DoT: The Duality of Time Postulate

For a quantum state, the von Neumann entropy is $S(\rho)=-k_B\operatorname{tr}(\rho\log\rho)$. Unitary evolution preserves it. Reduced entropy may change as correlations are exchanged with an environment, but it need not grow monotonically for every finite system and every preparation.

A useful controlled statement uses a completely positive trace-preserving semigroup $\Phi_t$ with a stationary reference state $\sigma$. Where the relative entropy is finite,

This follows from information contraction under the channel and becomes monotonic along the semigroup. It does not say that $S(\rho)$ itself must increase under cooling or relaxation to a nonuniform equilibrium. A thermodynamic interpretation must include heat exchange and the appropriate equilibrium structure.

The dual-time proposal can organize the origin of reduced descriptions and persistent records. To explain a universal thermodynamic arrow it must further explain the relevant boundary conditions, why records are stable, and why the reduced dynamics has the required positivity and statistical properties.