Pith. sign in

REVIEW 2 major objections 4 minor 64 references

Strong converse Exponents of Partially Smoothed Information Measures

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper determines exact strong converse exponents for partially smoothed quantum information measures, and shows they are not uniform across state classes.

desk verdict Genuinely new strong-converse exponents for partially smoothed measures, but the key minimax step in Proposition 9 is not fully justified and needs a fix before I'd rely on the privacy amplification exponent. read the letter →

arxiv 2505.06050 v1 pith:5O4FPB4H submitted 2025-05-09 quant-ph

Mario Berta , Yongsheng Yao This is my paper · ORCID
classification quant-ph MSC 81P4594A17 PACS 03.67.-a
keywords partiallysmoothedinformationmeasuresstrongconverseexponentsconditionalmin-entropymutualmax-informationRényidivergencesprivacyamplificationquantumdatacompressionintrinsicrandomness
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 derives the exact exponential rates at which partially smoothed mutual max-information and conditional min-entropy fail—that is, the rates at which their smoothing error converges to 1—for classical states and pure states. These rates, called strong converse exponents, quantify how quickly one-shot quantities deviate from their asymptotic limits when the rate parameter r crosses the mutual information or conditional entropy threshold. The main results give closed-form expressions for both trace and purified distance, and show that the trace-distance exponent takes different forms for classical and pure states, thereby disproving the natural conjecture that one formula covers all quantum states. As applications, the paper obtains exact strong converse exponents for blind quantum data compression, intrinsic randomness extraction, classical state splitting, and classical privacy amplification.

What carries the argument

The arguments run on type-theoretic empirical distribution calculus combined with Rényi divergence variational formulas. The central object is the variational expression (54), which rewrites the classical privacy-amplification exponent as an infimum over distributions $t$ and states $\{\tau^x_A\}$ of $2D(t\|p)+\mathbb{E}_{x\sim t}D(\tau^x_A\|\rho^x_A)+|r+\mathbb{E}_{x\sim t}D(\tau^x_A\|I_A)|_+$, obtained by swapping a supremum over Rényi order via Sion's minimax theorem. The operator Hölder inequality of Lemma 21 supplies the fidelity-to-Rényi bounds used in the optimality parts, while the achievability parts go through privacy amplification and quantum data compression reductions, with pinching and symmetric-state domination controlling the type-class decomposition.

What would settle it

A numerical check could settle the central exchange: choose a two-letter classical state, for instance $R=\{1,2\}$ with $p=(1/2,1/2)$ and $\rho^x_A$ a binary distribution, fix $r$ at an interior value, and compute both sides of (54) directly; any strict gap between the left-hand supremum-over-$\alpha$ and the right-hand infimum over $t$ and $\{\tau^x_A\}$ would refute Proposition 9.

Watch

Extended reading notes

Core claim

For classical states with bipartite distribution $p$, the trace-distance strong converse exponents are $\sup_{0\leq \alpha \leq 1}(1-\alpha)(r-\bar{H}_\alpha(A|R)_p)$ for the partially smoothed conditional min-entropy and $\sup_{0\leq \alpha \leq 1}(1-\alpha)(I_\alpha(R:A)_p-r)$ for the partially smoothed mutual max-information, while the purified-distance conditional min-entropy exponent is $\sup_{\frac12\leq\alpha\leq1}\inf_{t\in Q(R)}\{2D(t\|p)+\frac{1-\alpha}{\alpha}(r-\mathbb{E}_{x\sim t}H_\alpha(p(\cdot|x)))\}$. For a pure state with Schmidt coefficients $p$, both trace and purified distance give $\inf_{t\in Q(X)}\{2D(t\|p)+|r+H(t)|_+\}$ for the conditional min-entropy and $\sup_{\beta>1}\frac{\beta-1}{\beta}(2H_\beta(R)_\rho-r)$ for the mutual max-information. The paper proves that the classical trace-distance formulas cannot be converted into the pure-state formulas, so no single trace-distance formula holds for all quantum states.

