theorem 2 is the discovery: for a quantum protocol satisfying the threshold condition $(\eta_A < \eta^\star) \wedge (\eta_E < \eta^\star)$, when the authorized learner's noise lies below $\eta^\star$, the PAC lower bound $\delta^\star_A = e^{-\gamma_A |\Xi_{Q,A}|}$ is strictly smaller than $\delta^\star_E = e^{-\gamma_E |\Xi_{Q,E}|}$ with $\gamma = \epsilon^2(1-2\eta)^2/2$ and equal accuracies. An eavesdropper can therefore never be assured PAC quality $(\epsilon_E \le \epsilon_A) \wedge (\delta_E \le \delta_A)$ in the same data transmission, though a single lucky hypothesis is not ruled out. The condition depends only on authorized-side measurements, so the authorized learner can certify the eavesdropper's limitation from their own noisy dataset.

Load-bearing premise

The load-bearing premise is that the authorized learner's large-dataset quality $\gamma_A|\Xi_{Q,A}|$ exceeds the eavesdropper's whenever $\eta_A < \eta^\star$; the paper imports this noise-quality tradeoff from quantum key distribution security analyses instead of deriving it for the label-encoding protocol.

Editorial extensions

If this is right

  • An authorized learner can certify, from their own dataset size and noise level alone, that eavesdroppers are not guaranteed equally good PAC learning.
  • The guarantee is about assurance rather than absolute prohibition: an eavesdropper might still return a good hypothesis, but no PAC-style promise covers it.
  • Tighter noisy sample-complexity bounds would upgrade Theorem 2 from 'not ensured' to 'forbidden', as the authors state in the conclusion.
  • Lowering $\eta^\star$ by improving the encoding protocol would shrink the learning quality available to eavesdroppers.
  • CNN experiments show the authorized-vs-eavesdropper gap in learning probability grows as measured $\eta_A$ decreases.

Reading between the lines

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

  • Beyond the paper's claims, a natural next test is to derive the $\eta_A$–$\eta_E$ tradeoff analytically for the label-encoding protocol itself, rather than importing thresholds from quantum key distribution; that would determine whether Theorem 2 survives outside the QKD regime.
  • The learning-probability formulation could serve as a practical audit tool: a model provider can report a PAC-style certificate from the empirical cumulative distribution of data consumption, independent of any adversary.
  • Robust models with ample data will erode the numerical gap (the paper already observes this for NASNetLarge), so the security guarantee should be understood as worst-case PAC assurance, not a universal accuracy separation.
  • The same certification logic might extend to other encodings, such as continuous-variable labels or multi-level quantum labels, if a corresponding noise threshold can be identified.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper proposes a framework for guaranteeing that an authorized learner achieves better PAC learning outcomes than an eavesdropper, using quantum label encoding. It defines a 'learning probability', connects it to PAC confidence, derives a corollary relating dataset size, label noise, accuracy, and confidence (Eq. (7)), and states Theorem 2 claiming that if a protocol satisfies Eq. (15), the authorized learner can be guaranteed a better PAC guarantee than any eavesdropper. The authors implement a quantum label transmission protocol and test with CNNs on a cats/dogs image classification task, reporting learning-probability curves for authorized and eavesdropping learners under different noise levels.

Significance. If the main theorem were proved, this would be a striking result connecting quantum information-theoretic security with PAC learnability, with potential applications in secure machine learning. The paper deserves credit for making the central claim precise enough to be checked, for specifying a concrete protocol, and for quantitative experiments with 150 runs per condition across three CNN models. However, the theorem's key implication (Eq. (18)) is unproven; the suppression of the model-complexity term in Eqs. (6)-(7) invalidates the derived PAC bound; and the experiments largely restate that label noise degrades CNN accuracy. As it stands, the central contribution is not established.

