Pith. sign in

REVIEW 2 major objections 4 minor 33 references

The entanglement-assisted transmission capacity is a strong converse bound for identification

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read The paper proves that for every finite-dimensional quantum channel N, the identification capacity is bounded by the entanglement-assisted transmission capacity, $C_{\mathrm{ID}}(N)\leq C_E(N)$, with equality on low-noise channels and…

desk verdict A serious strong-converse result for identification capacity that deserves refereeing, with the main caveat being its dependence on an unproved companion bound. read the letter →

arxiv 2608.11000 v1 pith:ORDQGT64 submitted 2026-08-11 quant-ph cs.ITmath.ITmath.MG

Satvik Singh This is my paper · ORCID
classification quant-phcs.ITmath.ITmath.MG MSC 81P4581P4794A17
keywords identificationcapacityentanglement-assistedclassicalquantumchannelsstrongconverseboundGaussianmeanwidthchannelmax-informationtranspose-depolarizingsuperadditivity
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

Identification asks the receiver only whether the sent message equals a queried one, so the number of identifiable messages can grow doubly exponentially with blocklength. The paper's central claim is that this doubly exponential rate is still capped by the entanglement-assisted transmission capacity: $C_{\mathrm{ID}}(\mathcal{N})\le C_E(\mathcal{N})$ for every finite-dimensional quantum channel $\mathcal{N}$. That closes a previously open converse for identification and, for sufficiently low-noise channels, combines with known achievability to give the exact equality $C_{\mathrm{ID}}(\mathcal{N}) = C_E(\mathcal{N})$. For general channels the bound can be strict, and the paper exhibits transpose-depolarizing channels where $C_{\mathrm{ID}} < C_E$, leading also to strict superadditivity of the identification capacity.

What carries the argument

The workhorse is the Euclidean Gaussian mean width converse functional $\mu^*(\mathcal{N}) = \inf_W K(W)^2 \|Q_{\mathcal{N},W}\|_\infty$, where $W$ ranges over Euclidean structures on the output Hermitian space, $K(W)$ is the trace-norm domination cost, and $Q_{\mathcal{N},W}$ is the $W$-weighted singular operator of the channel. The new technical step is Lemma 3.2, which proves $\mu^*(\mathcal{N}) \le 2^{I_{\max}(\mathcal{N})+1}$ by choosing $W$ to be the arithmetic-mean metric induced by a feasible state in the channel max-information optimization, so that the domination cost is one and the singular operator is controlled by the Choi operator inequality. This connects a purely geometric packing bound to the channel smooth max-information, whose asymptotic equipartition property produces the single-letter entanglement-assisted capacity.

What would settle it

Construct any identification code for a fixed finite-dimensional channel whose double-exponential rate exceeds $C_E(\mathcal{N})$ by a fixed positive amount; the paper proves no such code exists, so one example would falsify Theorem 3.3. A more targeted check is to look for an ID code for the transpose-depolarizing channel at $q=d/(d+1)$ with rate above $\log d - \frac{d-1}{d}\log(d+1)$, which Proposition 3.5 rules out.

Watch

Extended reading notes

Core claim

The paper establishes that the entanglement-assisted classical transmission capacity $C_E$ is a strong converse bound for classical identification over quantum channels: for every finite-dimensional channel $\mathcal{N}$ and every admissible error pair, $\limsup_{n\to\infty} (1/n)\log\log N^{(n,\lambda_1,\lambda_2)}(\mathcal{N}) \le C_E(\mathcal{N})$. The proof converts any ID code into a packing of the channel's output image, applies the Euclidean Gaussian-mean-width converse to a slightly perturbed copy of the $n$-fold channel, bounds the resulting geometric functional by the channel's smooth max-information, and takes the blocklength limit through the asymptotic equipartition property. For low-noise channels, the known quantum-identification achievability result matches this upper bound, giving $C_{\mathrm{ID}} = C_E$. The transpose-depolarizing channel at $q=d/(d+1)$ provides strict separation, with $C_{\mathrm{ID}} \le \log d - \frac{d-1}{d}\log(d+1) < C_E = 1 + \log\frac{d}{d+1}$, and as $d$ grows the gap widens because $C_{\mathrm{ID}}$ tends to zero while $C_E$ tends to one. The same example yields strict superadditivity of $C_{\mathrm{ID}}$ when the channel is tensored with a sufficiently large noiseless channel.

