Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotically Optimal Quantum Universal Quickest Change Detection

Arick Grootveld, Haodong Yang, Nandan Sriranga, Biao Chen, Venkata Gandikota, Jason Pollack

TL;DR

The paper tackles quantum quickest change detection when the post-change quantum state is unknown, introducing a two-stage NWLA-QUSUM that first converts quantum data to classical statistics via block POVMs preserving $S(\sigma\|\rho)$ and then applies a windowed-CUSUM on the resulting data. It leverages Hayashi’s asymptotic measurement construction to realize a PVM yielding classical distributions that faithfully approximate the quantum divergence, and it proves that with carefully chosen block length $\ell$ and window size $w$, the resulting WADD scales as $\frac{\log \bar{T}_{FA}}{S(\sigma\|\rho)-\varepsilon_\ell}$ up to a vanishing correction, achieving asymptotic optimality under Lorden’s criterion. The analysis hinges on verifying two key conditions (KL-Loss and SecondMoment) for the kernel density estimator used in the NWLA step, showing these hold when the classical support size grows as $d=O(w^{1/2})$ and all bin probabilities satisfy $p_i=\Omega(w^{-1/2})$. The approach unifies quantum hypothesis testing and classical universal change detection to enable practical quantum universal QCD in finite dimensions, with potential impact on quantum sensing, communication, and error correction.

Abstract

This paper investigates the quickest change detection of quantum states in a universal setting: specifically, where the post-change quantum state is not known a priori. We establish the asymptotic optimality of a two-stage approach in terms of worst average delay to detection. The first stage employs block POVMs with classical outputs that preserve quantum relative entropy to arbitrary precision. The second stage leverages a recently proposed windowed-CUSUM algorithm that is known to be asymptotically optimal for quickest change detection with an unknown post-change distribution in the classical setting.

Asymptotically Optimal Quantum Universal Quickest Change Detection

TL;DR

The paper tackles quantum quickest change detection when the post-change quantum state is unknown, introducing a two-stage NWLA-QUSUM that first converts quantum data to classical statistics via block POVMs preserving and then applies a windowed-CUSUM on the resulting data. It leverages Hayashi’s asymptotic measurement construction to realize a PVM yielding classical distributions that faithfully approximate the quantum divergence, and it proves that with carefully chosen block length and window size , the resulting WADD scales as up to a vanishing correction, achieving asymptotic optimality under Lorden’s criterion. The analysis hinges on verifying two key conditions (KL-Loss and SecondMoment) for the kernel density estimator used in the NWLA step, showing these hold when the classical support size grows as and all bin probabilities satisfy . The approach unifies quantum hypothesis testing and classical universal change detection to enable practical quantum universal QCD in finite dimensions, with potential impact on quantum sensing, communication, and error correction.

Abstract

This paper investigates the quickest change detection of quantum states in a universal setting: specifically, where the post-change quantum state is not known a priori. We establish the asymptotic optimality of a two-stage approach in terms of worst average delay to detection. The first stage employs block POVMs with classical outputs that preserve quantum relative entropy to arbitrary precision. The second stage leverages a recently proposed windowed-CUSUM algorithm that is known to be asymptotically optimal for quickest change detection with an unknown post-change distribution in the classical setting.
Paper Structure (10 sections, 4 theorems, 44 equations, 1 figure)

This paper contains 10 sections, 4 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Classical Change Point Detection
  4. Quantum Change Point Detection
  5. NWLA-QUSUM Algorithm for Universal Change Point Detection
  6. Proofs
  7. Proof of NWLA-QUSUM Result
  8. Condition 1
  9. Condition 2
  10. Conclusions

Key Result

Theorem 1

Suppose we have a change point problem with a known pre-change distribution $P$ and an unknown post-change distribution $Q$. Assume $\hat{Q}^w_n$ is a kernel density estimator satisfying condition:KL_Loss and condition:SecondMoment with $\beta_1 = \frac{1}{2}$. Then taking $w = (\log\bar{T}_{FA})^{1

Figures (1)

  • Figure 1: Block diagram of NWLA-QUSUM. Measured quantum operators become classical random variables. A window of classical random variables is combined to estimate the density, which is used in the Classical NWLA CUSUM algorithm.

Theorems & Definitions (5)

  • Theorem 1: Theorem 2 of liang2024quickest
  • Theorem 2: Theorem 2 of fanizza2023ultimate
  • Theorem 3: Theorem 2 hayashi2001asymptotics
  • Theorem 4
  • proof