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arXiv:2603.09885

Optimal Universal Bounds for Quantum Divergences

Gilad Gour

quant-phmath-phSubmitted Mar 10, 2026• Announced Mar 11, 2026• 0 citations• Revised Oct 23, 2026 (v3)
v1 · Mar 10, 2026v2 · Mar 23, 2026v3 · Oct 23, 2026
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Abstract

We identify a universal structural principle underlying the smoothing of classical divergences: the optimizer of the smoothing problem is a clipped probability vector, independently of the specific divergence. This yields a divergence-independent characterization of all smoothed classical divergences and reveals a common geometric structure behind seemingly different quantities. Building on this structural insight, we derive optimal universal bounds for smoothed quantum divergences, including quantum Rényi divergences of arbitrary order and the hypothesis-testing divergence. Our inequalities relate divergences of different orders through bounds of the form $D_β^{\varepsilon} \le D_α+ \mathrm{correction}$ and $D_β^{\varepsilon} \ge D_α+ \mathrm{correction}$, and we prove that the correction terms are optimal among all universal, state-independent inequalities of this type. Consequently, our results strictly improve previously known bounds whenever those were suboptimal, and in cases where earlier bounds coincide with ours, our analysis establishes their optimality. In particular, we obtain optimal universal bounds for the hypothesis-testing divergence.

Comments
25 pages; major revisions, including substantially shorter proofs, new reductions to binary distributions, improved exposition, and additional references

