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arxiv: 2605.18726 · v1 · pith:DBP43JRKnew · submitted 2026-05-18 · 🪐 quant-ph · math-ph· math.MP

Quantum Shannon theory made robust: a tale of three protocols for almost i.i.d. sources

Filippo Girardi , Nilanjana Datta , Giacomo De Palma , Ludovico Lami This is my paper

Pith reviewed 2026-05-20 11:13 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords quantum Shannon theoryalmost i.i.d. sourceshypothesis testingdata compressionchannel codingclub distancediamond distancerobust protocols
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The pith

Robust protocols achieve optimal asymptotic rates for any almost i.i.d. quantum source in hypothesis testing, data compression and channel coding.

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

The paper establishes that protocols for hypothesis testing, data compression, and channel coding attain the same optimal asymptotic rates when the underlying resource is an arbitrary almost i.i.d. source rather than an exact i.i.d. one. It shows this is possible without knowing the precise form of any deviation from i.i.d. behavior, provided the deviation remains bounded in a new distance called the club distance. A reader would care because real quantum systems rarely satisfy the exact i.i.d. assumption that underpins standard rate calculations, so small unknown perturbations can otherwise cause protocols to fail. The work also defines almost i.i.d. processes as those whose club distance to true i.i.d. processes vanishes in the large-system limit.

Core claim

There exist protocols for hypothesis testing, data compression, and channel coding that achieve the optimal asymptotic rates when the i.i.d. resource is replaced by any arbitrary almost i.i.d. resource. An almost i.i.d. process is one for which the club distance to an i.i.d. process goes to zero in the appropriate limit. This robustness holds without requiring exact knowledge of the perturbation, as long as it is controlled in the club distance.

What carries the argument

The club distance, a variant of the diamond distance that quantifies deviations of a process from i.i.d. behavior in a manner that permits compensation by suitably chosen protocols without knowing the exact defect.

If this is right

  • Hypothesis testing protocols achieve the same error exponents on almost i.i.d. sources as on exact i.i.d. ones.
  • Data compression achieves the same rates for almost i.i.d. quantum states.
  • Channel coding achieves the same capacities for almost i.i.d. channels.
  • The club distance supplies a sufficient condition under which bespoke compensation is unnecessary.

Where Pith is reading between the lines

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

  • The same robustness technique may apply to other quantum tasks such as entanglement distillation.
  • Experimental implementations could use the club distance to certify tolerance to small statistical fluctuations.
  • Finite-size analysis could incorporate the club distance to bound how rapidly rates approach their asymptotic values.

Load-bearing premise

Unknown perturbations from i.i.d. behavior remain bounded in the club distance so that robust protocols can compensate without exact knowledge of their form.

What would settle it

An explicit almost i.i.d. source whose club distance to i.i.d. vanishes yet no protocol achieves the claimed optimal asymptotic rate for one of the three tasks.

Figures

Figures reproduced from arXiv: 2605.18726 by Filippo Girardi, Giacomo De Palma, Ludovico Lami, Nilanjana Datta.

