Two notions of free entropy differ even when both are finite
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy -∞< \chi< \chi^*< ∞. This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.
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MANUSCRIPT 01
A finite-entropy separation of microstates and nonmicrostates free entropy
We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively. We construct a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state such that -∞< \chi(X)\leq\chi^*(X)-\tfrac12< ∞. Here χ is the original microstates entropy with an operator-norm…
Full source abstract
We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively. We construct a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state such that -∞< \chi(X)\leq\chi^*(X)-\tfrac12< ∞. Here χ is the original microstates entropy with an operator-norm cutoff and a limsup over matrix sizes. The counterexample uses a large but fixed number of variables.
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