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We identify a universal structural principle underlying the smoothing of classical divergences: the optimizer of the smoothing problem is a clipped probability vector, independently of the specific divergence. This yields a divergence-independent characterization of all smoothed classical divergences and reveals a common geometric structure behind seemingly different quantities. Building on this structural insight, we derive optimal universal bounds for smoothed quantum divergences, including quantum Rényi divergences of arbitrary order and the hypothesis-testing divergence. Our inequalities relate divergences of different orders through bounds of the form Dβε≤Dα+correctionD_β^{\varepsilon} \le D_α+ \mathrm{correction}Dβε​≤Dα​+correction and Dβε≥Dα+correctionD_β^{\varepsilon} \ge D_α+ \mathrm{correction}Dβε​≥Dα​+correction, and we prove that the correction terms are optimal among all universal, state-independent inequalities of this type. Consequently, our results strictly improve previously known bounds whenever those were suboptimal, and in cases where earlier bounds coincide with ours, our analysis establishes their optimality. In particular, we obtain optimal universal bounds for the hypothesis-testing divergence.
We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.
One possibility of defining a quantum Rényi$\alpha −divergenceoftwoquantumstatesistooptimizetheclassicalReˊnyi-divergence of two quantum states is to optimize the classical Rényi−divergenceoftwoquantumstatesistooptimizetheclassicalReˊnyi\alpha −divergenceoftheirpost−measurementprobabilitydistributionsoverallpossiblemeasurements(measuredReˊnyidivergence),andmayberegularizethesequantitiesovermultiplecopiesofthetwostates(regularizedmeasuredReˊnyi-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi−divergenceoftheirpost−measurementprobabilitydistributionsoverallpossiblemeasurements(measuredReˊnyidivergence),andmayberegularizethesequantitiesovermultiplecopiesofthetwostates(regularizedmeasuredReˊnyi\alpha −divergence).AkeyobservationbehindthetheoremforthestrongconverseexponentofasymptoticbinaryquantumstatediscriminationisthattheregularizedmeasuredReˊnyi-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi−divergence).AkeyobservationbehindthetheoremforthestrongconverseexponentofasymptoticbinaryquantumstatediscriminationisthattheregularizedmeasuredReˊnyi\alpha −divergencecoincideswiththesandwichedReˊnyi-divergence coincides with the sandwiched Rényi−divergencecoincideswiththesandwichedReˊnyi\alpha −divergencewhen-divergence when−divergencewhen\alpha >1.Moreover,italsofollowsfromthesametheoremthattoachievethis,itissufficienttoconsider2−outcomemeasurements(tests)foranynumberofcopies(thisissomewhatsurprising,asachievingthemeasuredReˊnyi. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider 2-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi.Moreover,italsofollowsfromthesametheoremthattoachievethis,itissufficienttoconsider2−outcomemeasurements(tests)foranynumberofcopies(thisissomewhatsurprising,asachievingthemeasuredReˊnyi\alpha −divergencefor-divergence for−divergenceforncopiesmightrequireanumberofmeasurementoutcomesthatdivergesincopies might require a number of measurement outcomes that diverges incopiesmightrequireanumberofmeasurementoutcomesthatdivergesinn,ingeneral).Inviewofthis,itseemsnaturaltoexpectthesamewhen, in general). In view of this, it seems natural to expect the same when,ingeneral).Inviewofthis,itseemsnaturaltoexpectthesamewhen\alpha < 1;however,weshowthatthisisnotthecase.Infact,weshowthatevenforcommutingstates(classicalcase)theregularizedquantityattainableusing2−outcomemeasurementsisingeneralstrictlysmallerthantheReˊnyi; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using 2-outcome measurements is in general strictly smaller than the Rényi;however,weshowthatthisisnotthecase.Infact,weshowthatevenforcommutingstates(classicalcase)theregularizedquantityattainableusing2−outcomemeasurementsisingeneralstrictlysmallerthantheReˊnyi\alpha −divergence(whichisuniqueintheclassicalcase).Inthegeneralquantumcasethisshowsthattheabove“regularizedtest−measured”Reˊnyi-divergence (which is unique in the classical case). In the general quantum case this shows that the above “regularized test-measured” Rényi−divergence(whichisuniqueintheclassicalcase).Inthegeneralquantumcasethisshowsthattheabove“regularizedtest−measured”Reˊnyi\alpha −divergenceisnotevenaquantumextensionoftheclassicalReˊnyidivergencewhen-divergence is not even a quantum extension of the classical Rényi divergence when−divergenceisnotevenaquantumextensionoftheclassicalReˊnyidivergencewhen\alpha < 1,insharpcontrasttothe, in sharp contrast to the,insharpcontrasttothe\alpha >1$case.
One-shot information theory entertains a plethora of entropic quantities, such as the smooth max-divergence, hypothesis testing divergence, and information spectrum divergence, that characterize various operational tasks in quantum information theory and are used to analyze their asymptotic behavior. Tight inequalities between these quantities are thus of immediate interest. In this note, we use a minimax approach (appearing previously, for example, in the proofs of the quantum substate theorem), to simplify the quantum problem to a commutative one, which allows us to derive such inequalities. Our derivations are conceptually different from previous arguments and in some cases lead to tighter relations. We hope that the approach discussed here can lead to progress in open problems in quantum Shannon theory and exemplify this by applying it to a simple case of the joint smoothing problem.
Pinsker's inequality sets a lower bound on the Umegaki divergence of two quantum states in terms of their trace distance. In this work, we formulate corresponding estimates for a variety of quantum and classical divergences including fff-divergences like Hellinger and χ2χ^2χ2-divergences as well as Rényi divergences and special cases thereof like the Umegaki divergence, collision divergence, max divergence. We further provide a strategy on how to adapt these bounds to smoothed divergences.
The precise one-shot characterisation of operational tasks in classical and quantum information theory relies on different forms of smooth entropic quantities. A particularly important connection is between the hypothesis testing relative entropy and the smoothed max-relative entropy, which together govern many operational settings. We first strengthen this connection into a type of equivalence: we show that the hypothesis testing relative entropy is equivalent to a variant of the smooth max-relative entropy based on the information spectrum divergence, which can be alternatively understood as a measured smooth maxrelative entropy. Furthermore, we improve a fundamental lemma due to Datta and Renner that connects the different variants of the smoothed max-relative entropy, introducing a modified proof technique based on matrix geometric means. We use the unveiled connections and tools to strictly improve on previously known one-shot bounds and duality relations between the smooth max-relative entropy and the hypothesis testing relative entropy, sharpening also bounds that connect the max-relative entropy with Rényi divergences.
Quantum experiments without a definite causal order offer new ways to distinguish devices, estimate physical parameters, and communicate. We prove a common limit on these tasks: the geometric Rényi divergence at the output of a deterministic completely positive process cannot exceed the sum of the divergences of its individual channel uses. The inequality holds for every Rényi order in (0,1)∪(1,2](0,1)\cup(1,2](0,1)∪(1,2] and any number of independently replaceable channels. Parallel experiments with separate reference-assisted inputs approach the bound, giving an exact equality after optimization over processes. We also prove an additive bound on the right logarithmic derivative Fisher information of parameter-dependent channels under every fixed process in this class. A resulting Cramér--Rao bound limits locally unbiased estimation precision without assuming a definite causal order, and an erasure-probability example attains the bound. The proofs for the geometric Rényi divergence and RLD Fisher information follow from the requirement that every independent choice of channels produce a normalized output. We show how this requirement controls positive local operations, and how small channel perturbations convert the divergence inequality into a Fisher-information bound, including for singular channels. Further applications give finite-use bounds on channel discrimination and exclusion, together with communication and private-state-generation converses when the processes preserve the relevant comparison conditions.