Asymptotically Optimal Quantum Universal Quickest Change Detection

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.

💡 Research Summary

The paper addresses the problem of quickest change detection (QCD) for quantum states when the post‑change quantum state is completely unknown, i.e., a universal setting. Building on the classical Lorden formulation, the authors aim to minimize the worst‑case average detection delay (WADD) subject to a constraint on the mean time to false alarm (MTFA).

First, the authors review the classical QCD framework, recalling that the CUSUM test is asymptotically optimal when both pre‑ and post‑change distributions are known. They then present the non‑parametric window‑limited adaptive (NWLA) CUSUM algorithm, which estimates the unknown post‑change density using a sliding window of size w and a kernel density estimator. Two technical conditions are required for the estimator: (1) a KL‑loss that decays as C₁ w^{‑β₁} with β₁>½, and (2) a bounded second moment of the log‑likelihood ratio decaying as C₂ w^{‑β₂} with β₂<2. Under these conditions, NWLA‑CUSUM achieves the same asymptotic delay as the optimal CUSUM with known post‑change distribution.

In the quantum setting, the source emits quantum states ξₜ that are ρ before the change point ν and σ afterwards. The authors consider the most general measurement strategy: collect the states in blocks of length ℓ, perform a joint POVM on each block, and obtain classical outcomes X_{ℓ,M,t}. The key technical tool is a result from quantum hypothesis testing (Theorem 3) which guarantees the existence of a projection‑valued measure (PVM) M_ℓ that depends only on the known pre‑change state ρ and satisfies
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