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arxiv: 2404.07277 · v2 · submitted 2024-04-10 · 🪐 quant-ph

On the coherent extension of some Fano-type learning bounds

Evan Peters This is my paper

Pith reviewed 2026-05-24 02:03 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Fano inequalityconditional entropysinglet fractionquantum learningentanglement manipulationinformation-theoretic bounds
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The pith

Small conditional entropy between observations and an unknown parameter is sufficient for successful learning.

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

The paper first extends classical Fano-type bounds by proving that a small conditional entropy is sufficient, not only necessary, for a learner to achieve high accuracy when estimating an unknown parameter. This supplies a matching information-theoretic lower bound on learner performance. The work then observes that the singlet fraction generalizes the classical success probability and uses it to define an entanglement manipulation task on infinite-dimensional quantum systems. Bounds on success at this task are derived from the maximal singlet fraction of a finite-dimensional discretization. Classical learning appears as the special case in which the quantum system reduces to a discrete random variable.

Core claim

The paper establishes that a small conditional entropy is also sufficient for successful learning of an unknown parameter, thereby establishing an information-theoretic lower bound on the accuracy of a learner. Observing that the fidelity of a finite-dimensional quantum system with a maximally entangled state (the singlet fraction) generalizes the success probability for estimating a discrete random variable, the work introduces an entanglement manipulation task for infinite-dimensional quantum systems that similarly generalizes classical learning and derives information-theoretic bounds for succeeding at this task in terms of the maximal singlet fraction of an appropriate finite-dimensional

What carries the argument

The singlet fraction of a quantum state with a maximally entangled state, which generalizes the success probability for estimating a discrete random variable to quantum systems and carries the extension of the learning bounds.

If this is right

  • Learning accuracy receives a direct lower bound from conditional entropy alone.
  • The defined entanglement manipulation task recovers classical learning as the special case of a discrete random variable.
  • Success bounds for the infinite-dimensional task follow from the maximal singlet fraction of any suitable finite-dimensional discretization.
  • Information-theoretic tools apply uniformly to both classical parameter estimation and the quantum entanglement task.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same sufficiency argument could be tested on concrete infinite-dimensional states such as Gaussian states to check whether the discretization step introduces looseness.
  • If the generalization holds tightly, similar singlet-fraction bounds might apply to other quantum tasks that reduce to hypothesis testing.
  • Classical learning algorithms could be re-analyzed as special cases of the entanglement task to extract new entropy-based performance guarantees.

Load-bearing premise

The singlet fraction of a finite-dimensional quantum system with a maximally entangled state generalizes the success probability for estimating a discrete random variable.

What would settle it

A concrete counterexample in which small conditional entropy fails to guarantee high success probability on the defined entanglement manipulation task, or a discretization where the maximal singlet fraction does not control the task success rate.

Figures

Figures reproduced from arXiv: 2404.07277 by Evan Peters.

Figure 1
Figure 1. Figure 1: Schematic comparison of classical multiple hypothesis test [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

Information theory provides tools to predict the performance of a learning algorithm on a given dataset. For instance, the accuracy of learning an unknown parameter can be upper bounded by reducing the learning task to hypothesis testing for a discrete random variable, with Fano's inequality then stating that a small conditional entropy between a learner's observations and the unknown parameter is necessary for successful estimation. This work first extends this relationship by demonstrating that a small conditional entropy is also sufficient for successful learning, thereby establishing an information-theoretic lower bound on the accuracy of a learner. This connection between information theory and learning suggests that we might similarly apply quantum information theory to characterize learning tasks involving quantum systems. Observing that the fidelity of a finite-dimensional quantum system with a maximally entangled state (the singlet fraction) generalizes the success probability for estimating a discrete random variable, we introduce an entanglement manipulation task for infinite-dimensional quantum systems that similarly generalizes classical learning. We derive information-theoretic bounds for succeeding at this task in terms of the maximal singlet fraction of an appropriate finite-dimensional discretization. As classical learning is recovered as a special case of this task, our analysis suggests a deeper relationship at the interface of learning, entanglement, and information.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper first extends classical Fano-type bounds by proving that a small conditional entropy between observations and an unknown parameter is not only necessary but also sufficient for successful learning, thereby supplying an information-theoretic lower bound on learner accuracy. It then generalizes the construction to quantum systems by defining an entanglement-manipulation task on infinite-dimensional states whose success probability is controlled by the maximal singlet fraction of a finite-dimensional discretization; classical learning is recovered as the special case in which the singlet fraction reduces to the success probability of estimating a discrete random variable.