Load-bearing premise

The load-bearing premise is that the supremum over the Rényi order can be swapped with the infimum over distributions and states in formulas such as (54), even though the order domain is only $(0,1)$ and no compactness argument is supplied; if this exchange fails, the variational expressions and the achievability part of the privacy-amplification exponent lose their foundation.

Editorial extensions

If this is right

  • For classical states, the trace-distance exponents become Legendre transforms of Petz Rényi conditional entropy and mutual information, giving exact large-deviation rates for the partially smoothed measures.
  • For pure states, the trace- and purified-distance exponents coincide and take simple relative-entropy-plus-positive-part forms.
  • The trace-distance strong converse exponents are not uniform over all quantum states, since classical and pure states require different formulas.
  • Blind quantum data compression now has an exact strong converse exponent of $\sup_{\beta>1}\frac{\beta-1}{\beta}(2H_\beta(A)_\rho-2r)$, filling a gap left by the visible compression result.
  • Intrinsic randomness extraction, classical state splitting, and classical privacy amplification each acquire exact strong converse exponents as direct consequences.

Reading between the lines

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

  • A plausible next target is a mixed-state trace-distance formula that interpolates between the classical and pure expressions; the non-uniformity result suggests the exponent may require optimizing over decompositions of the mixed state into pure or classical pieces.
  • Because purified distance gives the same exponents as trace distance for pure states, the purified-distance classical formula (36) may be a candidate for a uniform expression across all states, though the paper does not claim this.
  • The classical privacy-amplification exponent (51) may extend to classical-quantum side information, since Lemma 21 is operator-valued; however, the infimum over types would then lack a classical interpretation, so a different proof would be needed.
  • The achievability proofs for data compression and intrinsic randomness suggest that matching finite-blocklength refinements beyond exponent order may follow from the same variational formulas.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper determines exact strong converse exponents for the partially smoothed mutual max-information and conditional min-entropy, for classical states (Theorem 4) and pure states (Theorem 5). It shows that the trace-distance exponents are not uniform across state classes (Remark 7, Proposition 28). As applications, it derives strong converse exponents for blind quantum data compression (Theorem 15), intrinsic randomness (Theorem 17), and classical state splitting (Theorem 19). A key technical ingredient is the exact strong converse exponent for classical privacy amplification (Theorem 8), obtained via a variational expression (Proposition 9) and a long achievability proof. The paper is technically dense and builds on several published lemmas from the authors and others.

Significance. The paper addresses a genuine open problem in one-shot quantum information theory. The main results — exact strong converse exponents for partially smoothed information measures in the classical and pure-state cases — are natural and likely useful; the observation that trace-distance exponents are not uniform across state classes is a notable conceptual point. The derivation of the classical privacy amplification exponent is a contribution of independent interest. The proofs are long and mostly well structured, and the paper is honest about its reliance on external results. However, the validity of several minimax swaps, especially in Proposition 9, is not fully justified as written, and since those swaps support the achievability directions of the central theorems, the soundness of the claimed exponents is not yet fully established.