Figure 1
Figure 1. Figure 1: Three different notions of almost i.i.d. processes A pictorial representation of the hierarchical relation between the different notions of almost i.i.d.-ness. As discussed in [5], the Mazzola–Sutter–Renner notion is the strictest, the one derived from Wasserstein distance is the intermediate one, and, eventually, weak almost i.i.d.- ness is the broadest. The classical probability source (𝑝˜𝑛)𝑛 and the pur… view at source ↗
Figure 2
Figure 2. Figure 2: From 𝑛 i.i.d. uses of a channel to an almost i.i.d. process. A common assumption when evaluating the transmission capabilities – namely, error prob￾ability and communication rate – of multiple uses of a given channel N is the possibility to access exact i.i.d. instances of the channel. As a consequence, coding theorems and strategies are intended for parallel uses of channels satisfying such idealised beha… view at source ↗
Figure 3
Figure 3. Figure 3: Almost i.i.d. hypothesis testing. The operational task of quantum hypothesis testing with almost i.i.d. states can be illustrate in two instances. Suppose either one or both the i.i.d. hypotheses 𝐻0 and 𝐻1 are replaced with almost i.i.d. hypotheses 𝐻˜ 0 and 𝐻˜ 1. The first question is the following: suppose we are aware of the nature of the defects for the individual sources 𝜌𝑛 and 𝜎𝑛; is it then possible … view at source ↗
Figure 4
Figure 4. Figure 4: Almost i.i.d. classical data compression. Let 𝑃 ∈ P(X) be a classical probability distribution. In the i.i.d. setting (a), the aim of data￾compression is to design a sequence of codes (E𝑛 , D𝑛) such that each sequence of symbols 𝑋 𝑛 = (𝑋1, . . . , 𝑋𝑛) gets compressed to a message 𝑚 ∈ [𝑀] with 𝑀 ≪ |X| 𝑛 ; then 𝑚 gets decompressed to the original sequence with vanishing error probability when 𝑋 𝑛 ∼ 𝑃 ×𝑛 . Su… view at source ↗
Figure 5
Figure 5. Figure 5: Almost i.i.d. quantum data compression. Let 𝜌 ∈ D(H𝐴) be a state. In the i.i.d. setting (a), a data compression code (E𝑛 , D𝑛) acts on the subsystem 𝐴 𝑛 of an arbitrary – possibly entangled – state Ψ𝐴𝑛𝑅 with marginal Tr𝑅 Ψ𝐴𝑛𝑅 = 𝜌 ⊗𝑛 by mapping it into a space K𝑛; then, for a good code, the decompressed state 𝜎𝐴𝑛𝑅 = [PITH_FULL_IMAGE:figures/full_fig_p034_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Communicating with an almost i.i.d. channel and shared randomness. Let N be a quantum channel. The protocol constructed in the proof of Theorem 24 lever￾ages a sequence of codes (E𝑛 , D𝑛), which only depend on N and which make use of shared randomness between the sender and the receiver, in order to achieve 𝐶(N ) – i.e. the classical capacity of N – as a reliable communication rate when the sequence (N ⊗𝑛 … view at source ↗
Figure 7
Figure 7. Figure 7: Schematic interpretation of almost i.i.d. processes. Writing the definition of the club distance between an i.i.d. channel W×𝑛 and a non i.i.d. channel W˜ (𝑛) in terms of couplings as in (168), it is possible to interpret the transformation of a random input 𝑋 𝑛 to a random output 𝑌˜ induced by W˜ (𝑛) as a two step transformation of 𝑋 𝑛 : first, the i.i.d. channel W×𝑛 is applied to 𝑋 𝑛 , yielding a output … view at source ↗
read the original abstract

The asymptotic rates of information-theoretic protocols - including error exponents, compression rates, and channel capacities - are traditionally defined under the idealised assumption that the underlying resource (state or channel) is independent and identically distributed (i.i.d.). Somewhat surprisingly, even slight departures from the exact i.i.d. structure can lead to a drastic breakdown of these protocols. The asymptotic rates of information theoretic protocols - error exponents, compression rates, capacities - were originally evaluated taking for granted that the underlying source (state or channel) is i.i.d. Differently from what we might expect at first glance, it is not hard to exhibit instances of protocols that may drastically fail when the i.i.d. assumption holds only approximately rather than exactly. If the precise nature of the perturbation from the i.i.d. regime is known (e.g. a pointwise defect), we could design a bespoke protocol that compensates for the defect (for example, by discarding the corrupted subsystem). However, in any realistic setting, neither can the i.i.d. behaviour of the system be precisely guaranteed, nor can the deviations from the ideal regime be determined exactly. In this paper we answer the following question: are there protocols that can still achieve the optimal asymptotic rates when the i.i.d. resource is replaced by any arbitrary almost i.i.d. resource along it? What is the nature of the unknown perturbation under which protocols like these are possible? We focus, in particular, on hypothesis testing, data compression, and channel coding. As a by-product of our analysis, we introduce the notion of club distance, as a variant of the well-known diamond distance, and of an almost i.i.d. process, which may be of independent interest.

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

3 major / 2 minor

Summary. The paper claims that protocols for quantum hypothesis testing, data compression, and channel coding exist which achieve the exact optimal asymptotic rates (Stein exponent, von Neumann entropy rate, Holevo capacity) when the underlying resource is replaced by an arbitrary almost i.i.d. process, where almost i.i.d. is formalized via a new 'club distance' (a variant of the diamond distance) that quantifies deviation from exact i.i.d. structure without requiring knowledge of the specific perturbation.

