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arXiv:2201.05534 · Journal of Mathematical Physics

Uniform continuity bound for sandwiched Rényi conditional entropy

Ashutosh S. Marwah, Frédéric Dupuis

Published May 1, 2022• 7 citations
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Abstract

We prove a simple uniform continuity bound for the sandwiched Rényi conditional entropy for α ∈ [1/2, 1) ∪ (1, ∞], which is independent of the dimension of the conditioning system.
DOI
10.1063/5.0088507

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We determine the sharp modulus of continuity, in trace distance, of the optimized Petz and sandwiched Rényi conditional entropies for every order α∈[12,1)α\in[\frac12,1)α∈[21​,1). If two bipartite states are within trace distance δδδ, then both conditional entropies differ by at most 11−αlog⁡[(1−ε)α+(D−1)1−αεα]\frac{1}{1-α} \log[(1-\varepsilon)^α +(D-1)^{1-α}\varepsilon^α]1−α1​log[(1−ε)α+(D−1)1−αεα], where ε:=min⁡{δ,1−1/D}\varepsilon := \min\{δ,1-1/D\}ε:=min{δ,1−1/D} and DDD is the effective dimension, given by the dimension of the first subsystem times the largest possible Schmidt rank. For every distance constraint δ∈[0,1]δ\in[0,1]δ∈[0,1], the bound is attained by an isotropic pair with a maximally entangled anchor. Taking α↑1α\uparrow1α↑1 recovers the recent sharp continuity bound of quantum conditional entropy by Berta et al. [arXiv:2607.24687]. The proof linearizes the relevant concave Rényi functional at a comparison point dictated by the isotropic equality family. Schmidt-rank domination extends the equality geometry to an arbitrary anchor state, after which trace-distance duality and a noncommutative calibration estimate control the perturbation and anchor term without weakening the sharp constant. The latter estimate requires matrix analysis and is assisted by ChatGPT 5.6 Sol.
We prove a tight uniform continuity bound for Arimoto’s version of the conditional$\alpha −Reˊnyientropyfortherange-Rényi entropy for the range−Reˊnyientropyfortherange\alpha \in [0, 1).Thisdefinitionoftheconditional). This definition of the conditional).Thisdefinitionoftheconditional\alpha −Reˊnyientropyisthemostnaturaloneamongthemultipleformswhichexistintheliterature,sinceitsatisfiestwodesirablepropertiesofaconditionalentropy,namely,thefactthatconditioningreducesentropy,andthattheassociatedreductioninuncertaintycannotexceedtheinformationgainedbyconditioning.Furthermore,ithasfoundinterestingapplicationsinvariousinformationtheoretictaskssuchasguessingwithsideinformationandsequentialdecoding.ThisconditionalentropyreducestotheconditionalShannonentropyinthelimit-Rényi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with side information and sequential decoding. This conditional entropy reduces to the conditional Shannon entropy in the limit−Reˊnyientropyisthemostnaturaloneamongthemultipleformswhichexistintheliterature,sinceitsatisfiestwodesirablepropertiesofaconditionalentropy,namely,thefactthatconditioningreducesentropy,andthattheassociatedreductioninuncertaintycannotexceedtheinformationgainedbyconditioning.Furthermore,ithasfoundinterestingapplicationsinvariousinformationtheoretictaskssuchasguessingwithsideinformationandsequentialdecoding.ThisconditionalentropyreducestotheconditionalShannonentropyinthelimit\alpha \to 1,andthisinturnallowsustorecovertherecentlyobtainedtightuniformcontinuityboundforthelatterfromourresult.Finally,weapplyourresulttoobtainatightuniformcontinuityboundfortheconditional, and this in turn allows us to recover the recently obtained tight uniform continuity bound for the latter from our result. Finally, we apply our result to obtain a tight uniform continuity bound for the conditional,andthisinturnallowsustorecovertherecentlyobtainedtightuniformcontinuityboundforthelatterfromourresult.Finally,weapplyourresulttoobtainatightuniformcontinuityboundfortheconditional\alpha −Reˊnyientropyofaclassical−quantumstate,for-Rényi entropy of a classical-quantum state, for−Reˊnyientropyofaclassical−quantumstate,for\alpha inthesamerangeasabove.Thisagainyieldsthecorrespondingknownboundfortheconditionalentropyofthestateinthelimitin the same range as above. This again yields the corresponding known bound for the conditional entropy of the state in the limitinthesamerangeasabove.Thisagainyieldsthecorrespondingknownboundfortheconditionalentropyofthestateinthelimit\alpha \to 1$.
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