major comments (2)
  1. [Section V.A, Proposition 9] In the proof of Proposition 9 (Eqs. (54)-(55)), the equality between the supremum over 0<s<1 of the infimum over t and {τ_x} and the infimum over t and {τ_x} of the supremum over 0<s<1 is justified by an application of Sion's minimax theorem. The domain of s is the open interval (0,1), which is not compact, and the paper does not supply a limiting argument. This is a load-bearing step: Proposition 9 underlies the achievability of the classical privacy amplification exponent (Theorem 8) and hence Eq. (36). The gap is repairable, because for fixed (t,{τ_x}) the expression is affine in s, so the supremum over (0,1) equals the supremum over the compact interval [0,1]; the authors should extend s to [0,1] and then apply Sion. As written, the proof is incomplete.
  2. [Section VI and Section VIII.B] The same non-compact minimax swap is used in the proof of Eq. (87) in Section VI (swap of sup_{0<s<1} and inf_{t∈Q(X)}) and in Proposition 18 (Eq. (124)) for the intrinsic randomness exponent. In both places the function is linear in the auxiliary parameter for fixed outer variables, so the same extension to the closed interval repairs the argument. The authors should add a remark or lemma stating the compactification once and refer to it at all three occurrences.
minor comments (4)
  1. [Introduction] The introduction names 'an improved Fuchs-van de Graaf inequality (Lemma 23 in Appendix)', but Lemma 23 is a fidelity inequality for projections; the improved Fuchs-van de Graaf inequality used later is Lemma 26. The cross-reference should be corrected.
  2. [Eq. (61)] The derivation in Eq. (61) invokes the Fannes-Audenaert inequality and Hölder's inequality without explicit statements or references; since the resulting constants enter the exponent bound, a citation and a brief statement of the inequalities would help the reader.
  3. [Theorem 4 and Theorem 8] The formulas in Eq. (36) and Eq. (51) take a supremum over the closed interval [1/2,1], including α=1, whereas the proofs are written only for 1/2<α<1. The authors should note the continuity of the relevant Rényi quantities in α to justify passage to the endpoint.
  4. [Proposition 12, Eq. (103)-(104)] In Eq. (103) and Eq. (104), the notation ϵ^P(ρ^{⊗n},nr) omits the subscripts ˙R^n:A^n; for consistency with the rest of the paper the full notation ϵ^P_{\dot R^n:A^n}(ρ^{⊗n},nr) should be used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction; the claimed exponents are derived from independent lemmas and the self-citations are not load-bearing as reductions.

full rationale

Walking the derivation chain, I find no load-bearing step in which an output is fed back as an input. Equation (34) follows from the established equality of partial and standard smoothing of the conditional min-entropy ([9, Prop. 2]) together with the known classical conditional smooth-min-entropy strong converse exponent ([33, Thm. 2]). Equation (35) is proved directly with the trace-distance identity [34, Lemma 2], the upper bound from [35, Remark 4.6], and the symmetric-state domination Lemma 1; the final Petz Renyi mutual information form uses Proposition 3(ix), not the theorem being proved. Equation (36) divides into an optimality proof via Lemma 21 and type counting, and an achievability proof through the separately established classical privacy-amplification exponent (Theorem 8), whose own proof uses Hayashi's extraction theorem and the variational Proposition 9 rather than assuming Eq. (36). The pure-state formula (37) is derived from type/projection reductions and Lemma 21; the trace-distance version then follows from the improved Fuchs-van de Graaf inequality Lemma 26, a parameter-free bound whose statement does not contain the target exponent. Equation (38) uses Proposition 12/Corollary 13 for optimality and Proposition 14 for achievability, with Proposition 14 resting on the external classical source-coding exponent (108). No fitted parameter is renamed as a prediction, and no uniqueness claim is imported from the authors' prior work. The self-citations ([1], [34], [55]) supply lemmas that are independent of the theorems being derived; they are parameter-free identities or inequalities, so they do not raise the circularity score. The substantive concern flagged by the skeptic, namely the appeal to Sion's minimax on the open interval s in (0,1) in Proposition 9 (and similarly in Propositions 18 and 28), is a rigor/correctness gap about compactness and not a circularity: even if the exchange is unjustified as written, the claimed exponents are not assumed in the premises of the step.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new free parameters or invented entities. It relies on standard results in quantum information theory and on several external theorems listed above. The central new formulas are derived from these tools.