Significance. If the central claims hold, the work provides a robustness framework for quantum Shannon theory that addresses realistic deviations from i.i.d. assumptions common in experiments, potentially enabling more practical protocol designs. The introduction of club distance and almost i.i.d. processes could be of independent interest for continuity arguments in quantum information.

major comments (3)
  1. [§2] §2 (Definitions): The definition of club distance and almost i.i.d. process must explicitly require that d_club(ρ^{(n)}, σ^{⊗n}) = o(n) (or a suitable normalization that vanishes after division by n) to ensure continuity of entropy and mutual information functionals preserves the exact i.i.d. rates without additive offsets; the current phrasing 'any arbitrary almost i.i.d. resource' leaves open whether bounded (non-vanishing) club distance is permitted, which would contradict the rate-optimality claim via standard continuity bounds.
  2. [§4] §4 (Hypothesis Testing): The error-exponent analysis for the hypothesis-testing protocol relies on the club-distance bound to recover the Stein exponent; the derivation should include an explicit inequality showing how the per-copy deviation vanishes in the limit, as a constant club-distance bound would yield a strictly smaller exponent by the continuity of the quantum relative entropy.
  3. [§5] §5 (Channel Coding): The achievability proof for the Holevo capacity under almost i.i.d. channels must demonstrate that the random coding argument and typicality sets remain valid when the channel is perturbed in club distance; without a quantitative bound on how the perturbation affects the output statistics, the capacity-achieving rate may incur a positive gap.
minor comments (2)
  1. [Abstract] The abstract contains two nearly identical paragraphs describing the breakdown of i.i.d. protocols; consolidate to avoid redundancy.
  2. [§2] Notation for the club distance should be introduced with a clear comparison to the diamond norm (e.g., via an explicit inequality relating the two) to aid readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments below and outline the revisions we plan to make to strengthen the paper.

read point-by-point responses

  1. Referee: [§2] §2 (Definitions): The definition of club distance and almost i.i.d. process must explicitly require that d_club(ρ^{(n)}, σ^{⊗n}) = o(n) (or a suitable normalization that vanishes after division by n) to ensure continuity of entropy and mutual information functionals preserves the exact i.i.d. rates without additive offsets; the current phrasing 'any arbitrary almost i.i.d. resource' leaves open whether bounded (non-vanishing) club distance is permitted, which would contradict the rate-optimality claim via standard continuity bounds.

    Authors: We appreciate the referee's careful attention to the definition in Section 2. The notion of almost i.i.d. process is defined such that the club distance satisfies d_club(ρ^{(n)}, σ^{⊗n}) = o(n) as n tends to infinity, which is necessary for the continuity arguments to yield the exact asymptotic rates. The phrasing 'arbitrary almost i.i.d. resource' is meant to refer to resources satisfying this vanishing normalized distance condition. To eliminate any potential ambiguity, we will revise the definition to explicitly state this requirement. revision: yes

  2. Referee: [§4] §4 (Hypothesis Testing): The error-exponent analysis for the hypothesis-testing protocol relies on the club-distance bound to recover the Stein exponent; the derivation should include an explicit inequality showing how the per-copy deviation vanishes in the limit, as a constant club-distance bound would yield a strictly smaller exponent by the continuity of the quantum relative entropy.

    Authors: We agree that an explicit inequality would clarify the argument. In the revised manuscript, we will insert a step in the proof of the hypothesis testing result that explicitly bounds the deviation in the relative entropy using the club distance and shows that the per-copy contribution vanishes in the asymptotic limit, thereby recovering the exact Stein exponent. revision: yes

  3. Referee: [§5] §5 (Channel Coding): The achievability proof for the Holevo capacity under almost i.i.d. channels must demonstrate that the random coding argument and typicality sets remain valid when the channel is perturbed in club distance; without a quantitative bound on how the perturbation affects the output statistics, the capacity-achieving rate may incur a positive gap.

    Authors: The proof in Section 5 adapts the standard random coding argument by using the club distance to control the deviation in the output state statistics. We will add a quantitative estimate showing that the perturbation in club distance leads to a vanishing effect on the typicality sets and error probabilities in the limit, ensuring that the achievable rate remains the Holevo capacity without a gap. revision: yes

Circularity Check

0 steps flagged

No circularity: new definitions and protocol constructions are self-contained

full rationale

The paper introduces the club distance (a variant of diamond distance) and the notion of almost i.i.d. processes as fresh mathematical tools, then constructs explicit protocols for hypothesis testing, data compression, and channel coding that achieve the i.i.d. asymptotic rates under these definitions. No load-bearing step reduces by construction to a fitted input, self-citation chain, or renamed known result; the central claims rest on independent proofs that the new distance controls continuity of the relevant information quantities in the asymptotic limit. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 2 invented entities

The central claim rests on the existence of protocols that tolerate arbitrary small perturbations quantified by a new distance; the abstract provides no explicit free parameters or axioms beyond standard quantum information assumptions.

invented entities (2)
  • club distance no independent evidence
    purpose: Variant of diamond distance to quantify closeness to i.i.d. structure for robustness analysis.
    Introduced as a new tool to handle unknown perturbations from exact i.i.d.
  • almost i.i.d. process no independent evidence
    purpose: Formal notion of sources that are close to but not exactly i.i.d.
    Defined to capture realistic deviations while allowing optimal protocol performance.

