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Academia Sinica（中央研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明单副本量子老虎机中谱测量加经典KL-UCB达到渐近实例最优，其遗憾下界由经典Burnetas-Katehakis泛函决定，并指出进一步改进需超越单副本测量类。
AI 中文摘要
我们研究多臂老虎机问题，其中每个臂提供未知的量子态，其平均奖励由已知效应$F$定义。每个新副本可通过任意自适应选择的POVM进行测量。我们证明了一个精确坍缩：从臂态到其混淆奖励半空间的测量相对熵等于$F$谱统计的经典Burnetas-Katehakis泛函，并且$F$的谱测量与一个由矩阵对数导数构建的显式最不利态一起，构成了底层测量博弈的鞍点。因此，在所有一致的单副本策略中，谱测量后接经典KL-UCB算法在渐近意义上是实例最优的，适用于每个有限维度以及在所述非退化假设下的每个奖励效应：自适应和随机测量设计无法改进前导对数遗憾系数，存储副本以供后续单副本测量也无法改进。一个自包含的有限时间界涵盖了一般效应和边界分布。任何进一步的改进必须来自单副本类之外的测量。对于具有任意量子存储器的一致策略，一个摊销相对熵论证给出了与Umegaki半空间散度的逆命题，并且两个每臂遗憾下界$1/K_{\mathrm{inf}}$和$1/D_{\mathrm{inf}}$当且仅当臂与$F$对易时重合。Umegaki下界是否可达涉及一个复合量子Stein问题以及一个到自适应遗憾的单独归约；我们将其作为开放问题提出，并附有两副本数值证据。
英文摘要
We study multi-armed bandits whose arms supply unknown quantum states and whose mean rewards are defined by a known effect $F$. Each fresh copy may be measured by an arbitrary, adaptively chosen POVM. We prove an exact collapse: the measured relative entropy from an arm state to its confusing reward half-space equals the classical Burnetas-Katehakis functional of $F$'s spectral statistics, and the spectral measurement of $F$ together with an explicit least-favorable state, built from the derivative of the matrix logarithm, forms a saddle point of the underlying measurement game. Consequently, spectral measurement followed by classical KL-UCB is asymptotically instance-optimal among all consistent single-copy policies, in every finite dimension and for every reward effect under the stated nondegeneracy assumptions: adaptive and randomized measurement design cannot improve the leading logarithmic regret coefficient, and neither can storing copies for later single-copy measurement. A self-contained finite-time bound covers general effects and boundary distributions. Any further improvement must come from measurements outside the single-copy class. For consistent policies with arbitrary quantum memory, an amortized relative-entropy argument gives a converse with the Umegaki half-space divergence, and the two per-arm regret floors, $1/K_{\mathrm{inf}}$ and $1/D_{\mathrm{inf}}$, coincide if and only if the arm commutes with $F$. Whether the Umegaki floor is attainable involves a composite quantum Stein problem and a separate reduction to adaptive regret; we pose it as an open problem, with two-copy numerical evidence.
Comments9 pages, 4 figures, 1 table
