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Concise Security Bounds for Practical Decoy-State Quantum Key Distribution
Charles Ci Wen Lim, Marcos Curty, Nino Walenta, Feihu Xu, Hugo Zbinden
TL;DR
The appendix develops finite-size security bounds for a two-decoy-state QKD method, addressing the gap between asymptotic bounds and observed finite-key statistics. It combines entropic uncertainty relations, finite-size decoy-state analysis, random sampling, and privacy amplification to obtain the main-text formulas.
Problem
Asymptotic decoy-state bounds involve quantities that are not directly applicable to observed finite-size statistics.
Method
The analysis combines entropic uncertainty relations with a novel finite-size analysis of three intensity levels, random sampling for phase errors, and two-universal hashing.
Results
The resulting finite-size relations are used to establish the formulas stated in the main text, including bounds on vacuum and single-photon events.
Takeaways & Limitations
The vacuum-event lower bound becomes tight when µ3 →0.
Abstract
from arXiv · showhide
Due to its ability to tolerate high channel loss, decoy-state quantum key distribution (QKD) has been one of the main focuses within the QKD community. Notably, several experimental groups have demonstrated that it is secure and feasible under real-world conditions. Crucially, however, the security and feasibility claims made by most of these experiments were obtained under the assumption that the eavesdropper is restricted to particular types of attacks or that the finite-key effects are neglected. Unfortunately, such assumptions are not possible to guarantee in practice. In this work, we provide concise and tight finite-key security bounds for practical decoy-state QKD that are valid against general attacks.
SUPPLEMENTARY MATERIAL
The supplementary material details security bounds based on entropic uncertainty relations and a novel finite-size analysis for the two-decoy-state method.
- The security analysis combines entropic uncertainty relations with a novel finite-size analysis for the two-decoy-state method.
A. Decoy-state analysis for three intensity levels
The analysis treats three intensity levels through a counterfactual photon-number formulation, relating observed finite-sample statistics to asymptotic event and error counts.
- A. Decoy-state analysis for three intensity levels: The three intensities satisfy µ1 > µ2 + µ3 and µ2 > µ3 ≥ 0, while the analysis uses a counterfactual photon-number description.Because Eve cannot distinguish intensity choices from the final prepared state, intensity selection can be viewed as occurring after detection.
- A. Decoy-state analysis for three intensity levels: For X-basis data, sX,n denotes detections observed when Alice sent n-photon states, and their total equals nX.
- A. Decoy-state analysis for three intensity levels: Hoeffding’s inequality relates finite-sample intensity-assigned counts nX,k to their asymptotic counterparts.The deviation term δ(nX, ε1) is the same for all intensity values k.
- A. Decoy-state analysis for three intensity levels: The same finite-sample relation is applied to expected and observed errors, connecting mX,k with the photon-number-resolved error counts.
- A. Decoy-state analysis for three intensity levels: These relations are later inserted into the decoy-state analysis to obtain bounds from observed statistics.
1. Lower-bound on the number of vacuum events
The vacuum-event bound is derived from the conditional intensity probabilities using Bayes’ rule and is tight as the third intensity approaches zero.
- 1. Lower-bound on the number of vacuum events: An analytical lower bound on sX,0 is obtained by exploiting the structure of the conditional probabilities pk|n.
- 1. Lower-bound on the number of vacuum events: Bayes’ rule is used to relate the conditional probabilities pk|n to the probability that Alice prepares an n-photon state.
- 1. Lower-bound on the number of vacuum events: The lower-bound expression for sX,0 contains a non-negative second term when µ2 > µ3, enabling its rewritten form.
- 1. Lower-bound on the number of vacuum events: The lower bound is tight when µ3 → 0.
2. Lower-bound on the number of single-photon events
The single-photon-event lower bound is developed through three concise steps, using the intensity constraint and the decomposition of multi-photon events before solving for sX,1.
- 2. Lower-bound on the number of single-photon events: The lower bound on the number of single-photon events is derived in three concise steps.
- 2. Lower-bound on the number of single-photon events: One step uses an inequality valid for n ≥ 2 under the condition µ2 + µ3 ≤ µ1.
- 2. Lower-bound on the number of single-photon events: Another step uses the fact that the sum of multi-photon events has a specific decomposition.
- 2. Lower-bound on the number of single-photon events: The derivation concludes by solving for sX,1.
3. Upper-bound on the number of single-photon errors
The analysis derives an upper bound on the number of single-photon errors using observed statistics associated with the third intensity level.
- An upper bound on the number of single-photon errors can be obtained using m∗X,µ3.
- The bound uses the quantity eµ2m∗X,µ3/pµ3.
4. Finite-size decoy-state analysis
The finite-size analysis converts asymptotic decoy-state bounds into bounds applicable to observed statistics through finite-size relations.
- The initial bounds are not directly applicable because Eqs. (4), (5), and (6) contain asymptotic quantities n∗X,k.
- Equations (1) and (2) resolve this issue by relating asymptotic quantities to observed statistics.
- Substituting the resulting quantities into Eqs. (4), (5), and (6) yields the formulas stated in the main text.
B. Secrecy analysis
The secrecy analysis bounds the raw key’s smooth min-entropy conditioned on Eve by accounting for revealed information, event types, and single-photon phase errors. Entropic uncertainty relations connect phase-error estimates to secrecy after privacy amplification.
- The analysis uses entropic uncertainty relations to bound the smooth min-entropy of the raw key conditioned on Eve’s information.
- Privacy amplification with two-universal hashing extracts an εsec-secret key of length ℓ from Alice’s raw key.
- The secrecy parameter is chosen through smoothing parameters proportional to εsec/(1−pabort).
- Conditional smooth min-entropy quantifies Eve’s uncertainty about Alice’s raw key and is bounded using protocol statistics.
- Error correction and error verification reduce the entropy bound by leakEC and log2 2/εcor bits, respectively.
- The raw key is decomposed into vacuum, single-photon, and multi-photon components, with the decomposition information included in Eve’s system.