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Reference graph

Works this paper leans on

91 extracted references · 91 canonical work pages · 2 internal anchors

  1. [1]

    Security of Quantum Key Distribution

    R.Renner.Securityofquantumkeydistribution. PhDthesis,ETHZurich,2005. Preprint arXiv:quant-ph/0512258. 3, 16

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  2. [2]

    F. G. S. L. Brandão and M. B. Plenio. A generalization of quantum Stein’s lemma. Commun. Math. Phys., 295(3):791–828, 2010. 3

    work page 2010

  3. [3]

    L.Lami.AsolutionofthegeneralizedquantumStein’slemma.IEEETrans.Inf.Theory, 71(6):4454–4484, 2025. 3

    work page 2025

  4. [4]

    Mazzola, D

    G. Mazzola, D. Sutter, and R. Renner. Almost-iid information theory.Preprint arXiv:2603.15792, 2026. 3, 15, 16, 27

    work page arXiv 2026

  5. [5]

    Girardi, G

    F. Girardi, G. De Palma, and L. Lami. New approaches to almost i.i.d. information theory, 2026. 3, 14, 15, 22

    work page 2026

  6. [6]

    H. Umegaki. Conditional expectation in an operator algebra. IV. Entropy and infor- mation.Kodai Math. Sem. Rep., 14(2):59–85, 1962. 6

    work page 1962

  7. [7]

    Cambridge University Press, Cambridge, 2010

    M.A.NielsenandI.L.Chuang.QuantumComputationandQuantumInformation: 10th Anniversary Edition. Cambridge University Press, Cambridge, 2010. 6

    work page 2010

  8. [8]

    F.BuscemiandN.Datta.Thequantumcapacityofchannelswitharbitrarilycorrelated noise.IEEE Trans. Inf. Theory, 56(3):1447–1460, 2010. 7

    work page 2010

  9. [9]

    Feinstein.A new basic theorem of information theory

    A. Feinstein.A new basic theorem of information theory. Massachusetts Institute of Technology, Research Laboratory of Electronics, Cambridge, MA, 1954. Tech. Rep. No. 282. 7

    work page 1954

  10. [10]

    Blackwell, L

    D. Blackwell, L. Breiman, and A. J. Thomasian. The capacity of a class of channels. Ann. Math. Statist., 30:1229–1241, 1959. 7

    work page 1959

  11. [11]

    Verdu and T

    S. Verdu and T. S. Han. A general formula for channel capacity.IEEE Trans. Inf. Theory, 40(4):1147–1157, 1994. 7

    work page 1994

  12. [12]

    Poor, and S

    Y.Polyanskiy, H.V. Poor, and S. Verdu. Channel coding rate in thefinite blocklength regime.IEEE Trans. Inf. Theory, 56(5):2307–2359, 2010. 7

    work page 2010

  13. [13]

    T.OgawaandH.Nagaoka.Anewproofofthechannelcodingtheoremviahypothesis testing in quantum information theory. InProc. IEEE Int. Symp. Inf. Theory (ISIT), page 73, 2002. 7

    work page 2002

  14. [14]

    Makinggoodcodesforclassical-quantumchannelcoding via quantum hypothesis testing.IEEE Trans

    T.OgawaandH.Nagaoka. Makinggoodcodesforclassical-quantumchannelcoding via quantum hypothesis testing.IEEE Trans. Inf. Theory, 53(6):2261–2266, 2007. 7

    work page 2007

  15. [15]

    Simpleandtighterderivationofachievabilityforclassicalcommunica- tion over quantum channels

    H.-C.Cheng. Simpleandtighterderivationofachievabilityforclassicalcommunica- tion over quantum channels. 4(4):040330, 2023. 7 54

    work page 2023

  16. [16]

    Hiai and D

    F. Hiai and D. Petz. The proper formula for relative entropy and its asymptotics in quantum probability.Comm. Math. Phys., 143(1):99–114, 1991. 7, 20, 27

    work page 1991

  17. [17]