major comments (5)
  1. [Sec. 2.1, Eqs. (6) and (7)] The derivation from Eq. (5) to Eq. (6) drops the model-complexity factor. From |Theta| >= M_{b,η} = [2/(ε^2(1−2η)^2)] ln(2|H|/δ), the correct implication is δ >= 2|H| exp(−ε^2(1−2η)^2|Theta|/2), not δ >= exp(−ε^2(1−2η)^2|Theta|/2). Consequently Eq. (7) is not a valid lower bound on δ, and every later comparison of δ* values in Eqs. (16) and (18) inherits this error.
  2. [Sec. 2.3, Theorem 2 proof, Eq. (18)] The proof asserts that 'from Eq. (14) and Eq. (15), we can prove that η_A < η⋆ ⇒ δ⋆_A < δ⋆_E', but this is not derived. Eq. (14) gives |Ξ_A| >= |Ξ_E|, and Eq. (15) gives η_A, η_E < η⋆ without ordering η_A and η_E. Since δ⋆_j = exp(−ε_j^2(1−2η_j)^2|Ξ_j|/2), with ε_E = ε_A the inequality δ⋆_A < δ⋆_E requires (1−2η_A)^2|Ξ_A| > (1−2η_E)^2|Ξ_E|, which neither Eq. (14) nor Eq. (15) implies. The proof substitutes the desired conclusion for a derivation.
  3. [Sec. 2.3, Theorem 2 statement] The theorem states that 'if η_A > η⋆ is secured from Ξ_Q,A, there is no condition that ensures LE becomes a (ϵ_E, δ_E)-PAC learner satisfying (ϵ_E <= ϵ_A) ∧ (δ_E <= δ_A)', but the proof uses η_A < η⋆ in Eq. (18). This is an internal inconsistency: the condition in the theorem and the condition in the proof are opposite. The statement must be corrected before the claim can be evaluated.
  4. [Sec. 3.1 and 3.2, experimental validation] The experimental validation is largely circular. The protocol generates Ξ_Q,A and Ξ_Q,E from the same noise model, and the simulations compare learners with η_A = 0.01, 0.03, and 0.05, all below η⋆ ≈ 0.11, while η_E is not measured (Sec. 3.1 states 'the value of η_E cannot be estimated in S_E(P)'). Thus the observed gap between LA and LE learning curves only demonstrates that lower label noise yields better CNN accuracy; it does not test the quantum information-disturbance tradeoff that Eq. (18) requires. Moreover, the text acknowledges cases where LE's learning probability 'appears similar to, or even higher than, those of LA' (Sec. 3.2, NNL with large data), which contradicts the guaranteed-superiority claim in Theorem 2 and is not reconciled.
  5. [Sec. 2.3, Proposition 1 and Sec. 3.1, Conjecture 1] The existence of a protocol P and threshold η⋆ satisfying Eq. (15) is imported from quantum key distribution results (Ref. [31]) and is not proven in this PAC learning framework. Conjecture 1, which links Holevo's condition to learning outcome quality, is explicitly a conjecture and is not demonstrated. Since Theorem 2 depends on both, the paper's central claim is conditional on unproved external assumptions that are not stated as theorems or formally connected to the PAC sample-complexity analysis.
minor comments (5)
  1. [Sec. 2.1, Eq. (4)] Theorem 1 is stated as an 'iff' with |Theta| >= (1/ε) ln(|H|/δ). This is the realizable-case sample complexity for a finite hypothesis class; the relationship to the constant 2 in Eq. (5) is not explained, and the sense in which it is tight should be clarified.
  2. [Sec. 3.2, Fig. 2 caption] The caption says '(a) for ε_T = 0.03 and (c) ε_T = 0.01', but the learning-probability panels are (b) and (c); the parenthetical labels are inconsistent with the figure.
  3. [Sec. 1, Introduction] There is a typo in 'Hilbet-space'; it should read 'Hilbert-space'.
  4. [Sec. 2.2, Definition 3 and Remark 1] The identification of the learning probability with the PAC confidence 1−δ is asserted via Remark 1, but the random test model in Eq. (8) is a heuristic. The connection between the rate parameter ξ and the PAC sample-complexity bound is not formalized, so calling P_L a 'measurable physical quantity' is stronger than what is demonstrated.
  5. [Sec. 2.3, Eq. (14)] The relation |Ξ_Q,E| <= |Ξ_Q,A| is described as 'generally assumed', but in the collective-attack simulations the paper sets |Θ_Q| = |Ξ_Q,A| = |Ξ_Q,E|, making Eq. (14) an equality. The dataset-size advantage in Eq. (14) is therefore not tested by the experiments.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 2's key implication Eq. (18) is asserted without derivation and reduces to an assumed noise ordering between authorized and eavesdropping learners, making the central PAC-superiority claim partially circular.

  1. self definitional [Section 2.3, Theorem 2 proof, Eq. (18)]
    "The proof of this theorem is straightforward. At first, let ϵE = ϵA. Then, from Eq. (14) and Eq. (15), we can prove that ηA < η⋆ ⇒ δ⋆A < δ⋆E."