Load-bearing premise

The universal bound inherits the validity of the Gaussian mean width converse from the author's earlier paper, which is cited but not proved here, and also the asymptotic equipartition property for smooth channel max-information; if either fails for the smoothed channels used in the proof, the conclusion $C_{\mathrm{ID}}\le C_E$ would not be established.

Editorial extensions

If this is right

  • For every finite-dimensional quantum channel, any identification code has double-exponential rate at most the entanglement-assisted transmission capacity, closing the missing universal converse.
  • On low-noise channels, identification capacity equals entanglement-assisted capacity: $C_{\mathrm{ID}} = Q_{\mathrm{ID},v} = Q_{\mathrm{ID}} = C_E$.
  • The bound is not tight in general: for the transpose-depolarizing channel at $q=d/(d+1)$, $C_{\mathrm{ID}} < C_E$, and the gap grows with dimension, with $C_{\mathrm{ID}}$ tending to zero while $C_E$ tends to one.
  • Identification capacity is strictly superadditive: for $m>(d+1)/2$, $C_{\mathrm{ID}}(D^T_{d/(d+1)}\otimes \mathrm{id}_m) > C_{\mathrm{ID}}(D^T_{d/(d+1)}) + C_{\mathrm{ID}}(\mathrm{id}_m)$.
  • The $C_E$ converse dominates the previously known capacity-based and state-weighted Gaussian converse bounds, as illustrated for the qubit depolarizing channel.

Reading between the lines

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

  • Extension: because identification capacity depends only on the channel's output image, the image-optimized upper bound $\bar{C}_{\mathrm{im}}$ defined in the paper may be tight in regimes where $C_E$ is loose; computing it for specific channels is left open.
  • Testable extension: Lemma 3.2 shows the arithmetic-mean metric is optimal among monotone metrics, but not among all Euclidean structures; a numerical search for a non-monotone $W$ with $\mu^*(\mathcal{N}) < 2^{I_{\max}(\mathcal{N})+1}$ would probe whether the smoothing step can be sharpened.
  • Implicit consequence: since channels with the same output image can have different $C_E$ values, the strict gap identifies a genuinely non-transmission component of identification capacity, so entanglement assistance alone does not account for the power of identification.
Share X Bluesky LinkedIn Reddit HN

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 claims two main results. First, it proves a universal strong converse bound for classical identification over every finite-dimensional quantum channel: C_ID(N) ≤ C_E(N), where C_E is the entanglement-assisted classical transmission capacity. The proof combines a one-shot Euclidean Gaussian mean width converse (Theorem 2.10, imported from the companion preprint [25]) with a new technical lemma (Lemma 3.2) relating the optimized geometric functional to the channel max-information via the Bures metric, followed by smoothing and an asymptotic equipartition property for the smooth max-information. Second, the paper shows that this upper bound can be strict: for the transpose-depolarizing channel at q = d/(d+1), C_ID < C_E. As a corollary, it obtains strict superadditivity of C_ID by tensoring this channel with a sufficiently large noiseless channel. For low-noise channels, the converse combines with Hayden-Winter achievability to yield C_ID = C_E.

Significance. If the proof is correct, this is a substantial advance: it settles an open question raised by Winter about whether C_ID ≥ C_E holds universally, and it does so in the negative direction, while providing the first characterization of identification capacity for a nontrivial class of quantum channels. The separation example is concrete and explicit, and the resulting strict superadditivity of C_ID is a new structural phenomenon. The proof strategy is also interesting in its own right: Lemma 3.2 establishes a clean connection between the geometric converse functional of [25] and the channel max-information, using the Bures metric, and the smoothing step is a natural way to convert one-shot geometric bounds into an asymptotic strong converse. The paper is also commendable for stating precise one-shot comparisons and for including a comparison with existing converse bounds. However, the central theorem is not self-contained: Theorem 2.10 and the key width estimate (2.51) are taken from an unreviewed companion preprint, and the proof of Theorem 3.3 applies that theorem to non-tensor-product smoothed channels, a case not covered by the theorem as stated in the present manuscript.

