Universal chain rules from entropic triangle inequalities

Ashutosh Marwah; Fr\'ed\'eric Dupuis

REVIEW 3 major objections 3 minor 32 references

Universal chain rules from entropic triangle inequalities

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · 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 a variant of the smooth min-entropy satisfies a universal additive chain rule for arbitrarily many registers, and uses the same machinery to prove an approximate entropy accumulation theorem that works for…

desk verdict Strong paper with genuinely new chain rules; the flagged Eq. (105) concern is a subscript typo, not a load-bearing flaw. read the letter →

arxiv 2412.06723 v1 pith:QXG6ZE2Y submitted 2024-12-09 quant-ph

Ashutosh Marwah , Frédéric Dupuis This is my paper · ORCID

classification quant-ph MSC 81P4594A17

keywords smoothmin-entropychainruleentropictriangleinequalityentropyaccumulationtheoremquantuminformationtheoryone-shotgeneralizedGolden-Thompsonsubstate

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 asks whether the smooth min-entropy, a one-shot measure of uncertainty, can be decomposed like the von Neumann entropy into a sum of conditional entropies of individual parts. The standard smooth min-entropy cannot: there are states where any such bound with a fixed smoothing parameter would give a linearly growing right-hand side and a bounded left-hand side. The paper proves that a close variant, $H^{\downarrow,\epsilon}_{\min}$, defined by comparing the state to $\mathbb{1}_A\otimes\rho_B$ rather than optimizing over $\sigma_B$, does admit such a universal chain rule for all $n$ and all $\epsilon\in(0,1)$, with a per-register loss that vanishes as $\epsilon\to0$ and a smoothing parameter on the left that is independent of $n$. Since $H^{\downarrow,\epsilon}_{\min}$ and the standard smooth min-entropy agree up to constants, this gives a way to lower-bound the standard smooth min-entropy by equally strong conditional terms. The same proof technique yields an approximate entropy accumulation theorem that does not require the state to be generated sequentially.

What carries the argument

The load-bearing machinery is the entropic triangle inequality, which lets a smooth min-entropy of $\rho$ be bounded by an unsmoothed min-entropy of an auxiliary state $\sigma$ minus a smooth max-relative entropy: $H^\delta_{\min}(A|B)_\rho \ge H_{\min}(A|B)_\sigma - D^\delta_{\max}(\rho\|\sigma)$. The paper constructs a suitable $\sigma$ from smoothed conditional states $\bar\rho^{(k)}_{A_1^kB}$ using an operator-exponential Ansatz (Eq. (76)). The generalized Golden-Thompson inequality controls the trace exponential that appears in the variational expression for the relative entropy, yielding a measured-relative-entropy bound $D_m(\rho\|\sigma)\le n z(\epsilon,\delta)$. The quantum substate theorem converts this into a smooth max-relative entropy bound, and the triangle inequality transfers the entropy estimate back to $\rho$. Quasi-concavity of $H^{\downarrow}_{\min}$ makes each partial conditional entropy of $\sigma$ inherit the desired lower bound; the same recipe, with a R\'enyi version of the triangle inequality, gives the unstructured approximate entropy accumulation theorem.

What would settle it

Compare the displayed transfer step: after setting $\nu=\epsilon^2/8$, the right-hand side of Eq. (105) must bound entropies of $\rho$, not of the perturbed state $\rho'$, for the theorem as stated. A reader can test this by applying the proof to a rank-deficient classical distribution, such as the one in the introduction where $B$ is uniform and $A_1^n$ is constant when $B=0$, and checking whether the claimed inequality holds with the stated smoothing parameters.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.4: for any normalized quantum state $\rho_{A_1^n B}$ with equal dimensions $|A_k|=|A|$ and any $\epsilon\in(0,1)$, setting $\mu=O((\epsilon\log(|A|/\epsilon))^{1/3})$ gives $$$H^{{\downarrow,2\mu+\epsilon/2}}$_{\min}(A_1^n|B)_\rho \ge \sum_{k=1}^n $H^{{\downarrow,\epsilon/2}}$_{\min}(A_k|$A_1^{{k-1}}$B)_\rho - n\mu - \frac{1}{\$mu^{2}$} - \log\frac{1}{1-\$mu^{2}$} - \log\left(\frac{2}{\$mu^{2}$}+\frac{1}{1-\mu}\right).$$ In words, the smooth min-entropy of the whole string can be additively decomposed into smooth conditional min-entropies of equal strength, up to a loss that grows only linearly in $n$ with a coefficient vanishing with $\epsilon$ and a constant independent of $n$. This is false for the conventional $H^\epsilon_{\min}$; the paper gives a simple classical counterexample. The content is precisely that the variant $H^{\downarrow,\epsilon}_{\min}$, which equals the conventional one up to constants, restores the von Neumann chain-rule structure. Using duality, the same statement becomes an upper chain rule for the smooth max-entropy as Corollary 3.5.

