Quantum Inaccessibility

Abstract

Loschmidt's paradox asks why macroscopic irreversibility is universal despite the time-reversal symmetry of microscopic dynamics. We argue that irreversibility is not a property of the dynamics but of accessibility: chaotic evolution drives phase-space structure below the quantum resolution scale ${\ell }_{\hslash }$, at a critical time $t_c=\hat{I}{»}^{-1}\ln (\hat{I}{´}_0/{\ell }_{\hslash })$, after which the time-reversed microstate exists as a valid solution of Hamilton's equations but cannot be selected by any physically admissible operation. The mechanism operates entirely within the semiclassical regime $t_c\le t_E$, where classical geometry is exact. This provides a dynamical resolution of the Loschmidt paradox. The quantum foundation is established using a Krylov-complexity framework: we prove that for any $H(t)=H(-t)$, the quantum Lyapunov exponent satisfies $\hat{I}{»}_L^{\mathrm{forward}}=\hat{I}{»}_L^{\mathrm{backward}}$. The arrow of time is not in the dynamics. The mechanism predicts sigmoid fidelity decay, logarithmic scaling of $t_c$ with $\hat{I}{»}^{-1}$, and ensemble-size independence of the inaccessibility threshold -- all consistent with three decades of Loschmidt echo experiments and confirmed in a stadium-billiard simulation reported here. Underlying everything: quantum mechanics conserves information exactly. Entropy, defined as the logarithm of the multiplicity $\hat{I}\text{c◯}$ -- the number of possibilities consistent with the available information -- can only increase when information becomes operationally inaccessible. The second law reflects not a breakdown of microscopic reversibility, but the dynamical inaccessibility of the information required to reverse it.

20 pages, 2 figures. SI- 4 pages