    With δ⋆ defined in Eq. (16) as e^{-γ|Ξ|}, γ = ϵ²(1−2η)²/2, the claimed inequality δ⋆A < δ⋆E is equivalent to γ_A|Ξ_Q,A| > γ_E|Ξ_Q,E|. Under the proof's setting ϵ_E = ϵ_A and Eq. (14) |Ξ_Q,E| ≤ |Ξ_Q,A|, this reduces to the requirement η_A < η_E (more precisely γ_A > γ_E). But Eq. (15) only states η_A < η⋆ and η_E < η⋆; it imposes no ordering between η_A and η_E. The implication (18) therefore smuggles the noise-ordering assumption into the theorem, and the conclusion that LE cannot be guaranteed (δ_E ≤ δ_A) is equivalent to that assumed ordering rather than derived from the protocol premises.

full rationale

Score 6: The central PAC-superiority claim (Theorem 2) is not derived from its stated premises. The proof's key implication, Eq. (18), is asserted 'from Eq. (14) and Eq. (15),' but those equations only give |Ξ_A| ≥ |Ξ_E| and η_A, η_E < η⋆. Since δ⋆ = exp(−ϵ²(1−2η)²|Ξ|/2), the inequality δ⋆_A < δ⋆_E is algebraically equivalent to γ_A|Ξ_A| > γ_E|Ξ_E|, which (for ϵ_E = ϵ_A and Eq. (14)) requires η_A < η_E. That noise ordering is not a consequence of the stated assumptions; it is the conclusion being assumed. The QKD information-disturbance tradeoff cited from Ref. [31] is invoked informally and is not translated into a formal PAC premise, so the theorem's guarantee reduces to an unverified input rather than a derivation. The CNN experiments are self-contained and reproduce the well-known effect that higher label noise degrades learning, but they do not independently establish the quantum security claim. I find no load-bearing self-citation: Ref. [19] supplies the protocol, but Eq. (15)'s existence is grounded in no-cloning/disturbance arguments. The theorem statement also contains an internal mismatch (η_A > η⋆ in the statement vs. η_A < η⋆ in the proof), which is a correctness defect noted separately from circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper does not introduce new physical entities. The conceptual 'learning probability' is a definition, not a postulated entity. The main load-bearing inputs are the free parameters (ξ, ε_T, η_A, η⋆) and imported standard results from learning theory and quantum information.

free parameters (4)
  • ξ (rate parameter in random test model)
    The learning probability model PL ≈ 1 − e^{−ξ|Θ|} in Eq. (8) introduces ξ as a rate parameter characterizing the random test model. It is not derived from first principles and is not fitted to data; it is an ad hoc model parameter.
  • ε_T (target accuracy threshold) = 0.01 and 0.03
    The CNN experiments set target inaccuracy thresholds ε_T = 0.01 and 0.03 to define successful learning. These are chosen constants, not derived.
  • η_A (authorized learner's noise) = 0.01, 0.03, 0.05
    In the CNN eavesdropping simulations, η_A is set to 0.01, 0.03, and 0.05. These values are selected by the authors to demonstrate the predicted gap and are not derived from the protocol.
  • η⋆ (critical noise threshold) = 0.11 (collective attacks), 0.154 (memoryless), 0.146 (individual)
    The critical threshold η⋆ is imported from quantum key distribution security analysis (Ref. [31]); it is an input parameter, not derived in this paper.
assumptions (5)
  • standard math PAC sample complexity theorem (Theorem 1) and noisy sample complexity bound (Eq. 5 from Angluin and Slonim)
    Section 2.1 relies on Theorem 1 and Eq. (5) as given; these are standard results in computational learning theory.
  • domain assumption The premise (Eq. 9) that any learning algorithm has PL ≥ P^rs_L, i.e., is at least as good as random selection
    Section 2.2, Eq. (9): the paper assumes all learning algorithms are superior to random selection, which underlies the use of the random test model as a worst-case bound.
  • standard math Quantum no-cloning theorem and information-disturbance tradeoff
    Proposition 1's validity is grounded in the no-cloning theorem (Refs. [32,33]) and the information-disturbance tradeoff (Refs. [29,34]).
  • domain assumption Assumptions A.1-A.3 about the eavesdropper's capabilities: no influence on input choices, no access to devices, obey quantum mechanics
    Section 2.3, items [A.1]-[A.3]: these assumptions bound the eavesdropper's power and are necessary for the security argument.
  • ad hoc to paper Conjecture 1 (Holevo condition): higher extractable information implies better learning outcome
    Section 3.1, Conjecture 1: the paper conjectures that IΘA ≥ max IΘE guarantees higher learning quality; this is intuitive but not proven and is used to identify η⋆.