Load-bearing premise

The proof must transfer the bound from full-rank states to arbitrary states even though the entropy involved is discontinuous; the paper does this with a small perturbation, and that transfer is the step the argument depends on.

Editorial extensions

If this is right

For every fixed $\epsilon>0$, the total smoothing cost on the left is $2\mu+\epsilon/2$, independent of $n$, so the decomposition remains meaningful for arbitrarily long registers.
Because $H^{\downarrow,\epsilon}_{\min}$ is within an additive $O(\log 1/\epsilon)$ of the conventional $H^{\epsilon}_{\min}$, the theorem yields lower bounds on the standard smooth min-entropy in terms of equally strong conditional terms.
Purification duality converts the min-entropy chain rule into an upper chain rule for the smooth max-entropy, allowing both entropies to be decomposed along the same lines.
The approximate entropy accumulation theorem applies to arbitrary states, including fully parallel ones, whose prefixes approximate outputs of channels that sample the side information $B_k$ independently; the bound is the sum of von Neumann conditional entropies minus $n\,\tilde O(\epsilon^{1/12})$ plus $\tilde O(\epsilon^{-5/12})$.
The paper states that this unstructured entropy accumulation result enables security proofs for parallel device-independent quantum key distribution in a companion work.
The proof exposes a general route from von Neumann entropy arguments to one-shot arguments: any decomposition that uses the chain rule plus continuity can be replaced by the universal smooth chain rule with explicit smoothing overhead.

Reading between the lines

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

The universal chain rule can act as a bridge: von Neumann entropy arguments built on the ordinary chain rule, such as proofs for approximately independent registers, should convert into one-shot statements with explicit smoothing overhead; the paper motivates this but does not state it as a general transfer theorem.
In cryptographic settings, the smoothing parameter dependence on the approximation error in the unstructured EAT could likely be decoupled by adding a random testing event, since the paper's own example shows that the dependence comes from correlated failure; the authors flag this as a direction for future work.
The same triangle-inequality-plus-substate recipe should apply to other one-shot information measures, such as $I_{\max}$ or multipartite mutual information, as the authors themselves anticipate.
If the non-full-rank transfer step is tightened, the numerical constants of the theorem may shift but the qualitative content, that smooth min-entropy can be decomposed with an $n$-independent smoothing parameter, should survive.

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.

Desk Editor's Note

read the letter

Dear colleague,

Here's my read on arXiv:2412.06723. The two main results are genuinely new and the proof strategy is sound. The stress-test concern about Eq. (105) is real as a textual issue, but I believe it's a subscript typo rather than a gap: Eq. (104) already performs the transfer from ρ′ back to ρ using P(ρ,ρ′) ≤ √(2ν), and Eq. (105) just plugs in ν = ε²/8. The right-hand side should be evaluated on ρ, not ρ′.

What's actually new: Theorem 3.4's universal chain rule for H^{↓,ε}_min gives an additive decomposition of smooth min-entropy for arbitrary n, with smoothing that doesn't grow with n. That's a real advance over [DBWR14] and [VDTR13], which lose O(log(1/δ)) per step. Theorem 4.1's unstructured approximate EAT is also new and removes the sequential-process requirement. The proofs are built from the generalized Golden–Thompson inequality, the quantum substate theorem, and the entropic triangle inequalities from [MD23]; the connections are clean, and I didn't see a circular step.

Soft spots: the non-full-rank extension is terse. The footnote about non-attainment of the H^{↓,ε}_min supremum is a legitimate technicality; a limiting argument with a per-round ε-approximation should patch it, but as written it's an assumption. Also, the authors themselves point out that Theorem 4.1's smoothing parameter depends on the approximation parameter, which limits immediate cryptographic use, but that's an honest limitation, not a flaw.