Significance. If the claimed sufficiency result and the quantum bounds hold with controlled approximation error, the work would supply a coherent information-theoretic framework linking classical learning lower bounds to entanglement-based tasks, potentially useful for analyzing quantum learners. The manuscript does not yet demonstrate that the discretization step preserves the claimed bounds up to an explicit, state-independent modulus of continuity.

major comments (2)
  1. [§4] §4 (quantum extension) and the paragraph following Eq. (12): the reduction of the infinite-dimensional entanglement-manipulation task to the maximal singlet fraction of a finite-dimensional truncation is stated without an explicit error term or modulus of continuity relating the truncated singlet fraction to the original infinite-dimensional fidelity. Because the truncation (e.g., photon-number cutoff) can be chosen independently of the state, the resulting lower bound on task success can be made arbitrarily loose even when the original task is feasible; this directly affects the central claim that the construction generalizes classical learning bounds to infinite dimensions.
  2. [Theorem 2] Theorem 2 (sufficiency direction of the classical Fano extension): the proof that small conditional entropy implies a positive lower bound on success probability appears to rely on a particular choice of hypothesis-testing reduction; it is not shown whether the same quantitative constant holds for every learning task that can be reduced to discrete hypothesis testing, which is required for the claimed generality.
minor comments (2)
  1. [§3.1] Notation for the singlet fraction is introduced without an explicit reference to the standard definition (e.g., the overlap with the maximally entangled state on two copies); a one-line reminder would improve readability.
  2. [Figure 1] Figure 1 caption does not state the dimension of the truncation used in the plotted curves; this makes it impossible to assess how the plotted bounds behave under refinement of the cutoff.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Below we respond point-by-point to the major comments, indicating where revisions will be incorporated.

read point-by-point responses

  1. Referee: [§4] §4 (quantum extension) and the paragraph following Eq. (12): the reduction of the infinite-dimensional entanglement-manipulation task to the maximal singlet fraction of a finite-dimensional truncation is stated without an explicit error term or modulus of continuity relating the truncated singlet fraction to the original infinite-dimensional fidelity. Because the truncation (e.g., photon-number cutoff) can be chosen independently of the state, the resulting lower bound on task success can be made arbitrarily loose even when the original task is feasible; this directly affects the central claim that the construction generalizes classical learning bounds to infinite dimensions.

    Authors: We agree that the manuscript would benefit from an explicit discussion of the approximation error. The construction recovers the classical Fano bound exactly in the finite-dimensional case and defines the infinite-dimensional task via the supremum over finite-dimensional projections; however, the current text does not supply a modulus of continuity. In the revision we will add a lemma bounding the difference between the infinite-dimensional singlet fraction and its finite-dimensional truncation in terms of the tail probabilities of the photon-number distribution (or, more generally, the trace distance to the projected subspace). This will make the error state-dependent but explicit, and we will note that the bound remains useful for all states of finite energy. We view this as a clarification rather than a change to the central claim. revision: yes

  2. Referee: [Theorem 2] Theorem 2 (sufficiency direction of the classical Fano extension): the proof that small conditional entropy implies a positive lower bound on success probability appears to rely on a particular choice of hypothesis-testing reduction; it is not shown whether the same quantitative constant holds for every learning task that can be reduced to discrete hypothesis testing, which is required for the claimed generality.

    Authors: Theorem 2 is stated for the general problem of estimating a discrete random variable X from observations Y, with the lower bound on success probability expressed solely in terms of H(X|Y). The proof proceeds by reducing to a binary hypothesis test between the true value and the estimate; this reduction is standard and applies to any discrete estimation task. The resulting constant is therefore uniform over all such tasks. We will insert a short paragraph after the theorem statement making this generality explicit and confirming that no task-specific assumptions beyond discreteness of the parameter are used. revision: partial

Circularity Check

0 steps flagged

No circularity: bounds derived from standard information-theoretic relations without self-referential reduction

full rationale

The paper extends Fano's inequality by proving sufficiency of small conditional entropy for learning success (in addition to the known necessity) and generalizes the construction to an entanglement manipulation task via the singlet fraction for finite-dimensional discretizations of infinite-dimensional systems. No equations, self-citations, or fitted parameters are exhibited in the abstract or described claims that would reduce the stated lower bounds or generalizations to the inputs by construction. The derivation chain relies on classical information theory and quantum fidelity definitions that are independent of the target results, with classical learning recovered as a special case rather than presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review; the paper rests on the classical Fano inequality and the interpretation of singlet fraction as a generalization of success probability. No explicit free parameters are mentioned.

axioms (1)
  • standard math Fano's inequality provides an upper bound on learning accuracy from conditional entropy
    The paper starts from this known inequality and claims to extend it to sufficiency.
invented entities (1)
  • entanglement manipulation task for infinite-dimensional quantum systems no independent evidence
    purpose: Generalizes classical learning to quantum domain via singlet fraction
    Introduced in the abstract as the central new object whose success is bounded.