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Pith. "Pith review of Ensuring superior learning outcomes and data security for authorized learner." pith.science (2026). https://pith.science/paper/EKP7QFEE

@misc{pith2026250100754,
  author       = {Pith},
  title        = {Pith review of: Ensuring superior learning outcomes and data security for authorized learner},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EKP7QFEE}},
  note         = {Machine review of arXiv:2501.00754}
}
read the original abstract

The learner's ability to generate a hypothesis that closely approximates the target function is crucial in machine learning. Achieving this requires sufficient data; however, unauthorized access by an eavesdropping learner can lead to security risks. Thus, it is important to ensure the performance of the "authorized" learner by limiting the quality of the training data accessible to eavesdroppers. Unlike previous studies focusing on encryption or access controls, we provide a theorem to ensure superior learning outcomes exclusively for the authorized learner with quantum label encoding. In this context, we use the probably-approximately-correct (PAC) learning framework and introduce the concept of learning probability to quantitatively assess learner performance. Our theorem allows the condition that, given a training dataset, an authorized learner is guaranteed to achieve a certain quality of learning outcome, while eavesdroppers are not. Notably, this condition can be constructed based only on the authorized-learning-only measurable quantities of the training data, i.e., its size and noise degree. We validate our theoretical proofs and predictions through convolutional neural networks (CNNs) image classification learning.

Figures

Figures reproduced from arXiv: 2501.00754 by the authors.

Figure 1
Figure 1. A schematic of the protocol P. LA prepares a qubit state |k⟩ (k = 0, 1, ±) and send it to D via CQ. The state |k⟩ is passed through a function F with a chosen x. F generate a pair (x, |c(x) ⊕ k⟩) when k = 0, 1, and (x, |k⟩) when k = ±. The output pair is returned to LA via CC and CQ. At this point, LA obtains a training data for k = 0, 1. For k = ±, LA performs a ˆσx measurement to check if the incoming qubit has be… view at source ↗
Figure 2
Figure 2. (a) First, we plot the graphs showing the average accuracy obtained to the size of the training data |ΘQ| used for each CNN model—DN, XC, and NNL. Generally, a larger size of |ΘQ| allows for more accurate learning, but improvements reach a plateau beyond a certain threshold depending on the used CNN model. The achievable accuracy ranks in order: NNL, XC, and DN. We run each of the three CNN models through 150 trials… view at source ↗
Figure 3
Figure 3. Histogram distributions of the learning accuracy (1 − ϵ) achieved for each model (DN, XC, and NNL) and available data sizes (|ΘQ| = 100, 1000, 5000, and 10000), with intervals of 0.01. Generally, as the size of the available dataset increases for all models, higher learning accuracy is achieved consistently, allowing for successful completion of the CNN learning. However, with insufficient training data, the distrib… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: We plot graphs of the learning probability PL(|ΘQ| , ϵT ) of LA and LE versus the noise level η for each pre-trained CNN model (DN, XC, and NNL). These graphs illustrate six cases (a)-(f), with target accuracy set to ϵT = 0.03 and 0.01, and data sizes |ΘQ| of 5000, 100…
Figure 5
Figure 5. Figure 5: For a noise level of η = 0.01, i.e., when LE’s data extraction from ΘQ is less aggressive to avoid detection, the histogram distributions of the learning accuracy (1 − ϵ) obtained by LA and LE through each pre-trained model (DN, XC, and NNL) are shown at intervals of 0…
Figure 6
Figure 6. Figure 6: For a noise level of η = 0.03, indicating a more aggressive eavesdropping by LE, the histogram distributions of the learning accuracy (1 − ϵ) obtained by LA and LE through each pre-trained model (DN, XC, and NNL) are shown at intervals of 0.01 for the data sizes |ΘQ| o…
Figure 7
Figure 7. Figure 7: For a noise level of η = 0.05 (i.e., when LE’s data extraction from ΘQ is relatively large), the histograms for LA’s and LE’s CNN learnings are shown. In the performance-optimized model, such as NNL, with ample training data, both LA and LE can achieve a similar learni…

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Works this paper leans on

39 extracted references · 38 canonical work pages

  1. [31]

    Bocquet A, All´ eaume R and Leverrier A 2011Journal of Physics A: Mathematical and Theoretical 45 025305

    work page

  2. [1]

    Biamonte J, Wittek P, Pancotti N, Rebentrost P, Wiebe N and Lloyd S 2017 Nature 549 195

    work page 2017

  3. [2]

    Ciliberto C, Herbster M, Ialongo A D, Pontil M, Rocchetto A, Severini S and Wossnig L 2018 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474 20170551

    work page 2018

  4. [3]