major comments (2)
  1. [§3.1, Eqs. (3.30)–(3.31)] The diamond-norm smoothing step in the proof of Theorem 3.3 contains a factor-of-two error. From the bound (1/2)‖Ñ_n − N^{⊗n}‖⋄ ≤ ε one obtains d_Tr(ω_i, ω̃_i) ≤ ε, but for every effect 0 ≤ D ≤ 1 the inequality |Tr((ω̃_i − ω_i)D)| ≤ ‖ω̃_i − ω_i‖₁ ≤ 2ε holds, not ε. Consequently (3.30) and (3.31) should read 1 − λ₁ − 2ε and λ₂ + 2ε, respectively, and the packing gap becomes Δ − 4ε rather than Δ − 2ε. The correct choice is ζ = (Δ − 4ε)/2, which is positive under the stated assumption ε < Δ/4. The final rate statement is unaffected because ε → 0, but the displayed proof must be corrected.
  2. [§3.1, Theorem 3.3; §2.3, Eq. (2.51)] The central bound C_ID(N) ≤ C_E(N) is obtained by applying Theorem 2.10 to the smoothed channel Ñ_n, which is an arbitrary CPTP map on A^{⊗n} → B^{⊗n}. However, Theorem 2.10 as stated is a theorem about the n-fold tensor power N^{⊗n} of a fixed single-letter channel N, and its right-hand side contains log d_A for the single-letter input dimension. The paper neither states nor proves the one-shot version needed in (3.33), in which the input dimension is d_A^n and the channel need not be a tensor product. Moreover, the load-bearing width estimate (2.51) is imported verbatim from the companion preprint [25] without proof; if that estimate has a hidden assumption, a wrong dimension factor, or a superlinear n dependence, the strong converse would not follow. I therefore ask that the authors either prove the required general one-shot Euclidean converse (including (2.51)) in an appendix, or reproduce the precise statement from [25] and verify that it covers arbitrary block channels. This is not a stylistic point; it is the load-bearing step of the main theorem.
minor comments (4)
  1. [§2.3, Eq. (2.52)] The notation in Theorem 2.10 uses log d_A, which is natural for a single-letter channel, but the proof of Theorem 3.3 later uses the same symbol for the logarithm of the block input dimension d_A^n. The paper should clarify that the one-shot theorem is being used in its block-length-one form with input dimension d_A^n, or restate Theorem 2.10 in a form that covers arbitrary channels.
  2. [§4, Theorem 4.2] The equality C_im(N) = max{C(N), Q_ID,v(N)} is justified by image-monotonicity of C and Q_ID,v, but the displayed argument does not explain why image containment I_1(M) ⊆ I_k(N) at single-block level implies the containment I_m(M) ⊆ I_{km}(N) needed for all tensor powers, especially when the inputs to the m blocks are entangled. This step should be spelled out or the claim weakened.
  3. [Definition 2.6] The notation g ∼ N(0, 1_m) for a standard Gaussian vector is nonstandard and slightly ambiguous; N(0, I_m) would be clearer.
  4. [Acknowledgements] The sentence 'Motivated by these discussions.' is a fragment and should be merged with the following sentence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.3 is a genuine reduction to independent one-shot bounds, and the same-author Theorem 2.10 is parameter-free external evidence rather than a definitional input.

