The Single-Copy Quantum Bandit Is Classical: An Exact Spectral Collapse
Abstract
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/Kinf and 1/Dinf, 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.
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BibTeX
@misc{ng2026singlecopy,
title = {{The Single-Copy Quantum Bandit Is Classical: An Exact Spectral Collapse}},
author = {Siu Hin Ng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40339},
eprint = {2609.40339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40339},
}
RIS
TY - UNPB
TI - The Single-Copy Quantum Bandit Is Classical: An Exact Spectral Collapse
AU - Siu Hin Ng
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40339
UR - https://arxiv.org/abs/2609.40339
AB - 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_inf and 1/D_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.
ER -