The Integrable Arrow of Time: Computational Reversibility and Physical Irreversibility in Many-Body Localized Systems

Many-Body Localized (MBL) systems present a unique paradigm for the arrow of time, distinct
from the standard thermodynamic arrow that arises in chaotic, thermalizing systems. The conven-
tional framework grounds irreversibility in the computational intractability (QMA-completeness) of
reversing a physical process, a hardness that emerges from the scrambling of quantum information.
MBL systems, by failing to scramble and thermalize, serve as a critical test of this thesis. This paper
formalizes the ”Integrable Arrow of Time” characteristic of MBL systems. We demonstrate that,
due to the emergence of quasi-local integrals of motion (l-bits), the evolution of an MBL system is
computationally reversible in principle; the problem of inferring the initial state is tractable for an
ideal quantum computer (in BQP). However, we then prove that for any realistic, resource-bounded
observer, this computational reversibility is physically inaccessible. The profound stability of the
l-bit structure provides a robust, local memory that enforces a unidirectional flow of time. This
work establishes that physical irreversibility does not strictly require computational intractability,
revealing a spectrum of computational arrows of time, from the QMA-hard thermodynamic limit to
the BQP-tractable integrable limit.