Smooth Min-entropy Lower Bounds for Approximation Chains
Abstract
For a state ρA1nB, we call a sequence of states (σA1kB(k))k=1n an approximation chain if for every 1≤k≤n, ρA1kB≈ϵσA1kB(k). In general, it is not possible to lower bound the smooth min-entropy of such a ρA1nB, in terms of the entropies of σA1kB(k) without incurring very large penalty factors. In this paper, we study such approximation chains under additional assumptions. We begin by proving a simple entropic triangle inequality, which allows us to bound the smooth min-entropy of a state in terms of the Rényi entropy of an arbitrary auxiliary state while taking into account the smooth max-relative entropy between the two. Using this triangle inequality, we create lower bounds for the smooth min-entropy of a state in terms of the entropies of its approximation chain in various scenarios. In particular, utilising this approach, we prove approximate versions of the asymptotic equipartition property and entropy accumulation. In the companion paper (Marwah and Dupuis in Proving Security of BB84 Under Source Correlations, 2024. arXiv:2402.12346 [quant-ph]), we show that the techniques developed in this paper can be used to prove the security of quantum key distribution in the presence of source correlations.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- September 2024
- DOI:
- 10.1007/s00220-024-05074-8
- arXiv:
- arXiv:2308.11736
- Bibcode:
- 2024CMaPh.405..211M
- Keywords:
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- Quantum Physics;
- Computer Science - Information Theory
- E-Print:
- Section on source correlations is split off into a separate paper