Bottom line: a strong paper that deserves a serious referee. I'd recommend acceptance after a minor revision that fixes the subscript in Eq. (105) and spells out the supremum/limiting argument. I'd cite it if I worked on one-shot entropy or parallel DIQKD.

Referee Report

3 major / 3 minor

Summary. The paper develops two new tools built on entropic triangle inequalities, the generalized Golden–Thompson inequality, and the quantum substate theorem: a universal chain rule for the smooth min-entropy variant H^{↓,ε}_{min}, and an unstructured approximate entropy accumulation theorem (EAT) that removes the sequential/Markov-chain structure of standard EAT. The universal chain rule (Theorem 3.4) states that for any normalized n-partite state and any ε∈(0,1), the smooth min-entropy of A_1^n given B is lower bounded by the sum of smooth min-entropies of each A_k given A_1^{k-1}B, minus nμ and additional μ-dependent terms, where μ=O((ε log(|A|/ε))^{1/3}). A dual statement for the smooth max-entropy is derived, and an alternative proof of the chain rule is given in Section 5, together with a testing version of the approximate EAT in Appendix E. The stated motivation is the security analysis of parallel device-independent QKD in a companion paper.

Significance. If the results are correct, they constitute a substantial advance in one-shot information theory. The universal chain rule bypasses the known impossibility of an additive chain rule for the conventional smooth min-entropy by using H^{↓,ε}_{min}, which is equivalent up to constant factors, and it yields a per-round loss that vanishes with ε while the smoothing parameter does not grow linearly with n. The unstructured approximate EAT is conceptually important because it relaxes both the sequential production assumption and the Markov-chain condition of [DFR20], and it is concretely used in the authors' companion DIQKD security proof. The paper is unusually explicit: the main proofs are written out in detail, the error terms are given in closed form, and the same chain rule is proved by two independent methods, which is a genuine strength. The main caveats are localized proof gaps around perturbing non-full-rank states, one of which appears to be a typographical error, and the dependence of the approximate EAT's smoothing parameter on the approximation parameter, which the authors themselves discuss.

major comments (3)

[Sec. 3.2, Eq. (105)] The step from Eq. (104) to Eq. (105) is not displayed correctly. Eq. (104) has the right-hand smoothed entropies evaluated on the original state ρ after reducing the smoothing parameter by √(2ν); setting ν=ε²/8 gives H^{↓,ε/2}_{min}(A_k|A_1^{k-1}B)_ρ, but Eq. (105) writes this term with subscript ρ′ instead of ρ. With ρ′ in place, the displayed chain of inequalities proves the bound only for the full-rank perturbed state, and the theorem as stated for arbitrary normalized ρ is not established. If the ρ′ in Eq. (105) is a typographical error for ρ, the proof closes; as printed, this is a load-bearing gap in the proof of the paper's first main result.
[Sec. 3.2, Theorem 3.4 proof and footnote 10] The proof sets λ_k = H^{↓,ε}_{min}(A_k|A_1^{k-1}B)_ρ and then chooses a state ˜ρ^{(k)} attaining this value and satisfying the operator inequality (88). As the footnote concedes, H^{↓}_{min} is discontinuous and the supremum defining H^{↓,ε}_{min} need not be attained. If λ_k is only taken within γ of the supremum, the subsequent chain of inequalities gives a sum of λ_k's and hence the final right-hand side loses nγ; this loss is not present in the displayed theorem. A rigorous treatment requires either proving attainment or explicitly adding the nγ error and then taking γ→0 after the bound is obtained. The same issue recurs in the classical proof of Section 5.1 and in Lemma 5.6, so the footnote's 'throughout this paper' assertion should be replaced by a uniform argument.
[Sec. 4, Proof of Theorem 4.1, Case 2] The extension from full-rank states to arbitrary states in Theorem 4.1 relies on the sentence 'use the continuity of Hmin' after approximating ρ by a full-rank state ρ′. No continuity statement or reference is supplied for the smooth min-entropy as a function of the state with fixed smoothing parameter. This is a standard and true fact, and here the right-hand side is independent of the approximation parameter after taking the limit, so the step is fixable; however, as written the proof of the theorem for non-full-rank states is incomplete. A Lipschitz bound for H^{δ}_{min} in trace distance, or an explicit limiting argument, should be added.