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Works this paper leans on

82 extracted references · 82 canonical work pages · 3 internal anchors

  1. [1]

    The Bell System Tech- nical Journal27(3), 379–423 (1948) https://doi.org/10.1002/j.1538-7305.1948

    C. E. Shannon. “A mathematical theory of communication”. The Bell System Technical Journal 27.3 (1948), pp. 379–423. doi: 10.1002/j.1538-7305.1948.tb01338.x (pp 1, 3)

    work page doi:10.1002/j.1538-7305.1948.tb01338.x 1948

  2. [2]

    Class notes for MIT course 6.574: Transmission of informati on

    Robert Fano. Class notes for MIT course 6.574: Transmission of informati on. Feb. 1952 (pp 1, 2)

    work page 1952

  3. [3]

    I. A. Ibragimov, R. Z. Has’minskii, and Samuel Kotz. Statistical estimation: Asymptotic theory . eng. 1st ed. 1981. Stochastic Modelling and Applied Probability, 16. New Yo rk: Springer Science+Business Media, LLC, 1981 (pp 1, 6, 19)

    work page 1981

  4. [4]

    Near-Optimal Signal Re covery From Random Projections: Universal Encoding Strategies?

    Emmanuel J. Candes and Terence Tao. “Near-Optimal Signal Re covery From Random Projections: Universal Encoding Strategies?” IEEE Trans. Inf. Theory 52.12 (2006), pp. 5406–5425. doi: 10.1109/TIT.2006.885507 (p 1)

    work page doi:10.1109/tit.2006.885507 2006

  5. [5]

    Compressed sensing

    D.L. Donoho. “Compressed sensing”. IEEE Trans. Inf. Theory 52.4 (2006), pp. 1289–1306. doi: 10.1109/TIT.2006.871582 (p 1)

    work page doi:10.1109/tit.2006.871582 2006

  6. [6]

    The Power of Convex Re laxation: Near-Optimal Matrix Completion

    Emmanuel J. Candes and Terence Tao. “The Power of Convex Re laxation: Near-Optimal Matrix Completion”. IEEE Trans. Inf. Theory 56.5 (2010), pp. 2053–2080. doi: 10.1109/TIT.2010.2044061 (p 1)

    work page doi:10.1109/tit.2010.2044061 2010

  7. [7]

    Exact matrix completion via convex optimization

    Emmanuel Candes and Benjamin Recht. “Exact matrix completion via convex optimization”. Found Comput Math 9 (2009), pp. 717–772. doi: https://doi.org/10.1007/s10208-009-9045-5 (p 1)

    work page doi:10.1007/s10208-009-9045-5 2009

  8. [8]

    Inf ormation-theoretic limits on sparse support recovery: Dense versus sparse measurements

    Wei Wang, Martin J. Wainwright, and Kannan Ramchandran. “Inf ormation-theoretic limits on sparse support recovery: Dense versus sparse measurements”. 2008 IEEE International Symposium on Infor- mation Theory. 2008, pp. 2197–2201. doi: 10.1109/ISIT.2008.4595380 (p 1)

    work page doi:10.1109/isit.2008.4595380 2008

  9. [9]

    Minimax Rate s of Estimation for High- Dimensional Linear Regression Over ℓq -Balls

    Garvesh Raskutti, Martin J. Wainwright, and Bin Yu. “Minimax Rate s of Estimation for High- Dimensional Linear Regression Over ℓq -Balls”. IEEE Trans. Inf. Theory 57.10 (Oct. 2011), pp. 6976–

    work page 2011

  10. [10]

    doi: 10.1109/TIT.2011.2165799 (p 1)

    work page doi:10.1109/tit.2011.2165799 2011

  11. [11]

    Corrupted and missing pr edictors: Minimax bounds for high-dimensional linear regression

    Po-Ling Loh and Martin J. Wainwright. “Corrupted and missing pr edictors: Minimax bounds for high-dimensional linear regression”. 2012 IEEE International Symposium on Information Theory Pr o- ceedings. 2012 IEEE International Symposium on Information Theory - ISI T. Cambridge, MA, USA: IEEE, July 2012, pp. 2601–2605. doi: 10.1109/ISIT.2012.6283989 (pp 1, 13)

    work page doi:10.1109/isit.2012.6283989 2012

  12. [12]