    Rebentrost P, Mohseni M and Lloyd S 2014 Physical review letters 113 130503

    work page 2014

  5. [4]

    Schuld M, Sinayskiy I and Petruccione F 2016 Physical Review A 94 022342

    work page 2016

  6. [5]

    Wang G 2017 Physical review A 96 012335

    work page 2017

  7. [6]

    Lloyd S, Mohseni M and Rebentrost P 2014 Nature physics 10 631

    work page 2014

  8. [7]

    Aaronson S 2015 Nature Physics 11 291

    work page 2015

Show all 39 references
  1. [8]

    Arunachalam S, Gheorghiu V, Jochym-O’Connor T, Mosca M and Srinivasan P V 2015 New Journal of Physics 17 123010

    2015

  2. [9]

    Tang E 2021 Physical Review Letters 127 060503

    2021

  3. [10]

    Schuld M and Killoran N 2019 Physical review letters 122 040504

    2019

  4. [11]

    Havl ´ ıˇ cek V, C´ orcoles A D, Temme K, Harrow A W, Kandala A, Chow J M and Gambetta J M 2019 Nature 567 209

    2019

  5. [12]

    Lloyd S, Schuld M, Ijaz A, Izaac J and Killoran N 2020 arXiv preprint arXiv:2001.03622

    2020 arXiv

  6. [13]

    Weigold M, Barzen J, Leymann F and Salm M 2021 IET Quantum Communication 2 141

    2021

  7. [14]

    Bang J, Lee S W and Jeong H 2015 Quantum Information Processing 14 3933

    2015

  8. [15]

    Sheng Y B and Zhou L 2017 Science Bulletin 62 1025

    2017

  9. [16]

    Liu N and Rebentrost P 2018 Physical Review A 97 042315

    2018

  10. [17]

    Du Y, Hsieh M H, Liu T, Tao D and Liu N 2021 Physical Review Research 3 023153

    2021

  11. [18]

    Llorens S, Sent ´ ıs G and Mu˜ noz-Tapia R 2024Quantum 8 1452

  12. [19]

    Song W, Lim Y, Kwon H, Adesso G, Wie´ sniak M, Paw lowski M, Kim J and Bang J 2021Physical Review A 103 042409

  13. [20]

    Harney C and Pirandola S 2022 PRX Quantum 3 010311

    2022

  14. [21]

    Valiant L G 1984 Communications of the ACM 27 1134

    1984

  15. [22]

    Langley P 1996 Elements of machine learning (Morgan Kaufmann)

    1996

  16. [23]

    Lee J S, Bang J, Hong S, Lee C, Seol K H, Lee J and Lee K G 2019 Physical Review A 99 012313

    2019

  17. [24]

    Song W, Wie´ sniak M, Liu N, Paw lowski M, Lee J, Kim J and Bang J 2021Quantum Information Processing 20 275 18

  18. [25]

    Kotsiantis S 2011 Artificial intelligence review 42 157

    2011

  19. [26]

    Angluin D and Slonim D K 1994 Machine Learning 14 7

    1994

  20. [27]

    Liu J, Hann C T and Jiang L 2023 Physical Review A 108 032610

    2023

  21. [28]

    Liu J and Jiang L 2024 IEEE Network

    2024

  22. [29]

    Fuchs C A and Peres A 1996 Physical Review A 53 2038

    1996

  23. [30]

    Fuchs C A, Gisin N, Griffiths R B, Niu C S and Peres A 1997 Physical Review A 56 1163

    1997

  24. [32]

    Scarani V, Iblisdir S, Gisin N and Ac ´ ın A 2005Reviews of Modern Physics 77 1225

  25. [33]

    Dang G F and Fan H 2007 Physical Review A—Atomic, Molecular, and Optical Physics 76 022323

    2007

  26. [34]

    Banaszek K 2001 Physical Review Letters 86 1366

    2001

  27. [35]

    Devetak I 2005 IEEE Transactions on Information Theory 51 44

    2005

  28. [36]

    Cai N, Winter A and Yeung R W 2004 problems of information transmission 40 318

    2004

  29. [37]

    1 (Springer Science & Business Media)

    Holevo A S 2011 Probabilistic and statistical aspects of quantum theory vol. 1 (Springer Science & Business Media)

    2011

  30. [38]

    Schumacher B and Westmoreland M D 1997 Physical Review A 56 131

    1997

  31. [39]

    2016 Journal of Machine Learning Research 17 1

    Chen X, Zhang Y et al. 2016 Journal of Machine Learning Research 17 1

    2016

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Reviewed August 10, 2026 · model on record in the stance chip above.