full rationale

I walked the derivation chain of Theorem 3.3. Lemma 3.2 is proved in-paper: starting from a feasible V_B for I_max, it constructs the Bures Euclidean structure W_{sigma,B} and directly computes K(W)=1 and ||Q_{N,W}||_infty <= 2 Tr V_B; this is a new self-contained link. Theorem 3.3 then applies Theorem 2.10 to the diamond-close channel tilde N_n, combines Lemma 3.2 with the smoothing AEP (Theorem 3.1, an external non-self-cited result [13]), and takes the n->infty and epsilon->0 limits. The only same-author input is Theorem 2.10 from [25], quoted as the one-shot form of [25, Thm 5.2]; it is parameter-free, with no fitted values and with stated assumptions that do not include C_ID <= C_E, so by the independence rule it is real evidence rather than a circular step. The strictness example (Prop. 3.5) follows from channel covariance and the proved converse, not from the claimed conclusion; no fitted parameter is renamed as a prediction. The comparison propositions are auxiliary and do not feed back into the main bound. Any concern about the correctness or provenance of [25, Thm 5.2] is a reliance or correctness-risk issue, not a circularity issue.

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

The central claim is derived from standard information-theoretic tools plus one non-textbook geometric bound (Theorem 2.10) imported from the author's own preprint [25]. No numbers are fitted to data, and no new physical entities are introduced. The external theorems are the main assumptions.

assumptions (6)
  • standard math Euclidean Gaussian mean width converse (Theorem 2.10) bounds ID code size in terms of the functional mu* defined in Eq. (2.53).
    Imported from [25] (same author's prior preprint); proof not reproduced here. Essential for the smoothing step in Theorem 3.3.
  • standard math Channel smooth max-information asymptotic equipartition property (Theorem 3.1, from [13]).
    Imported from published work [13]; used to convert one-shot bounds into the capacity C_E.
  • standard math Hayden-Winter quantum identification capacity formula Q_ID(N) = lim 1/n Q^(1)(N^{⊗n}) (Eq. (2.16)).
    Imported from [14]; used for the low-noise achievability lower bound C_ID ≥ C_E and for Corollary 3.6.
  • standard math Additivity of entanglement-assisted classical capacity C_E.
    Used in Corollary 3.4 and Corollary 3.6 to evaluate product channels; standard result from [29].
  • standard math Identification capacity of the noiseless identity channel: C_ID(id_m) = 2 log m (Eq. (1.1)).
    Used in Corollary 3.6 to establish strict superadditivity; known result from [30,4,11].
  • standard math Sudakov inequality for Gaussian mean width (Lemma 2.7).
    Textbook result, used in the Euclidean converse Theorem 2.10.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The entanglement-assisted transmission capacity is a strong converse bound for identification." pith.science (2026). https://pith.science/paper/ORDQGT64

@misc{pith2026260811000,
  author       = {Pith},
  title        = {Pith review of: The entanglement-assisted transmission capacity is a strong converse bound for identification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ORDQGT64}},
  note         = {Machine review of arXiv:2608.11000}
}
abstract

Classical identification via a noisy channel is a communication task in which the receiver is not required to reconstruct the full transmitted message, but only to decide whether it coincides with a message of interest. This relaxation allows the number of identifiable messages to grow doubly exponentially with the blocklength. For quantum channels, the resulting (doubly exponential) identification capacity $C_{\mathrm{ID}}$ can strictly exceed the ordinary (exponential) transmission capacity $C$. In this paper, we prove that the entanglement-assisted transmission capacity $C_E$ is a strong converse bound for this task: $C_{\mathrm{ID}}\leq C_E$. For sufficiently low-noise channels, this bound can also be achieved via the Hayden-Winter (quantum) identification + fingerprinting codes. This yields an exact characterization $C_{\mathrm{ID}}=C_E$ of identification capacity for such channels. However, for general channels, we prove that this upper bound can be strict. We exhibit an explicit family of transpose-depolarizing channels for which $C_{\mathrm{ID}}<C_E$. As a consequence, we also obtain the first example of strict superadditivity of the identification capacity $C_{\mathrm{ID}}$.

Figures

Figures reproduced from arXiv: 2608.11000 by the authors.