    T.OgawaandH.Nagaoka.StrongconverseandStein’slemmainquantumhypothesis testing.IEEE Trans. Inf. Theory, 46(7):2428–2433, 2000. 7, 20

    work page 2000

  18. [18]

    C. E. Shannon. A mathematical theory of communication.The Bell System Technical Journal, 27(3):379–423, 1948. 7, 8, 10

    work page 1948

  19. [19]

    Schumacher

    B. Schumacher. Quantum coding.Phys. Rev. A, 51:2738–2747, 1995. 9

    work page 1995

  20. [20]

    Schumacher and M

    B. Schumacher and M. D. Westmoreland. Sending classical information via noisy quantum channels.Phys. Rev. A, 56:131–138, 1997. 10

    work page 1997

  21. [21]

    A. S. Holevo. The capacity of the quantum channel with general signal states.IEEE Trans. Inf. Theory, 44(1):269–273, 1998. 10

    work page 1998

  22. [22]

    Asimplederivationofthecodingtheoremandsomeapplications.IEEE Transactions on Information Theory, 11(1):3–18, 1965

    R.Gallager. Asimplederivationofthecodingtheoremandsomeapplications.IEEE Transactions on Information Theory, 11(1):3–18, 1965. 10

    work page 1965

  23. [23]

    Springer,

    RobertGGallager.Informationtheoryandreliablecommunication,volume588. Springer,

    work page

  24. [24]

    Lowerboundstoerrorprobability for coding on discrete memoryless channels

    C.E.Shannon,R.G.Gallager,andE.R.Berlekamp. Lowerboundstoerrorprobability for coding on discrete memoryless channels. I.Information and Control, 10(1):65–103,

    work page

  25. [25]

    Lowerboundstoerrorprobability forcodingondiscretememorylesschannels.II.InformationandControl,10(5):522–552,

    C.E.Shannon,R.G.Gallager,andE.R.Berlekamp. Lowerboundstoerrorprobability forcodingondiscretememorylesschannels.II.InformationandControl,10(5):522–552,

    work page

  26. [26]

    Villani.Optimal Transport: Old and New

    C. Villani.Optimal Transport: Old and New. Grundlehren der mathematischen Wis- senschaften. Springer Berlin Heidelberg, 2008. 11

    work page 2008

  27. [27]

    Imprimerieroyale,1781

    G.Monge.Mémoiresurlathéoriedesdéblaisetdesremblais. Imprimerieroyale,1781. 11

    work page

  28. [28]

    L. V. Kantorovich. On the translocation of masses.J. Math. Sci., 133(4), 2006. 11

    work page 2006

  29. [29]

    D. S. Ornstein. An application of ergodic theory to probability theory.Ann. Probab., 1(1):43–58, 1973. 11

    work page 1973

  30. [30]

    Gray.Entropy and Information Theory

    R.M. Gray.Entropy and Information Theory. Engineering. Springer US, 2011. 11

    work page 2011

  31. [31]

    De Palma, M

    G. De Palma, M. Marvian, D. Trevisan, and S. Lloyd. The quantum Wasserstein distance of order 1.IEEE Trans. Inf. Theory, 67(10):6627–6643, 2021. 12, 13, 17, 18, 19

    work page 2021

  32. [32]

    De Palma and D

    G. De Palma and D. Trevisan.Quantum Optimal Transport: Quantum Channels and Qubits, page 203–239. Springer Nature Switzerland, 2024. 12, 19 55

    work page 2024

  33. [33]

    B. T. Kiani, G. De Palma, M. Marvian, Z.-W. Liu, and S. Lloyd. Learning quantum data with the quantum earth mover’s distance.Quantum Sci. Technol., 7(4):045002,

    work page

  34. [34]

    De Palma, M

    G. De Palma, M. Marvian, C. Rouzé, and D. Stilck França. Limitations of variational quantum algorithms: A quantum optimal transport approach.PRX Quantum, 4(1),

    work page

  35. [35]

    Hirche, C

    C. Hirche, C. Rouzé, and D. Stilck França. Quantum differential privacy: An infor- mation theory perspective.Preprint arXiv:2202.10717, 2023. 13

    work page arXiv 2023

  36. [36]

    Quantumconcentrationinequalities.Ann.HenriPoincaré, 23(9):3391–3429, 2022

    G.DePalmaandC.Rouzé. Quantumconcentrationinequalities.Ann.HenriPoincaré, 23(9):3391–3429, 2022. 13

    work page 2022

  37. [37]

