Smoothing Exponents and Decoupling in Semifinite von Neumann Algebras

We study the smoothing exponent of the max-relative entropy in semifinite von Neumann algebras.

Our main result gives an exact exponent formula in this setting.

The proof develops operator-algebraic replacements for the dimension-dependent tools used in finite-dimensional arguments.

These ingredients show that the smoothing exponent is governed by the underlying von Neumann algebraic structure rather than by matrix dimension estimates.

As an application, we formulate catalytic quantum information decoupling with a semifinite von Neumann algebraic reference system.

We prove an intrinsic layer-cake lemma for von Neumann algebras, which removes the countable spectrum assumption in the finite-dimensional proof and yields the corresponding semifinite estimate.

Consequently, the decoupling reliability exponent is described by the same sandwiched Rényi mutual information formula as in the finite-dimensional theory.