A new study demonstrates that the single-copy quantum bandit problem, where each "arm" supplies an unknown quantum state and rewards are defined by a known quantum effect, can be optimally solved using classical strategies. Researchers have proven that the measured relative entropy from an arm state to its "confusing reward half-space" is equivalent to the classical Burnetas-Katehakis functional, which is based on the spectral statistics of the quantum effect. This implies that the inherent quantum complexity of the problem collapses to a classical description in this specific scenario.

The work establishes that a spectral measurement of the quantum effect, combined with a classical KL-UCB (Kullback-Leibler Upper Confidence Bound) type algorithm, is asymptotically instance-optimal. This means that neither adaptive or randomized measurements, nor storing copies for later measurements, can improve the leading logarithmic regret coefficient. This result holds for any finite dimension and for every reward effect under the stated nondegeneracy assumptions, providing a self-contained finite-time bound for general effects and boundary distributions.

The key implication is that any further improvement in solving these types of quantum bandit problems must come from measurements outside the single-copy class. For consistent policies with arbitrary quantum memory, an amortized relative-entropy argument provides a converse with the Umegaki half-space divergence. The authors note that the per-arm regret floors, 1/Kᵢₙ𝖿 and 1/Dᵢₙ𝖿, coincide if and only if the arm commutes with the effect F. Whether the Umegaki floor is attainable remains an open problem, with two-copy numerical evidence suggesting it may require using two copies.