Figure 1
Figure 1. Schematic of classical message transmission over n uses of a noisy channel N . A message i ∈ [N] is encoded into a quantum state ρi , transmitted through n uses of the channel, and decoded by a POVM to produce an estimate ˆi. For a given n ∈ N, λ ∈ [0, 1), the maximum size of all (n, N, λ) codes for N is N(n,λ)(N ) := max{N : ∃(n, N, λ) classical transmission code for N }, (2.4) and the classical transmission capaci… view at source ↗
Figure 2
Figure 2. Schematic of classical message identification over n uses of a noisy quantum channel N . A message i ∈ [N] is encoded into a quantum state ρi , transmitted via n uses of the channel, and the receiver performs the binary test {Dj , 1 − Dj} to decide whether the transmitted message was j ∈ [N]. Definition 2.2 A (n, N, λ1, λ2) (classical) identification (ID) code for a quantum channel N : L(A) → L(B) is defined by pair… view at source ↗
Figure 3
Figure 3. Strong converse and achievability bounds on the classical identification capacity of the qubit depolarizing channel Dp (see Eq. (3.39)) for 0 ≤ p ≤ 1. The light green curve shows the CE converse bounds from Theorem 3.3. The blue curve shows the state-weighted Gaussian converse bound from [25, Theorem 3.8] (see (3.61)). The orange curve shows the Ellipsoid converse bound from [32]. The purple and red curves show the … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 11 canonical work pages

  1. [25]

    Gaussian mean width strong converse bound on the classical identification capacity of quantum channels

    Satvik Singh. Gaussian mean width strong converse bound on the classical identifica- tion capacity of quantum channels, 2026. URL:https://arxiv.org/abs/2606.05032, arXiv:2606.05032

    work page Pith review arXiv 2026

  2. [1]

    Ahlswede and G

    R. Ahlswede and G. Dueck. Identification via channels.IEEE Transactions on In- formation Theory, 35(1):15–29, 1989. URL:http://dx.doi.org/10.1109/18.42172, doi:10.1109/18.42172

    doi:10.1109/18.42172 1989

  3. [2]

    Ahlswede and A

    R. Ahlswede and A. Winter. Strong converse for identification via quantum channels. IEEE Transactions on Information Theory, 48(3):569–579, March 2002. URL:http: //dx.doi.org/10.1109/18.985947,doi:10.1109/18.985947

    doi:10.1109/18.985947 2002

  4. [3]

    American Mathematical Society, June 2015

    Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali Milman.Asymptotic Ge- ometric Analysis, Part I. American Mathematical Society, June 2015. URL:http: //dx.doi.org/10.1090/surv/202,doi:10.1090/surv/202

    doi:10.1090/surv/202 2015

  5. [4]

    Sandeep Pradhan, and Andreas Winter

    Touheed Anwar Atif, S. Sandeep Pradhan, and Andreas Winter. Quantum soft- covering lemma with applications to rate-distortion coding, resolvability and iden- tification via quantum channels.International Journal of Quantum Information, 22(05), 2024. URL:http://dx.doi.org/10.1142/S0219749924400136,doi:10.1142/ s0219749924400136

    doi:10.1142/s0219749924400136 2024

  6. [5]

    Sandwiched r´ enyi divergence satisfies data processing inequality.Journal of Mathematical Physics, 54(12), December 2013

    Salman Beigi. Sandwiched r´ enyi divergence satisfies data processing inequality.Journal of Mathematical Physics, 54(12), December 2013. URL:http://dx.doi.org/10.1063/ 1.4838855,doi:10.1063/1.4838855

    doi:10.1063/1.4838855 2013

  7. [6]

    Bennett, Peter W

    Charles H. Bennett, Peter W. Shor, John A. Smolin, and Ashish V. Thapliyal. Entanglement-assisted classical capacity of noisy quantum channels.Phys. Rev. Lett., 83:3081–3084, Oct 1999. URL:https://link.aps.org/doi/10.1103/PhysRevLett. 83.3081,doi:10.1103/PhysRevLett.83.3081

    doi:10.1103/physrevlett 1999

  8. [7]

    Bhatia.Positive Definite Matrices

    R. Bhatia.Positive Definite Matrices. Princeton Series in Applied Mathematics. Princeton University Press, 2015. URL:https://books.google.co.in/books?id= Y22YDwAAQBAJ. 25

    work page 2015

Show all 33 references
  1. [8]