minor comments (3)

[Sec. 4, Eq. (121)] The inequality (√2+1)√(2(ϵ+ν)) ≤ 4√ϵ+ν is false for small ν (for example, ϵ=ν=0.01 gives a left-hand side of about 0.48 and a right-hand side of about 0.41). The proof later lets ν→0, so the theorem's conclusion is unaffected if the bound is replaced by something such as 4(√ϵ+√ν); the definition of z(ϵ+ν,δ) and the subsequent constants should be adjusted accordingly.
[Sec. 5.3, around Eq. (192)] In the purified-distance estimate for η_{A_1^nB}, the last projector is written P^{(n)}_{A_1^nB} without the label g; it should be P^{(n,g)}_{A_1^nB} to match the definition of η. This is a typographical error but it makes the formula momentarily ambiguous.
[Sec. 2.2 and Sec. 3.3] The notation for the two equivalences between H^{ϵ}_{min}, H^{↓,ϵ}_{min}, and the max-entropy variants is dense, with constants such as log(2/ϵ² + 1/(1−ϵ)) appearing both in Eq. (37) and in the proof of Theorem 3.4. A short table or a summary of the exact constants used in each direction would improve readability and make it easier to check the final bounds.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the universal chain rule is derived from independent lemmas and external inequalities; the self-citations are prior proven results.

full rationale

The derivation of Theorem 3.4 constructs an auxiliary state σ via Lemma 3.1 (proved in this paper from the external generalized Golden–Thompson inequality [SBT17] and the variational expressions), then relates ρ to σ by the measured relative entropy bound D_m ≤ nz, converts to smooth max-relative entropy using the substate theorem, and applies the entropic triangle inequality of [MD23] to pass from H_min(σ) to H^µ_min(ρ). None of these steps assumes the chain rule being proved; the chain rule for H↓_min used on σ is the elementary unsmoothed identity from Tomamichel's book. The per-round quantities H^{↓,ε}_min(A_k|A_1^{k-1}B)_ρ are inputs, not outputs of a fit, and the theorem's smoothing and loss terms are explicit functions of ε and |A|. The non-full-rank extension in Eqs. (103)–(105) contains a labeling of ρ versus ρ′ that is a proof gap/correctness concern, not a circularity: even if Eq. (105) is taken as displayed, it does not assume the target inequality. The cited [MD23] lemmas are prior results with independent proofs and are not equivalent to the conclusions here. No circular step found.

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

The paper contains no fits to data and no invented physical entities. The central proofs rely on previously established inequalities and the finite-dimensional quantum formalism. The main self-cited ingredient is [MD23], which is a genuine prior lemma rather than an assumption of the target result.

assumptions (5)

standard math Generalised Golden-Thompson inequality (Theorem 2.3, from [SBT17])
Used in Lemma 3.1 to bound the trace of the exponential of a sum of Hermitian operators and to derive the measured-relative-entropy bound against the auxiliary state.
standard math Quantum substate theorem (Theorem 2.2, from [JRS02, JN11])
Converts measured-relative-entropy bounds into smooth max-relative entropy bounds in both main theorems.
standard math Entropic triangle inequalities (Lemma 2.1, from [MD23])
Self-cited prior result connecting smooth min-entropy, unsmoothed min-entropy, and smooth max-relative entropy; treated as a lemma rather than rederived.
domain assumption Finite-dimensional quantum information formalism
The paper assumes finite-dimensional registers, CPTP channels, subnormalized states, and purified distance throughout; all bounds depend on finite dimensions |A| and |B|.
standard math Projector lemma from [fu23], reproduced as Lemma 5.5
Supports the alternative proof in Section 5. The proof is reproduced, but the lemma originates from a MathOverflow answer rather than a peer-reviewed article.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Universal chain rules from entropic triangle inequalities." pith.science (2026). https://pith.science/paper/QXG6ZE2Y