    Quantumconcentrationinequalitiesandequivalence of the thermodynamical ensembles: An optimal mass transport approach.J

    G.DePalmaandD.Pastorello. Quantumconcentrationinequalitiesandequivalence of the thermodynamical ensembles: An optimal mass transport approach.J. Stat. Phys., 192(6), 2025. 13

    work page 2025

  38. [38]

    De Palma, T

    G. De Palma, T. Klein, and D. Pastorello. Classical shadows meet quantum optimal mass transport.J. Math. Phys., 65(9), 2024. 13

    work page 2024

  39. [39]

    Rouzé, D

    C. Rouzé, D. Stilck França, E. Onorati, and J. D. Watson. Efficient learning of ground and thermal states within phases of matter.Nat. Commun., 15(1), 2024. 13

    work page 2024

  40. [40]

    Rouzé and D

    C. Rouzé and D. Stilck França. Learning quantum many-body systems from a few copies.Quantum, 8:1319, 2024. 13

    work page 2024

  41. [41]

    Bardet, Á

    I. Bardet, Á. Capel, L. Gao, A. Lucia, D. Pérez-García, and C. Rouzé. Entropy decay for Davies semigroups of a one dimensional quantum lattice.Commun. Math. Phys., 405(2), 2024. 13

    work page 2024

  42. [42]

    Kochanowski, Á

    J. Kochanowski, Á. M. Alhambra, Á. Capel, and C. Rouzé. Rapid thermalization of dissipativemany-bodydynamicsofcommutingHamiltonians.Commun.Math.Phys., 406(8), 2025. 13

    work page 2025

  43. [43]

    Bakshi, A

    A. Bakshi, A. Liu, A. Moitra, and E. Tang. A Dobrushin condition for quantum Markov chains: Rapid mixing and conditional mutual information at high tempera- ture.Preprint arXiv:2510.08542, 2025. 13

    work page arXiv 2025

  44. [44]

    De Palma and D

    G. De Palma and D. Trevisan. The Wasserstein distance of order 1 for quantum spin systems on infinite lattices.Ann. Henri Poincaré, 24(12):4237–4282, 2023. 13

    work page 2023

  45. [45]

    Symmetryoflargephysicalsystemsimpliesindependenceofsubsystems

    R.Renner. Symmetryoflargephysicalsystemsimpliesindependenceofsubsystems. Nat. Phys., 3(9):645–649, 2007. 16

    work page 2007

  46. [46]

    Duvenhage and M

    R. Duvenhage and M. Mapaya. Quantum Wasserstein distance of order 1 between channels.Infin. Dimens. Anal. Quantum Probab. Relat. Top., 26(03), 2023. 17 56

    work page 2023

  47. [47]

    Eric A Carlen and Jan Maas. An analog of the 2-Wasserstein metric in non- commutativeprobabilityunderwhichtheFermionicFokker–Planckequationisgra- dient flow for the entropy.Communications in Mathematical Physics, 331(3):887–926,

    work page

  48. [48]

    Gradient flow and entropy inequalities for quantum Markovsemigroupswithdetailedbalance.JournalofFunctionalAnalysis,273(5):1810– 1869, 2017

    Eric A Carlen and Jan Maas. Gradient flow and entropy inequalities for quantum Markovsemigroupswithdetailedbalance.JournalofFunctionalAnalysis,273(5):1810– 1869, 2017. 18

    work page 2017

  49. [49]

    Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems.Journal of Statistical Physics, 178(2):319–378, 2020

    Eric A Carlen and Jan Maas. Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems.Journal of Statistical Physics, 178(2):319–378, 2020. 18

    work page 2020

  50. [50]

    Concentration of quantum states from quan- tum functional and transportation cost inequalities.Journal of Mathematical Physics, 60(1):012202, 2019

    Cambyse Rouzé and Nilanjana Datta. Concentration of quantum states from quan- tum functional and transportation cost inequalities.Journal of Mathematical Physics, 60(1):012202, 2019. 18

    work page 2019

  51. [51]

    Relating relative entropy, optimal transport andFisherinformation: AquantumHWIinequality.AnnalesHenriPoincaré,21:2115– 2150, 2020

    Nilanjana Datta and Cambyse Rouzé. Relating relative entropy, optimal transport andFisherinformation: AquantumHWIinequality.AnnalesHenriPoincaré,21:2115– 2150, 2020. 18

    work page 2020

  52. [52]

    Geometrical Bounds of the Irreversibility in Markovian Systems.arXiv preprint arXiv:2005.02871, 2020

