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Witness wedges in fidelity-deviation plane: separating teleportation advantage and Bell-inequality violation

Kyoungho Cho, Jeongho Bang

Abstract

We develop a unified framework to analyze $d$-dimensional quantum teleportation through the joint geometry of two complementary figures of merit: average fidelity $F$ (how well a protocol works on average) and fidelity deviation $D$ (how uniformly it works across the inputs). Technically, we formulate a representation-theoretical framework based on Schur-Weyl duality and permutation symmetry calculus that reduce the higher-moment Haar averages to a finite set of trace invariants of the composed correction unitaries. This yields closed-form expressions for $F$ and $D$ in arbitrary Hilbert-space dimension and delivers tight bounds that link the admissible deviation directly to the gap from the optimal average performance. In particular, any measured pair $(F, D)$ can be ported into a visibility estimate for isotropic channel resources, turning the $(F, D)$-plane into a calibrated diagnostic map. We further cast the teleportation advantage and CGLMP-inequality violation as two witnesses lines in the $(F,D)$ plane: one line certifies that $F$ beats the classical benchmark $2/(d{+}1)$, while the other line certifies the Bell nonlocality. Their identical slope but distinct intercepts expose a quantitative gap between "entangled yet local" and "genuinely nonlocal" resources.

Witness wedges in fidelity-deviation plane: separating teleportation advantage and Bell-inequality violation

Abstract

We develop a unified framework to analyze -dimensional quantum teleportation through the joint geometry of two complementary figures of merit: average fidelity (how well a protocol works on average) and fidelity deviation (how uniformly it works across the inputs). Technically, we formulate a representation-theoretical framework based on Schur-Weyl duality and permutation symmetry calculus that reduce the higher-moment Haar averages to a finite set of trace invariants of the composed correction unitaries. This yields closed-form expressions for and in arbitrary Hilbert-space dimension and delivers tight bounds that link the admissible deviation directly to the gap from the optimal average performance. In particular, any measured pair can be ported into a visibility estimate for isotropic channel resources, turning the -plane into a calibrated diagnostic map. We further cast the teleportation advantage and CGLMP-inequality violation as two witnesses lines in the plane: one line certifies that beats the classical benchmark , while the other line certifies the Bell nonlocality. Their identical slope but distinct intercepts expose a quantitative gap between "entangled yet local" and "genuinely nonlocal" resources.

Paper Structure

This paper contains 24 sections, 12 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: teleportation setting and figures of merit
  3. Noisy teleportation: unconditional output and single-shot fidelity
  4. Figures of merit: average fidelity and fidelity deviation
  5. Haar Integrals via Schur--Weyl Duality
  6. General framework: unitary twirls, pure-state moments, and permutation unitaries
  7. Explicit cases: $k=2$ and $k=4$
  8. Average fidelity and fidelity deviation: closed forms, bounds, and when deviation matters
  9. Average fidelity: closed form and range
  10. Fidelity deviation: exact expression and universal bounds
  11. Beyond $F$: when $D$ becomes critical
  12. Teleportation, Bell nonlocality, and the $(F,D)$ witness geometry
  13. Visibility thresholds in isotropic noise: $p_{\mathrm{BV}}$ and $p_c$, and their $(F,D)$ projections
  14. A Bell-nonlocality witness on the $(F,D)$ plane
  15. Teleportation advantage: a distinct witness and a distinct threshold
  16. ...and 9 more sections

Key Result

Theorem 1

Every isotropic channel resource with $p \le p_{\mathrm{BV}}(d)$ satisfies Thus, all $(F, D)$ from CGLMP-local resources lie in the closed wedge Hence, any $(F,D)\notin\mathcal{W}_{\mathrm{local}}(d)$ certifies $p>p_{\mathrm{BV}}(d)$ and thus CGLMP nonlocality.

Theorems & Definitions (19)

  • Theorem 1: CGLMP-violation wedge
  • Theorem 2: Teleportation-advantage wedge
  • Theorem 3: Strict inclusion of witnesses
  • Definition 1
  • Proposition 1: Invariance properties
  • Definition 2: $k$-th moment operator / $k$-fold unitary twirl
  • Definition 3: Unitary $t$-design via moment operators
  • Definition 4: $k$-th order commutant
  • Theorem 4: Schur-Weyl duality
  • Definition 5: Permutation operators
  • ...and 9 more