@misc{pith2026241206723,
  author       = {Pith},
  title        = {Pith review of: Universal chain rules from entropic triangle inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXG6ZE2Y}},
  note         = {Machine review of arXiv:2412.06723}
}

abstract

The von Neumann entropy of an $n$-partite system $A_1^n$ given a system $B$ can be written as the sum of the von Neumann entropies of the individual subsystems $A_k$ given $A_1^{k-1}$ and $B$. While it is known that such a chain rule does not hold for the smooth min-entropy, we prove a counterpart of this for a variant of the smooth min-entropy, which is equal to the conventional smooth min-entropy up to a constant. This enables us to lower bound the smooth min-entropy of an $n$-partite system in terms of, roughly speaking, equally strong entropies of the individual subsystems. We call this a universal chain rule for the smooth min-entropy, since it is applicable for all values of $n$. Using duality, we also derive a similar relation for the smooth max-entropy. Our proof utilises the entropic triangle inequalities for analysing approximation chains. Additionally, we also prove an approximate version of the entropy accumulation theorem, which significantly relaxes the conditions required on the state to bound its smooth min-entropy. In particular, it does not require the state to be produced through a sequential process like previous entropy accumulation type bounds. In our upcoming companion paper, we use it to prove the security of parallel device independent quantum key distribution.

Figures

Figures reproduced from arXiv: 2412.06723 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 11 canonical work pages

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...
[2]

Partially smoothed information measures

Anurag Anshu, Mario Berta, Rahul Jain, and Marco Tomamichel. Partially smoothed information measures. IEEE Transactions on Information Theory , 66(8):5022--5036, 2020. doi:10.1109/TIT.2020.2981573

work page arXiv 2020
[3]

Continuity of quantum conditional information

R Alicki and M Fannes. Continuity of quantum conditional information. Journal of Physics A: Mathematical and General , 37(5):L55–L57, Jan 2004. doi:10.1088/0305-4470/37/5/l01

work page doi:10.1088/0305-4470/37/5/l01 2004
[4]

Practical device-independent quantum cryptography via entropy accumulation

Rotem Arnon-Friedman, Fr \'e d \'e ric Dupuis, Omar Fawzi, Renato Renner, and Thomas Vidick. Practical device-independent quantum cryptography via entropy accumulation. Nature Communications , 9(1):459, 2018. doi:10.1038/s41467-017-02307-4

doi:10.1038/s41467-017-02307-4 2018
[5]

Simple and tight device-independent security proofs

Rotem Arnon-Friedman, Renato Renner, and Thomas Vidick. Simple and tight device-independent security proofs. SIAM Journal on Computing , 48(1):181--225, jan 2019. doi:10.1137/18m1174726

doi:10.1137/18m1174726 2019
[6]

Koenraad M. R. Audenaert. Telescopic relative entropy. In Dave Bacon, Miguel Martin-Delgado, and Martin Roetteler, editors, Theory of Quantum Computation, Communication, and Cryptography , pages 39--52, Berlin, Heidelberg, 2014. Springer Berlin Heidelberg

work page 2014
[7]

On variational expressions for quantum relative entropies

Mario Berta, Omar Fawzi, and Marco Tomamichel. On variational expressions for quantum relative entropies. Letters in Mathematical Physics , 107(12):2239--2265, 2017. doi:10.1007/s11005-017-0990-7

doi:10.1007/s11005-017-0990-7 2017
[8]

Positive Definite Matrices

Rajendra Bhatia. Positive Definite Matrices . Princeton University Press, Princeton, 2007. doi:10.1515/9781400827787

doi:10.1515/9781400827787 2007

Show all 32 references

[9]

One-shot decoupling

Fr \'e d \'e ric Dupuis, Mario Berta, J \"u rg Wullschleger, and Renato Renner. One-shot decoupling. Communications in Mathematical Physics , 328(1):251--284, 2014. doi:10.1007/s00220-014-1990-4

2014 doi
[10]

Entropy accumulation with improved second-order term

Fr\'ed\'eric Dupuis and Omar Fawzi. Entropy accumulation with improved second-order term. IEEE Transactions on Information Theory , 65(11):7596--7612, 2019. doi:10.1109/TIT.2019.2929564

2019
[11]

Entropy accumulation

Fr\'ed\'eric Dupuis, Omar Fawzi, and Renato Renner. Entropy accumulation. Communications in Mathematical Physics , 379(3):867--913, 2020. doi:10.1007/s00220-020-03839-5