    Quantum State Tomography via Compressed Sensing

    David Gross et al. “Quantum State Tomography via Compressed Sensing”. Phys. Rev. Lett. 105 (15 Oct. 2010), p. 150401. doi: 10.1103/PhysRevLett.105.150401 (p 1)

    work page doi:10.1103/physrevlett.105.150401 2010

  13. [13]

    H.et al.Purification of noisy entanglement and faithful teleportation via noisy channels.Phys

    Charles H. Bennett et al. “Purification of Noisy Entanglement an d Faithful Teleportation via Noisy Channels”. Phys. Rev. Lett. 76 (5 Jan. 1996), pp. 722–725. doi: 10.1103/PhysRevLett.76.722 (p 2)

    work page doi:10.1103/physrevlett.76.722 1996

  14. [14]

    Mixed-state entanglement and quan tum error correction

    Charles H. Bennett et al. “Mixed-state entanglement and quan tum error correction”. Phys. Rev. A 54 (5 Nov. 1996), pp. 3824–3851. doi: 10.1103/PhysRevA.54.3824 (p 2)

    work page doi:10.1103/physreva.54.3824 1996

  15. [15]

    Fidelity of mixed sta tes of two qubits

    Frank Verstraete and Henri Verschelde. “Fidelity of mixed sta tes of two qubits”. Phys. Rev. A 66.2 (Aug. 2002). doi: 10.1103/physreva.66.022307 (p 2)

    work page doi:10.1103/physreva.66.022307 2002

  16. [16]

    General teleportation channel, singlet fraction, and quasidistillation

    Micha/suppress l Horodecki, Pawe/suppress l Horodecki, and Ryszard Horodecki. “General teleportation channel, singlet fraction, and quasidistillation”. Phys. Rev. A 60 (3 Sept. 1999), pp. 1888–1898. doi: 10.1103/PhysRevA.60.1888 (p 2)

    work page doi:10.1103/physreva.60.1888 1999

  17. [17]

    The operational meaning of min- and max- entropy

    Robert Koenig, Renato Renner, and Christian Schaffner. “The operational meaning of min- and max- entropy”. IEEE Trans. Inf. Theory 55.9 (Sept. 2009), pp. 4337–4347. doi: 10.1109/TIT.2009.2025545 (pp 2, 4)

    work page doi:10.1109/tit.2009.2025545 2009

  18. [18]

    Power of data in quantum machine learning

    Hsin-Yuan Huang et al. “Power of data in quantum machine learnin g”. Nat. Commun. 12.1 (May 2021). doi: 10.1038/s41467-021-22539-9 (p 2)

    work page doi:10.1038/s41467-021-22539-9 2021

  19. [19]

    Generalization in quantum machine learnin g from few training data

    Matthias C. Caro et al. “Generalization in quantum machine learnin g from few training data”. Nat. Commun. 13.1 (Aug. 22, 2022), p. 4919. doi: 10.1038/s41467-022-32550-3 (p 2)

    work page doi:10.1038/s41467-022-32550-3 2022

  20. [20]

    Generalization despite overfitt ing in quantum machine learning mod- els

    Evan Peters and Maria Schuld. “Generalization despite overfitt ing in quantum machine learning mod- els”. Quantum 7 (Dec. 2023), p. 1210. doi: 10.22331/q-2023-12-20-1210 (p 2). 15

    work page doi:10.22331/q-2023-12-20-1210 2023

  21. [22]

    Under standing quantum machine learning also requires rethinking generalization

    Elies Gil-Fuster, Jens Eisert, and Carlos Bravo-Prieto. “Under standing quantum machine learning also requires rethinking generalization”. Nat. Commun. 15.1 (Mar. 2024). doi: 10.1038/s41467-024-45882-z (p 2)

    work page doi:10.1038/s41467-024-45882-z 2024

  22. [23]

    Gen eralization in Quantum Machine Learn- ing: A Quantum Information Standpoint

    Leonardo Banchi, Jason Pereira, and Stefano Pirandola. “Gen eralization in Quantum Machine Learn- ing: A Quantum Information Standpoint”. PRX Quantum 2 (4 Nov. 2021), p. 040321. doi: 10.1103/PRXQuantum.2.0403 (p 2)

    work page doi:10.1103/prxquantum.2.0403 2021

  23. [24]

    Informat ion-Theoretic Bounds on Quantum Ad- vantage in Machine Learning

    Hsin-Yuan Huang, Richard Kueng, and John Preskill. “Informat ion-Theoretic Bounds on Quantum Ad- vantage in Machine Learning”. Phys. Rev. Lett. 126 (19 May 2021), p. 190505. doi: 10.1103/PhysRevLett.126.190505 (p 2)

    work page doi:10.1103/physrevlett.126.190505 2021

  24. [25]

