Exact entropy production requires unknown time-reversed transition matrix

Determine the exact total entropy production S_tot for the context-to-answer topic transition in the LLM bipartite information engine by identifying the time-reversed transition matrix A^R that appears in the entropy production formula and satisfies the required row-stochastic and marginal constraints (p^{(a)} and p^{(c)}), so that S_tot can be computed exactly rather than via lower bounds.

Background

The paper models LLM answer generation as a stochastic transformation of topic distributions from context (p^{(c)}) to answer (p^{(a)}) using a forward transition matrix A, and defines total entropy production using the Kullback–Leibler divergence between forward and time-reversed paths. Computing this quantity requires the reverse transition matrix A^R of the time-reversed process.

Because A^R is not observed from data, the authors cannot compute the exact entropy production and instead propose minimizing over admissible A^R to obtain a lower bound. This leaves unresolved the determination of the exact entropy production for the given QCA triplet and transition dynamics, motivating methods to recover A^R or otherwise compute S_tot exactly.