    Tan Van Vu and Yoshihiko Hasegawa. Geometrical Bounds of the Irreversibility in Markovian Systems.arXiv preprint arXiv:2005.02871, 2020. 18

    work page arXiv 2005

  53. [53]

    The conditional Entropy Power Inequality for quantumadditivenoisechannels.JournalofMathematicalPhysics,59(12):122201,2018

    Giacomo De Palma and Stefan Huber. The conditional Entropy Power Inequality for quantumadditivenoisechannels.JournalofMathematicalPhysics,59(12):122201,2018. 18

    work page 2018

  54. [54]

    Fisherinformationandlogarithmic sobolev inequality for matrix-valued functions.Annales Henri Poincaré, 21(11):3409– 3478, 2020

    LiGao,MariusJunge,andNicholasLaRacuente. Fisherinformationandlogarithmic sobolev inequality for matrix-valued functions.Annales Henri Poincaré, 21(11):3409– 3478, 2020. 18

    work page 2020

  55. [55]

    Matricial Wasserstein-1 distance.IEEE control systems letters, 1(1):14–19, 2017

    YongxinChen,TryphonTGeorgiou,LipengNing,andAllenTannenbaum. Matricial Wasserstein-1 distance.IEEE control systems letters, 1(1):14–19, 2017. 19

    work page 2017

  56. [56]

    Vector and matrix op- timal mass transport: theory, algorithm, and applications.SIAM Journal on Scientific Computing, 40(5):A3675–A3698, 2018

    Ernest K Ryu, Yongxin Chen, Wuchen Li, and Stanley Osher. Vector and matrix op- timal mass transport: theory, algorithm, and applications.SIAM Journal on Scientific Computing, 40(5):A3675–A3698, 2018. 19

    work page 2018

  57. [57]

    Matrix optimal mass transport: a quantum mechanical approach.IEEE Transactions on Automatic Control, 63(8):2612–2619, 2018

    Yongxin Chen, Tryphon T Georgiou, and Allen Tannenbaum. Matrix optimal mass transport: a quantum mechanical approach.IEEE Transactions on Automatic Control, 63(8):2612–2619, 2018. 19

    work page 2018

  58. [58]

    Wasserstein geometry of quantum states and optimal transport of matrix-valued measures

    Yongxin Chen, Tryphon T Georgiou, and Allen Tannenbaum. Wasserstein geometry of quantum states and optimal transport of matrix-valued measures. InEmerging Applications of Control and Systems Theory, pages 139–150. Springer, 2018. 19 57

    work page 2018

  59. [59]

    JuliánAgredo.AWasserstein-typedistancetomeasuredeviationfromequilibriumof quantumMarkovsemigroups.OpenSystems&InformationDynamics,20(02):1350009,

    work page

  60. [60]

    On exponential convergence of generic quantum Markov semigroups in a Wasserstein-type distance.International Journal of Pure and Applied Mathematics, 107(4):909–925, 2016

    J Agredo. On exponential convergence of generic quantum Markov semigroups in a Wasserstein-type distance.International Journal of Pure and Applied Mathematics, 107(4):909–925, 2016. 19

    work page 2016

  61. [61]

    Foundationofquantumoptimaltransportandapplications.Quantum Information Processing, 19(1):25, 2020

    KazukiIkeda. Foundationofquantumoptimaltransportandapplications.Quantum Information Processing, 19(1):25, 2020. 19

    work page 2020

  62. [62]

    On the mean field and classical limitsofquantummechanics.CommunicationsinMathematicalPhysics,343(1):165–205,

    François Golse, Clément Mouhot, and Thierry Paul. On the mean field and classical limitsofquantummechanics.CommunicationsinMathematicalPhysics,343(1):165–205,

    work page

  63. [63]

    Towards optimal transport for quantum densities

    Emanuele Caglioti, François Golse, and Thierry Paul. Towards optimal transport for quantum densities. preprint, Dec 2018. 19

    work page 2018

  64. [64]

    François Golse. The quantum N-body problem in the mean-field and semiclassical regime.Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376(2118):20170229, 2018. 19

    work page 2018

  65. [65]

    The Schrödinger equation in the mean-field and semiclassical regime.Archive for Rational Mechanics and Analysis, 223(1):57–94, 2017

    François Golse and Thierry Paul. The Schrödinger equation in the mean-field and semiclassical regime.Archive for Rational Mechanics and Analysis, 223(1):57–94, 2017. 19

    work page 2017

  66. [66]