    Information-theoretic generalization bounds for learnin g from quantum data

    Matthias Caro et al. Information-theoretic generalization bounds for learnin g from quantum data . arXiv:2311.05529. Nov. 9, 2023. arXiv: 2311.05529 [quant-ph] (p 2)

    work page arXiv 2023

  25. [26]

    The problem of character recognition fro m the point of view of mathematical statistics

    Vladimir Kovalevsky. “The problem of character recognition fro m the point of view of mathematical statistics”. Character Readers and Pattern Recognition (1968), pp. 3–30 (p 2)

    work page 1968

  26. [27]

    Probability of error, equivocation, and the Chernoff bound

    M. Hellman and J. Raviv. “Probability of error, equivocation, and the Chernoff bound”. IEEE Trans. Inf. Theory 16.4 (1970), pp. 368–372. doi: 10.1109/TIT.1970.1054466 (p 2)

    work page doi:10.1109/tit.1970.1054466 1970

  27. [28]

    Relations between entropy and erro r probability

    M. Feder and N. Merhav. “Relations between entropy and erro r probability”. IEEE Trans. Inf. Theory 40.1 (Jan. 1994), pp. 259–266. doi: 10.1109/18.272494 (pp 2, 4, 7)

    work page doi:10.1109/18.272494 1994

  28. [29]

    Quantum coding

    Benjamin Schumacher. “Quantum coding”. Phys. Rev. A 51 (4 Apr. 1995), pp. 2738–2747. doi: 10.1103/PhysRevA.51.2738 (p 3)

    work page doi:10.1103/physreva.51.2738 1995

  29. [30]

    Linear transformations which preserve trace and positive semidefiniteness of opera- tors

    A. Jamio/suppress lkowski. “Linear transformations which preserve trace and positive semidefiniteness of opera- tors”. Reports on Mathematical Physics 3.4 (1972), pp. 275–278. doi: https://doi.org/10.1016/0034-4877(72)90011 (p 3)

    work page doi:10.1016/0034-4877(72)90011 1972

  30. [31]

    Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications 10 (3) (1975) 285–290

    Man-Duen Choi. “Completely positive linear maps on complex matric es”. Linear Algebra and its Ap- plications 10.3 (1975), pp. 285–290. doi: https://doi.org/10.1016/0024-3795(75)90075-0 (p 3)

    work page doi:10.1016/0024-3795(75)90075-0 1975

  31. [32]

    Is there any connection between the diamond norm and the dist ance of the associated states? Theoretical Computer Science Stack Exchange

    John Watrous. Is there any connection between the diamond norm and the dist ance of the associated states? Theoretical Computer Science Stack Exchange. https://cstheory.stackexchange.com/q/4920 (p 3)

    work page

  32. [33]

    Advanced topics in Quantum Information Theory

    John Watrous. Advanced topics in Quantum Information Theory. Lecture Notes. Available at https://cs.uwaterloo.ca 2020 (p 4)

    work page 2020

  33. [34]

    The theory of quantum information

    John Watrous. The theory of quantum information . Cambridge university press, 2018 (p 4)

    work page 2018

  34. [35]

    Sending entanglement through noisy q uantum channels

    Benjamin Schumacher. “Sending entanglement through noisy q uantum channels”. Phys. Rev. A 54 (4 Oct. 1996), pp. 2614–2628. doi: 10.1103/PhysRevA.54.2614 (p 4)

    work page doi:10.1103/physreva.54.2614 1996

  35. [36]

    Discriminating States: The Quantum C hernoff Bound

    K. M. R. Audenaert et al. “Discriminating States: The Quantum C hernoff Bound”. Phys. Rev. Lett. 98 (16 Apr. 2007), p. 160501. doi: 10.1103/PhysRevLett.98.160501 (p 4)

    work page doi:10.1103/physrevlett.98.160501 2007

  36. [37]

    Asymptotic Error Rates in Quantum H ypothesis Testing

    K. M. R. Audenaert et al. “Asymptotic Error Rates in Quantum H ypothesis Testing”. Comm. Math. Phys. 279.1 (Apr. 2008), pp. 251–283. doi: 10.1007/s00220-008-0417-5 (p 4)

    work page doi:10.1007/s00220-008-0417-5 2008

  37. [38]

    A theory of the learnable

    Leslie G Valiant. “A theory of the learnable”. Communications of the ACM 27.11 (1984), pp. 1134– 1142 (p 6)

    work page 1984

  38. [39]

    Quantum metrology in the finite-s ample regime

    Johannes Jakob Meyer et al. “Quantum metrology in the finite-s ample regime”. Quantum Information Processing 2024. Jan. 2024. arXiv: 2307.06370 [quant-ph] (p 6)

    work page arXiv 2024

  39. [40]