assumptions (6)
  • domain assumption Equality of partially smoothed and standard smoothed conditional min-entropy for classical states ([9, Proposition 2])
    Invoked at Eq. (41) to convert the partially smoothed measure to the standard one; if false, Eq. (34) fails.
  • domain assumption Exact strong converse exponent for standard smoothed conditional min-entropy ([33, Theorem 2])
    Used directly at Eq. (43) to obtain the classical trace-distance exponent; the result is taken from an arXiv preprint and not reproved.
  • standard math Operator Hölder inequality (Lemma 21, from [52])
    Central to the optimality parts for purified distance and privacy amplification (Eqs. (52), (73), (85), (98)).
  • standard math Universal symmetric state dominance (Lemma 1)
    Used in the lower bounds for the classical mutual max-information (Eq. (47)) and in Proposition 12 (Eq. (100)).
  • domain assumption Hayashi's strong converse estimate for privacy amplification ([39, Theorem 1])
    Used in the achievability part of Theorem 8 to obtain a hash sequence with vanishing relative entropy for an auxiliary state tau (Eq. (57)).
  • ad hoc to paper Sion's minimax theorem applied with non-compact s-domain in Proposition 9
    The paper asserts the swap at Eq. (55) without addressing compactness of (0,1); this is the flagged weakest assumption.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Strong converse Exponents of Partially Smoothed Information Measures." pith.science (2026). https://pith.science/paper/5O4FPB4H

@misc{pith2026250506050,
  author       = {Pith},
  title        = {Pith review of: Strong converse Exponents of Partially Smoothed Information Measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5O4FPB4H}},
  note         = {Machine review of arXiv:2505.06050}
}
read the original abstract

Partially smoothed information measures are fundamental tools in one-shot quantum information theory. In this work, we determine the exact strong converse exponents of these measures for both pure quantum states and classical states. Notably, we find that the strong converse exponents based on trace distance takes different forms between pure and classical states, indicating that they are not uniform across all quantum states. Leveraging these findings, we derive the strong converse exponents for quantum data compression, intrinsic randomness extraction, and classical state splitting. A key technical step in our analysis is the determination of the strong converse exponent for classical privacy amplification, which is of independent interest.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 61 canonical work pages

  1. [1]

    IEEE Trans

    Anshu, A., Berta, M., Jain, R., Tomamichel, M.: Partially smoothed information measures. IEEE Trans. Inf. Theory 66(8), 5022–5036 (2020)

    work page 2020

  2. [2]

    IEEE Trans

    Tomamichel, M., Hayashi, M.: A hierarchy of information quantities for finite block length analysis of quantum tasks. IEEE Trans. Inf. Theory 59(11), 7693–7710 (2013)

    work page 2013

  3. [3]

    Alice takes a sample from qi Yn

    work page

  4. [4]

    With probability P′ XnYn(yn|xn) 2KnqYn(yn) she accepts the sample, sends i to Bob and exits the whole loop

    work page

  5. [5]

    With probability 1 − P′ XnYn(yn|xn) 2KnqYn(yn) she updates i→i + 1 and goes to Step 2 (End While), 32

    work page

  6. [6]

    If i> 2Rn, Alice sends 2Rn + 1 to Bob

    work page

  7. [7]

    If j >2Rn, Bob outputs a sample distributed as qYn

    Bob receives Alice’s message, which we call j. If j >2Rn, Bob outputs a sample distributed as qYn. Else he outputs the sample from qj Yn. Based on the same arguments as in the proof of [1, Theorem 4], the number of bits communicated is log(2Rn + 1) =nr and the output distribution is equal to P′′ XnYn = X xn∈X×n Pn X(xn)|xn⟩⟨xn|⊗ ((1−λn)P′ XnYn(·|xn) +λnqY...

    work page

  8. [8]

    Berta, M., Christandl, M., Renner, R.: The quantum reverse Shannon theorem based on one-shot information theory. Commun. Math. Phys. 306(3), 579–615 (2011)

    work page 2011

Show all 64 references
  1. [9]

    IEEE Trans

    Cao, M., Ramakrishnan, N., Berta, M., Tomamichel, M.: Channel simulation: Finite blocklengths and broadcast channels. IEEE Trans. Inf. Theory. (2020)