    Wave packets and the quadratic Monge– Kantorovich distance in quantum mechanics.Comptes Rendus Mathematique, 356(2):177–197, 2018

    François Golse and Thierry Paul. Wave packets and the quadratic Monge– Kantorovich distance in quantum mechanics.Comptes Rendus Mathematique, 356(2):177–197, 2018. 19

    work page 2018

  67. [67]

    Quantum Optimal Transport is Cheaper.Journal of Statistical Physics, 2020

    Emanuele Caglioti, François Golse, and Thierry Paul. Quantum Optimal Transport is Cheaper.Journal of Statistical Physics, 2020. 19

    work page 2020

  68. [68]

    Quantum monge-kantorovichproblemandtransportdistancebetweendensitymatrices.Phys- ical Review Letters, 129(11), 2022

    Shmuel Friedland, Michał Eckstein, Sam Cole, and Karol Życzkowski. Quantum monge-kantorovichproblemandtransportdistancebetweendensitymatrices.Phys- ical Review Letters, 129(11), 2022. 19

    work page 2022

  69. [69]

    Strongkantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits, 2026

    GergelyBunth,JózsefPitrik,TamásTitkos,andDánielVirosztek. Strongkantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits, 2026. 19

    work page 2026

  70. [70]

    Quantum wasserstein generative adversarial networks

    Shouvanik Chakrabarti, Huang Yiming, Tongyang Li, Soheil Feizi, and Xiaodi Wu. Quantum wasserstein generative adversarial networks. InAdvances in Neural Infor- mation Processing Systems, pages 6781–6792, 2019. 19

    work page 2019

  71. [71]

    Quantum optimal transport with quantum channels.Annales Henri Poincaré, 22(10):3199–3234, 2021

    Giacomo De Palma and Dario Trevisan. Quantum optimal transport with quantum channels.Annales Henri Poincaré, 22(10):3199–3234, 2021. 19 58

    work page 2021

  72. [72]

    Wasserstein dis- tances and divergences of order𝑝by quantum channels, 2026

    Gergely Bunth, József Pitrik, Tamás Titkos, and Dániel Virosztek. Wasserstein dis- tances and divergences of order𝑝by quantum channels, 2026. 19

    work page 2026

  73. [73]

    Metricpropertyof quantum wasserstein divergences.Physical Review A, 110(2), August 2024

    GergelyBunth, JózsefPitrik, TamásTitkos, andDánielVirosztek. Metricpropertyof quantum wasserstein divergences.Physical Review A, 110(2), August 2024. 19

    work page 2024

  74. [74]

    Triangle inequality for a quantum wasserstein divergence, 2025

    Melchior Wirth. Triangle inequality for a quantum wasserstein divergence, 2025. 19

    work page 2025

  75. [75]

    Balance between quantum Markov semi- groups.Annales Henri Poincaré, 19(6):1747–1786, 2018

    Rocco Duvenhage and Machiel Snyman. Balance between quantum Markov semi- groups.Annales Henri Poincaré, 19(6):1747–1786, 2018. 19

    work page 2018

  76. [76]

    On quantum versions of the classical Wasserstein distance.Stochastics, 89(6-7):910–922, 2017

    J Agredo and Franco Fagnola. On quantum versions of the classical Wasserstein distance.Stochastics, 89(6-7):910–922, 2017. 19

    work page 2017

  77. [77]

    The Monge distance between quan- tum states.Journal of Physics A: Mathematical and General, 31(45):9095, 1998

    Karol Zyczkowski and Wojeciech Slomczynski. The Monge distance between quan- tum states.Journal of Physics A: Mathematical and General, 31(45):9095, 1998. 19

    work page 1998

  78. [78]

    The Monge metric on the sphere and geometryofquantumstates.JournalofPhysicsA:MathematicalandGeneral,34(34):6689,

    Karol Zyczkowski and Wojciech Slomczynski. The Monge metric on the sphere and geometryofquantumstates.JournalofPhysicsA:MathematicalandGeneral,34(34):6689,

    work page

  79. [79]

    Cambridge University Press, 2017

    Ingemar Bengtsson and Karol Życzkowski.Geometry of Quantum States: An Introduc- tion to Quantum Entanglement. Cambridge University Press, 2017. 19

    work page 2017

  80. [80]

    J. Maas, S. Rademacher, T. Titkos, and D. Virosztek.Optimal Transport on Quantum Structures. Bolyai Society Mathematical Studies. Springer Nature Switzerland, 2024. 19

    work page 2024

Showing first 80 references.