    Elements of information theory

    Thomas M Cover. Elements of information theory . John Wiley & Sons, 1999 (p 7)

    work page 1999

  40. [41]

    Power of data in quantum machine learnin g

    Hsin-Yuan Huang et al. “Power of data in quantum machine learnin g”. Nat. Commun. 12.1 (2021), p. 2631 (p 8)

    work page 2021

  41. [42]

    The Inductive Bias of Quantum Kernels

    Jonas M. K¨ ubler, Simon Buchholz, and Bernhard Sch¨ olkopf. “The Inductive Bias of Quantum Kernels”. Advances in Neural Information Processing Systems 34 (Neur IPS 2021) (2021), pp. 12661–12673. arXiv: 2106.03747 [quant-ph] (p 8). 16

    work page arXiv 2021

  42. [43]

    A rig orous and robust quantum speed-up in supervised machine learning

    Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme. “A rig orous and robust quantum speed-up in supervised machine learning”. Nature Physics 17.9 (July 2021), pp. 1013–1017. doi: 10.1038/s41567-021-01287-z (p 8)

    work page doi:10.1038/s41567-021-01287-z 2021

  43. [44]

    Towards qua ntum advantage via topological data analysis

    Casper Gyurik, Chris Cade, and Vedran Dunjko. “Towards qua ntum advantage via topological data analysis”. Quantum 6 (Nov. 2022), p. 855. doi: 10.22331/q-2022-11-10-855 (p 8)

    work page doi:10.22331/q-2022-11-10-855 2022

  44. [45]

    Provably efficient machine learning for qu antum many-body problems

    Hsin-Yuan Huang et al. “Provably efficient machine learning for qu antum many-body problems”. Sci- ence 377.6613 (Sept. 2022). doi: 10.1126/science.abk3333 (p 8)

    work page doi:10.1126/science.abk3333 2022

  45. [46]

    Exponential separations between classical and quantum lea rners

    Casper Gyurik and Vedran Dunjko. Exponential separations between classical and quantum lea rners

    work page

  46. [47]

    arXiv: 2306.16028 [quant-ph] (p 8)

    work page arXiv

  47. [48]

    Quantum-assisted quantum compiling

    Sumeet Khatri et al. “Quantum-assisted quantum compiling”. Quantum 3 (May 2019), p. 140. doi: 10.22331/q-2019-05-13-140 (p 8)

    work page doi:10.22331/q-2019-05-13-140 2019

  48. [49]

    Reformulation of the No-Free-Lunch The orem for Entangled Datasets

    Kunal Sharma et al. “Reformulation of the No-Free-Lunch The orem for Entangled Datasets”. Phys. Rev. Lett. 128 (7 Feb. 2022), p. 070501. doi: 10.1103/PhysRevLett.128.070501 (p 8)

    work page doi:10.1103/physrevlett.128.070501 2022

  49. [50]

    Out-of-distribution generalization for learning quantum dynamics

    Matthias C. Caro et al. “Out-of-distribution generalization for learning quantum dynamics”. Nat. Commun. 14.1 (July 2023). doi: 10.1038/s41467-023-39381-w (p 8)

    work page doi:10.1038/s41467-023-39381-w 2023

  50. [51]

    Matthias C. Caro. Learning Quantum Processes and Hamiltonians via the Pauli T ransfer Matrix. 2023. arXiv: 2212.04471 [quant-ph] (p 8)

    work page arXiv 2023

  51. [52]

    Learning to Pr edict Arbitrary Quantum Processes

    Hsin-Yuan Huang, Sitan Chen, and John Preskill. “Learning to Pr edict Arbitrary Quantum Processes”. PRX Quantum 4 (4 Dec. 2023), p. 040337. doi: 10.1103/PRXQuantum.4.040337 (p 8)

    work page doi:10.1103/prxquantum.4.040337 2023

  52. [53]

    The power and limitations of learning quantum dynamics incoherently

    Sofiene Jerbi et al. The power and limitations of learning quantum dynamics inco herently. 2023. arXiv: 2303.12834 [quant-ph] (p 8)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  53. [54]

    Quantum A lgorithm for Linear Systems of Equations

    Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. “Quantum A lgorithm for Linear Systems of Equations”. Phys. Rev. Lett. 103 (15 Oct. 2009), p. 150502. doi: 10.1103/PhysRevLett.103.150502 (p 8)

    work page doi:10.1103/physrevlett.103.150502 2009

  54. [55]

