Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Beyond AME: A Novel Connection between Quantum Secret Sharing Schemes and $k$-Uniform States

Xuhong Liu, Shuai Shao

TL;DR

This work extends the link between quantum secret sharing (QSS) and $k$-uniform states beyond threshold/absolutely maximally entangled (AME) states by focusing on homogeneous access structures. The authors prove that every pure $3$-homogeneous QSS scheme induces a $3$-uniform QSS state, establishing a necessary condition that, however, is not sufficient by providing counterexamples. They deploy this connection to classify pure QSS schemes for up to $n=7$ players, showing that the unique $3$-homogeneous scheme on seven players arises from the Steane code (Fano plane), while other small-$n$ cases are ruled out or aligned with AME-based threshold structures. The results illuminate how $k$-uniform states inform non-threshold QSS design and set a foundation for future classifications of more intricate access structures.

Abstract

We study the connection between quantum secret sharing (QSS) schemes and $k$-uniform states of qubits beyond the equivalence between threshold QSS schemes and AME states. Specifically, we focus on homogeneous access structures and show that $3$-uniformity is a necessary but not sufficient condition for constructing a $3$-homogeneous QSS scheme using states of qubits. This gives a novel connection between non-threshold QSS schemes and $k$-uniform states. As an application of our result, we classify QSS schemes for up to 7 players and provide explicit characterizations of their existence. Our results offer new insights into the role of $k$-uniform states in the design of QSS schemes (not necessarily threshold) and provide a foundation for future classifications of QSS schemes with more complex structures.

Beyond AME: A Novel Connection between Quantum Secret Sharing Schemes and $k$-Uniform States

TL;DR

This work extends the link between quantum secret sharing (QSS) and -uniform states beyond threshold/absolutely maximally entangled (AME) states by focusing on homogeneous access structures. The authors prove that every pure -homogeneous QSS scheme induces a -uniform QSS state, establishing a necessary condition that, however, is not sufficient by providing counterexamples. They deploy this connection to classify pure QSS schemes for up to players, showing that the unique -homogeneous scheme on seven players arises from the Steane code (Fano plane), while other small- cases are ruled out or aligned with AME-based threshold structures. The results illuminate how -uniform states inform non-threshold QSS design and set a foundation for future classifications of more intricate access structures.

Abstract

We study the connection between quantum secret sharing (QSS) schemes and -uniform states of qubits beyond the equivalence between threshold QSS schemes and AME states. Specifically, we focus on homogeneous access structures and show that -uniformity is a necessary but not sufficient condition for constructing a -homogeneous QSS scheme using states of qubits. This gives a novel connection between non-threshold QSS schemes and -uniform states. As an application of our result, we classify QSS schemes for up to 7 players and provide explicit characterizations of their existence. Our results offer new insights into the role of -uniform states in the design of QSS schemes (not necessarily threshold) and provide a foundation for future classifications of QSS schemes with more complex structures.

Paper Structure

This paper contains 11 sections, 24 theorems, 52 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and definitions
  4. Quantum secret sharing schemes
  5. Quantum secret sharing states
  6. $k$-Uniform, absolutely maximal entangled states and known results
  7. 3-Homogeneous QSS States Imply 3-Uniform States
  8. Pure QSS Schemes with Small Numbers of Players
  9. Classification of homogeneous QSS schemes for $n\leq 7$
  10. Classification of general QSS schemes for $n\leq 6$
  11. Concluding Remarks

Key Result

Lemma 2

For a single qudit system with the density $\rho$, $S(\rho)\leq \log d$. The equality holds when $\rho$ is maximally mixed, i.e. $\rho=\frac{\mathbb{I}_{d\times d}}{d}$ where $\mathbb{I}_{d\times d}$ is the $d$-by-$d$ identity matrix.

Figures (1)

  • Figure 1: The Fano plane.

Theorems & Definitions (55)

  • Definition 1: von Neumann entropy
  • Lemma 2
  • Lemma 3
  • Definition 4: Mutual information
  • Definition 5: Quantum secret sharing (QSS) schemes
  • Definition 6: Minimal access structure
  • Lemma 7: PhysRevLett.83.648smith2000quantumsecretsharinggeneral
  • Remark 1
  • Lemma 8
  • proof
  • ...and 45 more