Concentration inequality using unconfirmed knowledge

Abstract

We give a concentration inequality based on the premise that random variables take values within a particular region. The concentration inequality guarantees that, for any sequence of correlated random variables, the difference between the sum of conditional expectations and that of the observed values takes a small value with high probability when the expected values are evaluated under the condition that the past values are known. Our inequality outperforms other wellknown inequalities, e.g. the Azuma-Hoeffding inequality, especially in terms of the convergence speed when the random variables are highly biased. This high performance of our inequality is provided by the key idea in which we predict some parameters and adopt the predicted values in the inequality.

Finite-key analysis of loss-tolerant quantum key distribution based on random sampling theory

“…We note that a novel concentration inequality for sums of dependent random variables has been recently uploaded to a preprint server by Kato [24]. This result can be regarded as an improved version of Azuma's inequality that is much tighter when the success probability of the random variables is low.…”

“…Azuma's inequalityAccording to Azuma's inequality[10],Pr Λ n − N u=1 Pr(ξ u = 1|F u−1 ) ≥ b √ N ≤ exp − b u = 1|F u−1 ) − Λ n ≥ b √ N ≤ exp − b 2hand sides to to ε A and solving for b, we have thatN u=1 Pr (ξ u = 1|ξ 1 , ..., ξ u−1 ) ≤ Λ N + ∆ A , Λ N ≤ N u=1 Pr (ξ u = 1|ξ 1 , ..., ξ u−1 ) + ∆ A ,(F2)except with probability at most ε A for each of the bounds, where ∆ A = 2N ln ε −1 A .b. Kato's inequalityAccording to Kato's inequality[24], for any n, and any a, b such that b ≥ |a|,Pr N u=1 Pr(ξ u = 1|F u−1 ) − Λ N ≥ b + a 2Λ N N − replacing ξ l → 1 − ξ land a → −a in Eq. (F3), we also derive[25] PrΛ N − N u=1 Pr(ξ u = 1|F u−1 ) ≥ b + a 2Λ N isolating Λ N in Eq.…”

Numerical Method for Finite-size Security Analysis of Quantum Key Distribution

“…P , we may apply concentration inequalities for dependent variables such as the Azuma's inequality or Kato's inequality [27]. Suppose we obtain an inequality in the following form,…”

“…Theorem 1. (Kato's inequality [27]) Let {X (u) } be a list of random variables, and {F u−1 } be a filtration that identifies random variables {X (1) , X (2) , • • • X (u−1) }. Suppose 0 ≤ X (u) ≤ 1 for any u, then for any n ∈ N, a ∈ R and b ≥ 0, we have the relation…”

“…Considering the dual problem of the SDP given in [15,16], we can obtain an operator inequality for the phase error operator within one round of quantum communication, which is independent of the quantum state. By applying concentration inequalities [25][26][27], it is further transformed into an upper bound of the number of phase error occurrence in finite-size case under general attacks. One can directly substitute the experimental statistics into our framework and obtain secure finite-size key rates, without knowing any techniques of security analysis.…”