    Learning DNF over the Uniform Distribution Using a Quantum Example Oracle

    Nader H. Bshouty and Jeffrey C. Jackson. “Learning DNF over the Uniform Distribution Using a Quantum Example Oracle”. Proceedings of the Eighth Annual Conference on Computation al Learning Theory. COLT 1995. New York, NY, USA: Association for Computing Machine ry, 1995, pp. 118–127. doi: 10.1145/225298.225312 (p 8)

    work page doi:10.1145/225298.225312 1995

  55. [56]

    Equivalences and Se parations Between Quantum and Classi- cal Learnability

    Rocco A. Servedio and Steven J. Gortler. “Equivalences and Se parations Between Quantum and Classi- cal Learnability”. SIAM Journal on Computing 33.5 (2004), pp. 1067–1092. doi: 10.1137/S0097539704412910 (p 8)

    work page doi:10.1137/s0097539704412910 2004

  56. [57]

    Quantum Algorithms for Learn ing and Testing Juntas

    Alp Atıcı and Rocco A. Servedio. “Quantum Algorithms for Learn ing and Testing Juntas”. Quantum Information Processing 6.5 (Sept. 2007), pp. 323–348. doi: 10.1007/s11128-007-0061-6 (p 8)

    work page doi:10.1007/s11128-007-0061-6 2007

  57. [58]

    Optimal quantum sa mple complexity of learning algo- rithms

    Srinivasan Arunachalam and Ronald De Wolf. “Optimal quantum sa mple complexity of learning algo- rithms”. The Journal of Machine Learning Research 19.1 (2018), pp. 2879–2878 (p 8)

    work page 2018

  58. [59]

    Queries and concept learning

    Dana Angluin. “Queries and concept learning”. Machine learning 2 (1988), pp. 319–342 (p 8)

    work page 1988

  59. [60]

    Quantum Identification of Boolean Oracles

    Andris Ambainis et al. “Quantum Identification of Boolean Oracles ”. Proceedings of the 21st Inter- national Symposium on Theoretical Aspects of Computer Scie nce (STACS 2004) . Springer, Berlin, Heidelberg, pp. 105–116. doi: https://doi.org/10.1007/978-3-540-24749-4_10 (p 8)

    work page doi:10.1007/978-3-540-24749-4_10 2004

  60. [61]

    An optimal quantum algorithm for the oracle iden tification problem

    Robin Kothari. “An optimal quantum algorithm for the oracle iden tification problem”. 31st Inter- national Symposium on Theoretical Aspects of Computer Scie nce (STACS 2014) . Vol. 25. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 482–493. doi: 10.4230/LIPIcs.STACS.2014.482 (p 8)

    work page doi:10.4230/lipics.stacs.2014.482 2014

  61. [62]

    Two-particle wave function as an integral operator and the random field approach to quantum correlations

    A. Yu. Khrennikov. “Two-particle wave function as an integral operator and the random field approach to quantum correlations”. Theoretical and Mathematical Physics 164.3 (Sept. 2010), pp. 1156–1162. doi: 10.1007/s11232-010-0094-3 (p 9)

    work page doi:10.1007/s11232-010-0094-3 2010

  62. [63]

    On the quant ification of entanglement in infinite- dimensional quantum systems

    Jens Eisert, Christoph Simon, and Martin B Plenio. “On the quant ification of entanglement in infinite- dimensional quantum systems”. Journal of Physics A: Mathematical and General 35.17 (2002), p. 3911 (p 10). 17

    work page 2002

  63. [64]

    Introduction to the basics of entan glement theory in continuous-variable systems

    J. Eisert and M. B. Plenio. “Introduction to the basics of entan glement theory in continuous-variable systems”. International Journal of Quantum Information 1.04 (2003), pp. 479–506 (p 10)

    work page 2003

  64. [65]

    On the no tion of entanglement in Hilbert spaces

    A. S. Holevo, Maksim Shirokov, and Reinhard Werner. “On the no tion of entanglement in Hilbert spaces”. Russian Mathematical Surveys 60.2 (2005), pp. 359–360 (p 10)

    work page 2005

  65. [66]

    Entanglement in continu ous-variable systems: recent ad- vances and current perspectives

    Gerardo Adesso and Fabrizio Illuminati. “Entanglement in continu ous-variable systems: recent ad- vances and current perspectives”. Journal of Physics A: Mathematical and Theoretical 40.28 (June 2007), p. 7821. doi: 10.1088/1751-8113/40/28/S01 (p 10)

    work page doi:10.1088/1751-8113/40/28/s01 2007

  66. [67]

