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Certification of High-Dimensional Entanglement Within the Resource Theory of Buscemi Nonlocality

Xian Shi

Abstract

High-dimensional entanglement, captured by the Schmidt number, underpins advantages in quantum information tasks, yet a unified resource-theoretic description across different Buscemi-type operational objects has been missing. Here we develop a convex framework that treats bipartite states, distributed measurements, and teleportation instruments generated from shared entanglement on equal footing. For a fixed Schmidt-number threshold k, we introduce robustness-based monotones for each class of objects and prove a quantitative collapse: the Schmidt-number robustness of a bipartite state coincides with the maximal robustness achievable by any distributed measurement or teleportation instrument derived from that state. Consequently, within Buscemi-type operational frameworks, these objects do not carry independent high-dimensional resources but are governed by a single robustness-based monotone. We further provide a direct operational interpretation by relating this unique quantifier to the optimal advantage in entanglement-assisted state discrimination games. Our results complete a unified resource-theoretic characterization of high-dimensional entanglement across states, measurements, and quantum devices.

Certification of High-Dimensional Entanglement Within the Resource Theory of Buscemi Nonlocality

Abstract

High-dimensional entanglement, captured by the Schmidt number, underpins advantages in quantum information tasks, yet a unified resource-theoretic description across different Buscemi-type operational objects has been missing. Here we develop a convex framework that treats bipartite states, distributed measurements, and teleportation instruments generated from shared entanglement on equal footing. For a fixed Schmidt-number threshold k, we introduce robustness-based monotones for each class of objects and prove a quantitative collapse: the Schmidt-number robustness of a bipartite state coincides with the maximal robustness achievable by any distributed measurement or teleportation instrument derived from that state. Consequently, within Buscemi-type operational frameworks, these objects do not carry independent high-dimensional resources but are governed by a single robustness-based monotone. We further provide a direct operational interpretation by relating this unique quantifier to the optimal advantage in entanglement-assisted state discrimination games. Our results complete a unified resource-theoretic characterization of high-dimensional entanglement across states, measurements, and quantum devices.

Paper Structure

This paper contains 5 sections, 6 theorems, 53 equations, 2 figures.

Table of Contents

  1. Supplemental Materials
  2. Properties of $\mathrm{R}_{kDM}(\mathbb{M})$
  3. Properties of $\mathrm{R}_{sc}(\mathsf{\Lambda})$.
  4. Proof of Theorem \ref{['t2']}
  5. Proof of Theorem \ref{['t5']}

Key Result

Theorem 1

Assume $\rho_{A'B'}$ is a bipartite state and fix $k$. Then Here the first maximization is over all POVMs $M_A=\{M^{AA'}_{a}\}$ used in the teleportation instrument defined in Eq. (6), and the second maximization is over all local bipartite POVMs $M_A=\{M^{AA'}_{a}\}$ and $M_B=\{M^{B'B}_{b}\}$ used in the distributed measurement defined in Eq. (pm).

Figures (2)

  • Figure 1: The relations among three objects: high dimensional states, higher dimensional distributed measurements and higher dimensional teleportation instruments. In Theorem \ref{['t2']}, we show the equivalence relations among the above three objects with the help of the robustness-based quantifier.
  • Figure 2: The Schmidt-number robustness of $\rho_{iso}$.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • Remark 2
  • Corollary 1: Free-object equivalence
  • Example 3
  • Theorem 2
  • Definition 4
  • Definition 5
  • ...and 3 more