Mathematical Physics

A quantum-Fisher-optimal measurement may resolve exponentially many eigenvectors even when its information-bearing score needs only a small outcome alphabet. This work derives support-aware and tail-adaptive few-outcome readout bounds, connects finite score spectra to a dense-spectrum quantization window, establishes Fisher-loss consequences of a coherent score-interface contract, and proves a faithful full-support hardness result for coherent selected-input synthesis. Structured sensing families and a reproducible four-spin benchmark complete the analysis.

For faithful finite-dimensional state pairs, globally optimized measured relative entropy is additive at any fixed copy number n >= 2 exactly when the states commute. This Letter proves the criterion through an explicit phase-optimized two-copy ascent direction, separates coherent collective access from the known fresh-copy classical-feed-forward bound, gives a faithful qubit construction, and states a thermodynamic corollary only for the Gibbs-prepared isolated unitary-drive setting.

At low temperature, cooling maps and thermal operations have identical transition power for a two-level system. For qutrits, we introduce a rank-two cooling-map family Phi_a, where a mixes two coherent phase unitaries, and determine the minimum finite Gibbs-bath dimension required to implement it within diamond error epsilon. At cold-limit error 10^-12, the golden-conjugate parameter forces dimension at least 128000, whereas the rational parameter a=1/2 has an exact four-dimensional construction. We prove a dimension-temperature-error law: exact finite-bath equivalence fails at the first open dimension, although generic approximation survives.

A systematic tutorial and semi-review develops a density-of-states-consistent stochastic thermodynamics for highly underdamped Hamiltonian systems from established Kramers energy diffusion. Positive homogeneous Hamiltonians organize exact energy, heat, entropy-production, work, protocol, cycle, and interacting graph-cage results by effective degrees of freedom and the relaxation rate.

Can two independently prepared quantum systems require a joint measurement to read a fluctuation with the least possible variance? We prepare two copies of either candidate state. A measured Pearson quantity compares the outcome distributions of the two candidate preparations. Fully-PPT detectors include measurements made separately on each copy. The class maximum is the product of the one-copy optima, attained by measuring an optimal score on each copy. Noncommuting states allow an unrestricted joint measurement to do strictly better. We then fix a state-calibrated collective score as a theoretical witness. Its spectral readout adds no variance, but every unbiased fully-PPT readout must add some. A moment inequality strengthens this necessary bound; the exact restricted minimum remains unknown. Driven and equilibrium models illustrate the bounds without a device demonstration. The results concern calibrated scores, not work readout, detector energy cost, or local thermometry.