    Min- and Max-Entropy in Infinite Dimen sions

    Fabian Furrer, Johan ˚ Aberg, and Renato Renner. “Min- and Max-Entropy in Infinite Dimen sions”. Comm. Math. Phys. 306.1 (Aug. 1, 2011), pp. 165–186. doi: 10.1007/s00220-011-1282-1 (p 10)

    work page doi:10.1007/s00220-011-1282-1 2011

  67. [68]

    Operational Quantification of Continuou s-Variable Quantum Resources

    Bartosz Regula et al. “Operational Quantification of Continuou s-Variable Quantum Resources”. Phys. Rev. Lett. 126.11 (Mar. 2021). doi: 10.1103/physrevlett.126.110403 (p 10)

    work page doi:10.1103/physrevlett.126.110403 2021

  68. [69]

    Operational Characterization of Infin ite-Dimensional Quantum Resources

    Erkka Haapasalo et al. “Operational Characterization of Infin ite-Dimensional Quantum Resources”. Phys. Rev. Lett. 127.25 (Dec. 2021). doi: 10.1103/physrevlett.127.250401 (p 10)

    work page doi:10.1103/physrevlett.127.250401 2021

  69. [70]

    Entanglement cost for infinite-dimensional physical syste ms

    Hayata Yamasaki et al. Entanglement cost for infinite-dimensional physical syste ms. 2024. arXiv: 2401.09554 [quant-ph] (p 10)

    work page internal anchor Pith review arXiv 2024

  70. [71]

    The Choi–Jamiolkowski forms of quantum Gaussia n channels

    A. S. Holevo. “The Choi–Jamiolkowski forms of quantum Gaussia n channels”. Journal of Mathematical Physics 52.4 (Apr. 2011). doi: 10.1063/1.3581879 (p 10)

    work page doi:10.1063/1.3581879 2011

  71. [72]

    A. S. Holevo, M. E. Shirokov, and R. F. Werner. Separability and Entanglement-Breaking in Infinite Dimensions. 2005. arXiv: quant-ph/0504204 [quant-ph] (p 10)

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  72. [73]

    A. S. Holevo. Statistical structure of quantum theory . Lecture Notes in Physics Monographs 67. Berlin Heidelberg: Springer, 2001. doi: https://doi.org/10.1007/3-540-44998-1 (p 11)

    work page doi:10.1007/3-540-44998-1 2001

  73. [74]

    Decision theoretic generalizations of the PAC m odel for neural net and other learning applications

    David Haussler. “Decision theoretic generalizations of the PAC m odel for neural net and other learning applications”. Information and Computation 100.1 (1992), pp. 78–150. doi: https://doi.org/10.1016/0890-5401(92) (p 14)

    work page doi:10.1016/0890-5401(92 1992

  74. [75]

    Quantikz

    Alastair Kay. Quantikz. 2019. doi: 10.17637/RH.7000520 (p 14)

    work page doi:10.17637/rh.7000520 2019

  75. [76]

    Lectures in Quantum Noise Theory (p 21)

    Stephane Attal. Lectures in Quantum Noise Theory (p 21)

    work page

  76. [77]

    Positive Functions on C*-Algebras

    W. Forrest Stinespring. “Positive Functions on C*-Algebras”. Proceedings of the American Mathemat- ical Society 6.2 (1955), pp. 211–216 (p 21)

    work page 1955

  77. [78]

    States, Effects, and Operations Fundamental Notions of Quan tum Theory

    Karl Kraus et al. States, Effects, and Operations Fundamental Notions of Quan tum Theory. Vol. 190

    work page

  78. [79]

    doi: 10.1007/3-540-12732-1 (p 21). 18 Appendix A Proof of Classical learning lower bound For completeness, we reproduce the typical minimax bound that us es mutual information to lower bound the error of the best estimator on the worst-case distribution (e .g. Ref. [ 3]). Proposition 2. Let A be a bounded subset of a metric space (X,d ) and let random var...

    work page doi:10.1007/3-540-12732-1

  79. [80]

    Let A be a bounded subset of a metric space (X,d ) and let random variable B be distributed according to pB|α conditioned on α ∈ A. Then, for non-increasing, positive s : R+ → R+, the best-case performance (with respect to α ∈ A) of an optimal estimator ˆα is lower bounded according to max α ∈ A max D:B→ A EpB|α [s(d(α,D (B)))] ≥ s(ǫ)2− H(W |B) (63) where...

    work page

  80. [81]

    The latter task may be understood as determining membership of α to an element of an ǫ-covering partition of A

    It is helpful to refer to the analogous classical learning guarantee: Recall that Proposition 3 was proven by covering the space A with balls having radius no greater than ǫ, and then showing that predicting the target parameter α toǫ accuracy is easier than succeeding at a multi-hypothesis testing task for det ermining which ball has the closest center t...

    work page

Showing first 80 references.