    Springer New York, 1997

    Rajendra Bhatia.Matrix Analysis. Springer New York, 1997. URL:http://dx.doi. org/10.1007/978-1-4612-0653-8,doi:10.1007/978-1-4612-0653-8

    1997 doi

  2. [9]

    Braunstein and Carlton M

    Samuel L. Braunstein and Carlton M. Caves. Statistical distance and the geometry of quantum states.Physical Review Letters, 72(22):3439–3443, 1994.doi:10.1103/ PhysRevLett.72.3439

    1994

  3. [10]

    Quantum finger- printing.Physical Review Letters, 87(16), 2001

    Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum finger- printing.Physical Review Letters, 87(16), 2001. URL:http://dx.doi.org/10.1103/ PhysRevLett.87.167902,doi:10.1103/physrevlett.87.167902

    2001 doi

  4. [11]

    Springer Nature Switzerland, 2025

    Pau Colomer, Christian Deppe, Holger Boche, and Andreas Winter.Zero-Entropy Encoders and Simultaneous Decoders in Identification via Quantum Channels, page 478–502. Springer Nature Switzerland, 2025. URL:http://dx.doi.org/10.1007/ 978-3-031-82014-4_18,doi:10.1007/978-3-031-82014-4_18

    2025 doi

  5. [12]

    Cover and Joy A

    Thomas M. Cover and Joy A. Thomas.Elements of Information Theory. Wiley, April

  6. [13]

    Quantum channel simu- lation and the channel’s smooth max-information.IEEE Transactions on Information Theory, 66(4):2129–2140, 2020.arXiv:1807.05354,doi:10.1109/TIT.2019.2943858

    Kun Fang, Xin Wang, Marco Tomamichel, and Mario Berta. Quantum channel simu- lation and the channel’s smooth max-information.IEEE Transactions on Information Theory, 66(4):2129–2140, 2020.arXiv:1807.05354,doi:10.1109/TIT.2019.2943858

    2020 arXiv

  7. [14]

    Weak decoupling duality and quantum identifi- cation.IEEE Transactions on Information Theory, 58(7):4914–4929, July 2012

    Patrick Hayden and Andreas Winter. Weak decoupling duality and quantum identifi- cation.IEEE Transactions on Information Theory, 58(7):4914–4929, July 2012. URL: http://dx.doi.org/10.1109/TIT.2012.2191695,doi:10.1109/tit.2012.2191695

    2012

  8. [15]

    From quasi-entropy to various quantum information quan- tities.Publications of the Research Institute for Mathematical Sciences, 48(3):525–542, 2012.doi:10.2977/PRIMS/79

    Fumio Hiai and D´ enes Petz. From quasi-entropy to various quantum information quan- tities.Publications of the Research Institute for Mathematical Sciences, 48(3):525–542, 2012.doi:10.2977/PRIMS/79

    2012 doi

  9. [16]

    A.S. Holevo. The capacity of the quantum channel with general signal states.IEEE Transactions on Information Theory, 44(1):269–273, 1998.doi:10.1109/18.651037

    1998 doi

  10. [17]

    Springer Berlin Heidelberg, 1991

    Michel Ledoux and Michel Talagrand.Probability in Banach Spaces. Springer Berlin Heidelberg, 1991. URL:http://dx.doi.org/10.1007/978-3-642-20212-4,doi:10. 1007/978-3-642-20212-4

    1991 doi

  11. [18]

    Andrew Lesniewski and Mary Beth Ruskai. Monotone Riemannian metrics and rela- tive entropy on noncommutative probability spaces.Journal of Mathematical Physics, 40(11):5702–5724, 1999.arXiv:math-ph/9808016,doi:10.1063/1.533053

    1999 arXiv

  12. [19]

    PhD thesis, Bielefeld University., 1999

    Peter L¨ ober.Quantum channels and simultaneous ID coding. PhD thesis, Bielefeld University., 1999. URL:https://pub.uni-bielefeld.de/record/2303327. 26

    1999

  13. [20]

