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On the Optimality of a Quantum Key Distribution

Georgi Bebrov

TL;DR

The paper defines a formal notion of optimality for quantum key distribution by maximizing the total efficiency $\mathfrak{E}$ over protocol parameters in the asymptotic regime. It shows that combining extremely biased preparation/measurement bases ($p \to 1$) with a completely compressed classical channel (channel squeezing) can yield asymptotically optimal QKD, where the classical channel becomes negligible and the quantum channel approaches capacity. The authors derive BB84-specific optimality expressions, compare them to standard BB84, and demonstrate improved efficiency, then introduce two optimal QKD protocols—BB84-like and twin-field–like—that leverage both biased bases and $\mathfrak{compr}_k$ with $k \to \infty$. They provide a theoretical framework for computing optimality and propose compression as a practical tool to enhance QKD performance, potentially guiding the design of future QKD systems. The work emphasizes that optimal QKD is achievable by integrating capacity-reaching quantum channels with completely compressed classical channels, offering a concrete path to higher security-key-generation efficiency in the asymptotic limit.

Abstract

Quantum key distribution (QKD) systems require optimal performance of both quantum and classical channels - utilizing as few as possible qubits and bits for establishing as many as possible key bits. Here we report a way to determine if a quantum key distribution model (or protocol) operates in an optimal behavior. This is accomplished by introducing a quantity, called optimality, which is the maximum over the total efficiency of a QKD under any circumstances (any values of QKD parameters). The optimality definition is given for the asymptotic operation of a QKD system - when infinitely many quantum systems are transferred/used in a quantum key distribution protocol or a quantum key distribution system is used infinitely many times. A way to attain the optimality is considered\textemdash implementation of a completely efficient QKD system (a combination of capacity-reaching quantum channel and a completely compressed classical channel) is presented. Optimal versions of BB84-QKD and twin-field QKD are introduced.

On the Optimality of a Quantum Key Distribution

TL;DR

The paper defines a formal notion of optimality for quantum key distribution by maximizing the total efficiency over protocol parameters in the asymptotic regime. It shows that combining extremely biased preparation/measurement bases () with a completely compressed classical channel (channel squeezing) can yield asymptotically optimal QKD, where the classical channel becomes negligible and the quantum channel approaches capacity. The authors derive BB84-specific optimality expressions, compare them to standard BB84, and demonstrate improved efficiency, then introduce two optimal QKD protocols—BB84-like and twin-field–like—that leverage both biased bases and with . They provide a theoretical framework for computing optimality and propose compression as a practical tool to enhance QKD performance, potentially guiding the design of future QKD systems. The work emphasizes that optimal QKD is achievable by integrating capacity-reaching quantum channels with completely compressed classical channels, offering a concrete path to higher security-key-generation efficiency in the asymptotic limit.

Abstract

Quantum key distribution (QKD) systems require optimal performance of both quantum and classical channels - utilizing as few as possible qubits and bits for establishing as many as possible key bits. Here we report a way to determine if a quantum key distribution model (or protocol) operates in an optimal behavior. This is accomplished by introducing a quantity, called optimality, which is the maximum over the total efficiency of a QKD under any circumstances (any values of QKD parameters). The optimality definition is given for the asymptotic operation of a QKD system - when infinitely many quantum systems are transferred/used in a quantum key distribution protocol or a quantum key distribution system is used infinitely many times. A way to attain the optimality is considered\textemdash implementation of a completely efficient QKD system (a combination of capacity-reaching quantum channel and a completely compressed classical channel) is presented. Optimal versions of BB84-QKD and twin-field QKD are introduced.

Paper Structure

This paper contains 10 sections, 1 theorem, 29 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. BB84-QKD
  3. Optimality
  4. Optimality Evaluation
  5. Compression
  6. Compression definition
  7. Compression results
  8. Optimal QKD protocols
  9. Summary
  10. Example of channel squeezing

Key Result

Theorem 1

A quantum key distribution protocol (considering only state preparation, state measurement, and sifting procedures) could asymptotically approach the optimal quantum key distribution provided that both biased preparation/measurement bases ($p\rightarrow \text{1}$) and compression process $\mathfrak{

Figures (2)

  • Figure 1: Total efficiency comparison between optimal BB84-QKD (presented in previous subsection) and standard BB84-QKD. Optimal BB84-QKD ($\mathfrak{O}^{\text{BB84}}(L)$)---dashed plot; Standard BB84-QKD ($\mathfrak{E}^{\text{BB84}}(L)$)---solid plot.
  • Figure 2: Relation between the compression coefficient $\sigma$ and the degree of compression $k$. The plot is obtained for $n$ = 10$^{\text{30}}$. Due to lack of computation resources, we calculate $\sigma$ only for $k$ = $\text{2,3,...,24}$.

Theorems & Definitions (3)

  • Definition 1: Channel squeezing
  • Theorem 1
  • proof