Global vs. Product Observables in Bipartite Quantum Systems: The Sharp Bound

What the math gives to ML

The paper gives a sharp, dimension-dependent comparison between probing a bipartite matrix with arbitrary global observables and probing it only with product observables. The transferable asset is a certified approximation principle: product probes cannot lose more than a factor of \(\sqrt{2}\min\{n,m\}\) relative to the full trace-norm signal, and the constant is optimal. This suggests an explicit diagnostic and training regularizer for multimodal or two-stream networks that measures how much of a learned interaction tensor is accessible through separable bilinear probes. The most practical first experiment is to use alternating spectral-norm-constrained optimization to estimate the product norm and compare product-only critics, global critics, and a norm-gap regularizer.