    2020

  2. [10]

    IEEE Trans

    Fang, K., Wang, X., Tomamichel, M., Berta, M.: Quantum channel simulation and the channel’s smooth max-information. IEEE Trans. Inf. Theory. 66(4), 2129–2140 (2019)

    2019

  3. [11]

    Faist, P., Berta, M., Brandap, F.: Thermodynamic implementations of quantum processes. Commun. Math. Phys. 384(3), 1709–1750 (2021)

    2021

  4. [12]

    Anshu, A., Devabathini, V.K., Jain, R.: Quantum communication using coherent rejection sampling. Phys. Rev. Lett. 119, 120506 (2017)

    2017

  5. [13]

    Li, K., Yao, Y.: Reliability function of quantum information decoupling via the sandwiched R´ enyi divergence. Commun. Math. Phys. 405, 160–189 (2024)

    2024

  6. [14]

    Abdelhadi, D., Renes, J.: On the second-order asymptotics of the partially smoothed conditional min- entropy & application to quantum compression. IEEE J. Sel. Areas Inf. Theory. 1(2), 416–423 (2020)

    2020

  7. [15]

    Hayashi, M.: Optimal sequence of quantum measurements in the sense of Stein’s lemma in quantum hypothesis testing. J. Phys. A: Math. Gen. 35, 10759 (2002)

    2002

  8. [16]

    Hayashi, M.: Universal coding for classical-quantum channel. Commun. Math. Phys. 289(3), 1087–1098 (2009)

    2009

  9. [17]

    Christandl, M., K¨ onig, R., Renner, R.: Postselection technique for quantum channels with applications to quantum cryptography. Phys. Rev. Lett. 102, 020504 (2009)

    2009

  10. [18]

    Hayashi, M., Tomamichel, M.: Correlation detection and an operational interpretation of the R´ enyi mutual information. J. Math. Phys. 57(10), 102201 (2016)

    2016

  11. [19]

    Mosonyi, M., Ogawa, T.: Strong converse exponent for classical-quantum channel coding. Commun. Math. Phys. 355(1), 373–426 (2017)

    2017

  12. [20]

    Tohoku Math

    Umegaki, H.: Conditional expectation in an operator algebra. Tohoku Math. J. 6(2), 177–181 (1954)

    1954

  13. [21]

    D´ enes, P.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23(1), 57–65 (1986)

    1986

  14. [22]

    M¨ uller-Lennert, M., Dupuis, F., Szehr, O., Fehr, S., Tomamichel, M.: On quantum R´ enyi entropies: a new generalization and some properties. J. Math. Phys. 54(12), 122203 (2013)

    2013

  15. [23]

    Wilde, M.M., Winter, A., Yang, D.: Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched R´ enyi relative entropy. Commun. Math. Phys.331(2), 593– 622 (2014)

    2014

  16. [24]

    IEEE Trans

    Cheng, H.-C., Hanson, E.P., Datta, N., Hsieh, M.-H.: Non-asymptotic classical data compression with quantum side information. IEEE Trans. Inf. Theory 67(2), 902–930 (2020)

    2020

  17. [25]

    Li, K., Yao, Y.: Operational interpretation of the sandwiched R´ enyi divergence of order 1/2 to 1 as strong converse exponents. Commun. Math. Phys. 405(2), 22–67 (2024)

    2024

  18. [26]

    Mosonyi, M., Ogawa, T.: Quantum hypothesis testing and the operational interpretation of the quan- tum R´ enyi relative entropies. Commun. Math. Phys.334(3), 1617–1648 (2015)

    2015

  19. [27]

    Gupta, M.K., Wilde, M.M.: Multiplicativity of completely bounded p-norms implies a strong converse for entanglement-assisted capacity. Commun. Math. Phys. 334(2), 867–887 (2015)

    2015

  20. [28]