    On quantum R´ enyi entropies: A new generalization and some proper- ties.Journal of Mathematical Physics, 54(12), December 2013

    Martin M¨ uller-Lennert, Fr´ ed´ eric Dupuis, Oleg Szehr, Serge Fehr, and Marco Tomamichel. On quantum R´ enyi entropies: A new generalization and some proper- ties.Journal of Mathematical Physics, 54(12), December 2013. URL:http://dx.doi. org/10.1063/1.4838856,doi:10.1063/1.4838856

    2013 doi

  14. [21]

    Monotone metrics on matrix spaces.Linear Algebra and its Applications, 244:81–96, 1996.doi:10.1016/0024-3795(94)00211-8

    D´ enes Petz. Monotone metrics on matrix spaces.Linear Algebra and its Applications, 244:81–96, 1996.doi:10.1016/0024-3795(94)00211-8

    1996 doi

  15. [22]

    Introduction to quantum Fisher information

    D´ enes Petz and C˘ at˘ alin Ghinea. Introduction to quantum Fisher information. In Miguel Orszag and Rolando Rebolledo, editors,Quantum Probability and Related Topics, vol- ume 27 ofQP–PQ: Quantum Probability and White Noise Analysis, pages 261–281. World Scientific, Singapor...

    2011 arXiv

  16. [23]

    Westmoreland

    Benjamin Schumacher and Michael D. Westmoreland. Sending classical information via noisy quantum channels.Phys. Rev. A, 56:131–138, Jul 1997. URL:https://link. aps.org/doi/10.1103/PhysRevA.56.131,doi:10.1103/PhysRevA.56.131

    1997 doi

  17. [24]

    C. E. Shannon. A mathematical theory of communication.Bell System Technical Journal, 27(3):379–423, July 1948. URL:http://dx.doi.org/10.1002/j.1538-7305. 1948.tb01338.x,doi:10.1002/j.1538-7305.1948.tb01338.x

    1948

  18. [26]

    V. N. Sudakov. Gaussian random processes and measures of solid angles in Hilbert space.Dokl. Akad. Nauk SSSR, 197(1):43–45, 1971. URL:https://mathscinet.ams. org/mathscinet-getitem?mr=0288832

    1971

  19. [27]

    Cambridge University Press, September 2018

    Roman Vershynin.High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press, September 2018. URL:http://dx.doi. org/10.1017/9781108231596,doi:10.1017/9781108231596

    2018 doi

  20. [28]

    Cambridge University Press, 2018

    John Watrous.The Theory of Quantum Information. Cambridge University Press, 2018

    2018

  21. [29]

    Wilde.Quantum Information Theory

    Mark M. Wilde.Quantum Information Theory. Cambridge University Press, November 2016. URL:http://dx.doi.org/10.1017/9781316809976,doi:10.1017/ 9781316809976

    2016 doi

  22. [30]

    Quantum and classical message identification via quantum channels

    Andreas Winter. Quantum and classical message identification via quantum channels. In Osamu Hirota, editor,Quantum Information, Statistics, Probability: Dedicated to Alexander S. Holevo on the Occasion of His 60th Birthday, pages 171–188. Rinton Press, Princeton, NJ, 2004. Rep...

    2004 arXiv

  23. [31]

    Springer Berlin Heidelberg, 2013

    Andreas Winter.Identification via Quantum Channels, page 217–233. Springer Berlin Heidelberg, 2013. URL:http://dx.doi.org/10.1007/978-3-642-36899-8_9,doi: 10.1007/978-3-642-36899-8_9

    2013 doi

  24. [32]

    Strong converse bounds on the classical identification capacity of the qubit depolarizing channel, 2026

    Liuhang Ye, Bjarne Bergh, and Nilanjana Datta. Strong converse bounds on the classical identification capacity of the qubit depolarizing channel, 2026. URL:https://arxiv. org/abs/2603.29987,arXiv:2603.29987. 28

    2026

  25. [2005]

    URL:http://dx.doi.org/10.1002/047174882X,doi:10.1002/047174882x

    doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.