2020 doi
[12]

Does approximate equality of quantum states imply operator inequality in a large subspace? MathOverflow https://mathoverflow.net/q/451757, 2023

fedja (https://mathoverflow.net/users/1131/fedja). Does approximate equality of quantum states imply operator inequality in a large subspace? MathOverflow https://mathoverflow.net/q/451757, 2023. (version: 2023-08-01)

2023
[13]

Renes, and David Sutter

Raban Iten, Joseph M. Renes, and David Sutter. Pretty good measures in quantum information theory. In 2017 IEEE International Symposium on Information Theory (ISIT) , page 3195–3199. IEEE Press, 2017. doi:10.1109/ISIT.2017.8007119

2017
[14]

Short proofs of the quantum substate theorem, 2011, 1103.6067 http://arxiv.org/abs/1103.6067

Rahul Jain and Ashwin Nayak. Short proofs of the quantum substate theorem, 2011, 1103.6067 http://arxiv.org/abs/1103.6067 . doi:10.48550/ARXIV.1103.6067

2011 doi arXiv
[15]

R. Jain, J. Radhakrishnan, and P. Sen. Privacy and interaction in quantum communication complexity and a theorem about the relative entropy of quantum states. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings. , pages 429--438, 2002. doi:1...

2002 arXiv
[16]

The operational meaning of min- and max-entropy

Robert K\"onig, Renato Renner, and Christian Schaffner. The operational meaning of min- and max-entropy. IEEE Transactions on Information Theory , 55(9):4337–4347, Sep 2009. doi:10.1109/tit.2009.2025545

2009
[17]

Brown, Jun Zhang, Roger Colbeck, Jingyun Fan, Qiang Zhang, and Jian-Wei Pan

Wen-Zhao Liu, Ming-Han Li, Sammy Ragy, Si-Ran Zhao, Bing Bai, Yang Liu, Peter J. Brown, Jun Zhang, Roger Colbeck, Jingyun Fan, Qiang Zhang, and Jian-Wei Pan. Device-independent randomness expansion against quantum side information. Nature Physics , 17(4):448--451, 2021. doi:10...

2021 doi
[18]

Smooth min-entropy lower bounds for approximation chains, 2023, 2308.11736 http://arxiv.org/abs/2308.11736

Ashutosh Marwah and Fr\'ed\'eric Dupuis. Smooth min-entropy lower bounds for approximation chains, 2023, 2308.11736 http://arxiv.org/abs/2308.11736

2023 arXiv
[19]

Proving security of parallel device independent QKD using approximate entropy accumulation

Ashutosh Marwah and Fr\'ed\'eric Dupuis. Proving security of parallel device independent QKD using approximate entropy accumulation. Manuscript in preparation , 2024

2024
[20]

Proving security of BB84 under source correlations, 2024, 2402.12346 http://arxiv.org/abs/2402.12346

Ashutosh Marwah and Frédéric Dupuis. Proving security of BB84 under source correlations, 2024, 2402.12346 http://arxiv.org/abs/2402.12346 . ://arxiv.org/abs/2402.12346

2024 arXiv
[21]

Generalised entropy accumulation, 2022, 2203.04989 http://arxiv.org/abs/2203.04989

Tony Metger, Omar Fawzi, David Sutter, and Renato Renner. Generalised entropy accumulation, 2022, 2203.04989 http://arxiv.org/abs/2203.04989 . doi:10.48550/ARXIV.2203.04989

2022 doi arXiv
[22]

A variational expression for the relative entropy

D \'e nes Petz. A variational expression for the relative entropy. Communications in Mathematical Physics , 114(2):345--349, 1988. doi:10.1007/BF01225040

1988 doi
[23]

Security of Quantum Key Distribution

Renato Renner. Security of Quantum Key Distribution . PhD thesis, 2006, quant-ph/0512258 http://arxiv.org/abs/quant-ph/0512258

2006 arXiv
[24]

Universally composable privacy amplification against quantum adversaries

Renato Renner and Robert K \"o nig. Universally composable privacy amplification against quantum adversaries. In Joe Kilian, editor, Theory of Cryptography , pages 407--425, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg

2005
[25]

Multivariate trace inequalities

David Sutter, Mario Berta, and Marco Tomamichel. Multivariate trace inequalities. Communications in Mathematical Physics , 352(1):37--58, 2017

2017
[26]

Quantum coding

Benjamin Schumacher. Quantum coding. Phys. Rev. A , 51:2738--2747, Apr 1995. doi:10.1103/PhysRevA.51.2738

1995 doi
[27]

Quantum Information Processing with Finite Resources

Marco Tomamichel. Quantum Information Processing with Finite Resources . Springer International Publishing, 2016. doi:10.1007/978-3-319-21891-5

2016 doi
[28]

Leftover hashing against quantum side information

Marco Tomamichel, Renato Renner, Christian Schaffner, and Adam Smith. Leftover hashing against quantum side information. In IEEE International Symposium on Information Theory , pages 2703 --2707, June 2010. doi:10.1109/ISIT.2010.5513652

2010
[29]

Chain rules for smooth min- and max-entropies

Alexander Vitanov, Fr\'ed\'eric Dupuis, Marco Tomamichel, and Renato Renner. Chain rules for smooth min- and max-entropies. IEEE Transactions on Information Theory , 59(5):2603--2612, 2013. doi:10.1109/TIT.2013.2238656

2013
[30]

The Theory of Quantum Information

John Watrous. The Theory of Quantum Information . Cambridge University Press, 2018. doi:10.1017/9781316848142

2018 doi
[31]

Mark M. Wilde. Quantum Information Theory . Cambridge University Press, 2013. doi:10.1017/CBO9781139525343

2013 doi
[32]

Tight uniform continuity bounds for quantum entropies: Conditional entropy, relative entropy distance and energy constraints

Andreas Winter. Tight uniform continuity bounds for quantum entropies: Conditional entropy, relative entropy distance and energy constraints. Communications in Mathematical Physics , 347(1):291--313, 2016. doi:10.1007/s00220-016-2609-8

2016 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

[2] [2]

Partially smoothed information measures

Anurag Anshu, Mario Berta, Rahul Jain, and Marco Tomamichel. Partially smoothed information measures. IEEE Transactions on Information Theory , 66(8):5022--5036, 2020. doi:10.1109/TIT.2020.2981573

work page arXiv 2020

[3] [3]

Continuity of quantum conditional information

R Alicki and M Fannes. Continuity of quantum conditional information. Journal of Physics A: Mathematical and General , 37(5):L55–L57, Jan 2004. doi:10.1088/0305-4470/37/5/l01

work page doi:10.1088/0305-4470/37/5/l01 2004

[4] [4]

Practical device-independent quantum cryptography via entropy accumulation

Rotem Arnon-Friedman, Fr \'e d \'e ric Dupuis, Omar Fawzi, Renato Renner, and Thomas Vidick. Practical device-independent quantum cryptography via entropy accumulation. Nature Communications , 9(1):459, 2018. doi:10.1038/s41467-017-02307-4

doi:10.1038/s41467-017-02307-4 2018

[5] [5]

Simple and tight device-independent security proofs

Rotem Arnon-Friedman, Renato Renner, and Thomas Vidick. Simple and tight device-independent security proofs. SIAM Journal on Computing , 48(1):181--225, jan 2019. doi:10.1137/18m1174726

doi:10.1137/18m1174726 2019

[6] [6]

Koenraad M. R. Audenaert. Telescopic relative entropy. In Dave Bacon, Miguel Martin-Delgado, and Martin Roetteler, editors, Theory of Quantum Computation, Communication, and Cryptography , pages 39--52, Berlin, Heidelberg, 2014. Springer Berlin Heidelberg

work page 2014

[7] [7]

On variational expressions for quantum relative entropies

Mario Berta, Omar Fawzi, and Marco Tomamichel. On variational expressions for quantum relative entropies. Letters in Mathematical Physics , 107(12):2239--2265, 2017. doi:10.1007/s11005-017-0990-7

doi:10.1007/s11005-017-0990-7 2017

[8] [8]

Positive Definite Matrices

Rajendra Bhatia. Positive Definite Matrices . Princeton University Press, Princeton, 2007. doi:10.1515/9781400827787

doi:10.1515/9781400827787 2007