An Exact Product Limit on Collective Quantum Readout

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.