    Cooney, T., Mosonyi, M., Wilde, M.M.: Strong converse exponents for a quantum channel discrimina- tion problem and quantum-feedback-assisted communication. Commun. Math. Phys. 344(3), 797–829 (2016)

    2016

  21. [29]

    IEEE Tran

    Li, K., Yao, Y.: Strong converse exponent for entanglement-assisted communication. IEEE Tran. Inf. Theory. 70(7), 5017–5029 (2024)

    2024

  22. [30]

    IEEE Trans

    Li, K., Yao, Y., Hayashi, M.: Tight exponential analysis for smoothing the max-relative entropy and for quantum privacy amplification. IEEE Trans. Inf. Theory 69(3), 1680–1694 (2023) 37

    2023

  23. [31]

    arXiv:2112.04475 [quant-ph]

    Li, K., Yao, Y.: Reliable simulation of quantum channels. arXiv:2112.04475 [quant-ph]

    arXiv

  24. [32]

    Datta, N.: Max-relative entropy of entanglement, alias log robustness. Int. J. Quantum. 7(02), 475–491 (2009)

    2009

  25. [33]

    Tomamichel, M., Berta, M., Hayashi, M.: Relating different quantum generalizations of the conditional R´ enyi entropy. J. Math. Phys.55(8), 082206 (2014)

    2014

  26. [34]

    S.: Sandwiched R´ enyi divergence satisfies data processing inequality

    Beigi. S.: Sandwiched R´ enyi divergence satisfies data processing inequality. J. Math. Phys. 54(12), 122202 (2013)

    2013

  27. [35]

    Mckinlay, A., Tomamichel, M.: Decomposition rules for quantum R´ enyi mutual information with an application to information exclusion relations. J. Math. Phys. 61(7), 072202 (2020)

    2020

  28. [36]

    Frank, R., Lieb, E.: Monotonicity of a relative R´ enyi entropy. J. Math. Phys. 54(12), 122201 (2013)

    2013

  29. [37]

    Cam- bridge University Press, Cambridge (2011)

    Csisz´ ar, I., K¨ orner, J.: Information Theory: Coding Theorems for Discrete Memoryless Systems. Cam- bridge University Press, Cambridge (2011)

    2011

  30. [38]

    arXiv:2202.11090 [quant-ph]

    Salzmann, A., Datta, N.: Total insecurity of communication via strong converse for quantum privacy amplification. arXiv:2202.11090 [quant-ph]

    arXiv

  31. [39]

    arXiv:2410.07051 [quant-ph]

    Oufkir, A., Cao, M., Cheng, H., Berta, M.: Exponents for shared randomness-assisted channel simula- tion. arXiv:2410.07051 [quant-ph]

    arXiv

  32. [40]

    Hiai, F., Mosonyi, M., Ogawa, T.: Error exponents in hypothesis testing for correlated states on a spin chain. J. Math. Phys. 49, 032112 (2008)

    2008

  33. [41]

    Devetak, I., Winter, A.: Distillation of secret key and entanglement from quantum states. Proc. R. Soc. A. 461(2053), 207–235 (2005)

    2005

  34. [42]

    arXiv:2202.11590 [quant-ph]

    Shen, Y., Gao, L., Cheng, H.: Optimal second-order rates for quantum soft covering and privacy amplification. arXiv:2202.11590 [quant-ph]

    arXiv

  35. [43]

    arXiv:2202.10263 [quant-ph]

    Shen, Y., Gao, L., Cheng, H.: Strong converse for privacy amplification against quantum side informa- tion. arXiv:2202.10263 [quant-ph]

    arXiv

  36. [44]

    Hayashi, M.: Precise evaluation of leaked information with secure randomness extraction in the presence of quantum attacker. Commun. Math. Phys. 333(1), 335–350 (2015)

    2015

  37. [45]

    Bell Syst

    Shannon, C.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948)

    1948

  38. [46]

    Schumacher, B.: Quantum coding. Phys. Rev. A. J. 51(4), 2738 (1995)

