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Faithful Simulation of Broadcast Measurements

Anders Høst-Madsen

TL;DR

This work addresses reducing communication for distributed quantum measurements by enabling faithful simulation of Charlie's POVM when only functions of the outcome are needed by Alice and Bob. The authors develop a sequential measurement protocol in which Charlie first yields $x_A$ to Alice and then applies a conditional measurement to obtain $x_B$, and they derive an achievable rate region for $(R_A,R_B)$ (with shared randomness rates $S_A,S_B$). The construction combines typicality-based random-bin schemes, conditional subspace projections, and operator Chernoff bounds to ensure the simulated statistics are faithful on a purification, with two Bob-decoding options yielding different rate expressions. The results extend faithful simulation theory to a two-user functional-splitting setting and show optimal performance in trivial cases such as identical functions or independent outcomes, potentially reducing communication for distributed quantum function computation.

Abstract

In this paper a central server Charlie has access to a quantum system C and measures it with a POVM $\{Λ_x\}$. Alice and Bob are only interested in the partial results $g_A(x)$ respectively $g_B(x)$. Alice, Bob, and Charlie share common randomness and Alice and Bob only need to faithfully simulate their measurements. The paper develops to achievable regions for the amount of communication needed to Alice and Bob.

Faithful Simulation of Broadcast Measurements

TL;DR

This work addresses reducing communication for distributed quantum measurements by enabling faithful simulation of Charlie's POVM when only functions of the outcome are needed by Alice and Bob. The authors develop a sequential measurement protocol in which Charlie first yields to Alice and then applies a conditional measurement to obtain , and they derive an achievable rate region for (with shared randomness rates ). The construction combines typicality-based random-bin schemes, conditional subspace projections, and operator Chernoff bounds to ensure the simulated statistics are faithful on a purification, with two Bob-decoding options yielding different rate expressions. The results extend faithful simulation theory to a two-user functional-splitting setting and show optimal performance in trivial cases such as identical functions or independent outcomes, potentially reducing communication for distributed quantum function computation.

Abstract

In this paper a central server Charlie has access to a quantum system C and measures it with a POVM . Alice and Bob are only interested in the partial results respectively . Alice, Bob, and Charlie share common randomness and Alice and Bob only need to faithfully simulate their measurements. The paper develops to achievable regions for the amount of communication needed to Alice and Bob.
Paper Structure (6 sections, 1 theorem, 42 equations)

This paper contains 6 sections, 1 theorem, 42 equations.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Faithful simulation
  4. Proof part I
  5. Proof part II: communication
  6. Proof part III: Faithful simulation

Key Result

Theorem 1

With sequential measurement (Alice before Bob), the rate region for Alice is Bob has two possible rate regions. First, Bob also needs to know Alice's common randomness with Charlie. Second, Bob does not need to know Alice's and Charlie's shared randomness. Here $R$ is a reference system purifying $\rho$.

Theorems & Definitions (1)

  • Theorem 1