    1995

  39. [47]

    Barnum, H., Fuchs, C., Jozsa, R., Schumacher, B.: General fidelity limit for quantum channels. Phys. Rev. A. J. 54(6), 4707 (1996)

    1996

  40. [48]

    Tomamichel, M.: Quantum Information Processing with Finite Resources: Mathematical Foundations, vol. 5. Springer, Berlin (2015)

    2015

  41. [49]

    Audenaert, K.M., Calsamiglia, J., Munoz-Tapia, R., Bagan, E., Masanes, L., Acin, A., Verstraete, F.: Discriminating states: the quantum Chernoff bound. Phys. Rev. Lett. 98, 160501 (2007)

    2007

  42. [50]

    Nussbaum, M., Szko la, A.: The Chernoff lower bound for symmetric quantum hypothesis testing. Ann. Statist. 37(2), 1040–1057 (2009)

    2009

  43. [51]

    arXiv:quant-ph/0611289 (2006)

    Nagaoka, H.: The converse part of the theorem for quantum Hoeffding bound. arXiv:quant-ph/0611289 (2006)

    2006 arXiv

  44. [52]

    Hayashi, M.: Error exponent in asymmetric quantum hypothesis testing and its application to classical- quantum channel coding. Phys. Rev. A 76, 062301 (2007)

    2007

  45. [53]

    Audenaert, K.M., Nussbaum, M., Szko la, A., Verstraete, F.: Asymptotic error rates in quantum hy- pothesis testing. Commun. Math. Phys. 279(1), 251–283 (2008)

    2008

  46. [54]

    arXiv:2207.08899 [quant-ph]

    Renes, J.M.: Achievable error exponents of data compression with quantum side information and communication over symmetric classical-quantum channels. arXiv:2207.08899 [quant-ph]

    arXiv

  47. [55]

    transition probability

    Uhlmann, A.: The “transition probability” in the state space of a ∗-algebra. Rep. Math. Phys. J. 9(2), 273–279 (1976)

    1976

  48. [56]

    Hayashi, M.: Exponents of quantum fixed-length pure-state source coding. Phys. Rev. A. J. 66(3), 032321 (2002)

    2002

  49. [57]

    Wang, X., Wilde, M.M.: Resource theory of asymmetric distinguishability. Phys. Rev. Research 1, 033170 (2019)

    2019

  50. [58]

    Leditzky, F., Wilde, M.M., Datta, N.: Strong converse theorems using R´ enyi entropies. J. Math. Phys. 57(8), 082202 (2016)

    2016

  51. [59]

    arXiv:1203.2142 [quant-ph]

    Tomamichel, M.: A framework for non-asymptotic quantum information theory. arXiv:1203.2142 [quant-ph]

    arXiv

  52. [60]

    arXiv:2410.12576 [quant-ph] 38

    Berta, M., Yao, Y.: Strong converse exponent of quantum dichotomies. arXiv:2410.12576 [quant-ph] 38

    arXiv

  53. [61]

    arXiv:1707.03619 [quant-ph]

    Anshu, A., Jain, R., Warsi, A.: A unified approach to source and message compression. arXiv:1707.03619 [quant-ph]

    arXiv

  54. [62]

    IEEE Tran

    Vembu, S., Verd´ u, S.: Generating random bits from an arbitrary source: Fundamental limits. IEEE Tran. Inf. Theory. 41(5), 1322–1332 (2002)

    2002

  55. [63]

    IEEE Tran

    Hayashi, M.: Second-order asymptotics in fixed-length source coding and intrinsic randomness. IEEE Tran. Inf. Theory. 54(10), 4619–4637 (2008)

    2008

  56. [64]

    IEEE Tran

    Hayashi, M.: Tight exponential analysis of universally composable privacy amplification and its appli- cations. IEEE Tran. Inf. Theory. 59(11), 7728–7746 (2013)

    2013

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.