Two-state generator extraction: property currents and a two-layer arrow of time in pre- and post-selected quantum dynamics

Research Question and Approach

This paper investigates the nature of friction and the arrow of time in quantum systems when the observer's description is extended from causal (past-conditioned) to time-symmetric (pre- and post-selected) dynamics. Using two-state generator extended dynamic mode decomposition (gEDMD), the author analyzes how weak values—which obey exact symmetry theorems—behave under different conditioning. The study employs a reflection involution on the ensemble to uniquely decompose window-fitted friction into an antisymmetric component (carrying boundary-condition physics) and a symmetric component (generated by the differencing scheme).

The Two-Layer Arrow of Time

The research identifies a dichotomy in the arrow of time. At the level of coherent modes, the antisymmetric component dominates, causing the arrow of time to reverse at the midpoint of the conditioning window. Conversely, at the fluctuation level, the symmetric component generated by the differencing scheme dominates, masking the reversal. The author shows that this is a matter of degree: the scheme-induced component is significantly larger at the fluctuation layer. When an exact-derivative baseline is used, both layers reverse, proving that the apparent immunity of the fluctuation layer is an artifact of the inference process rather than the ensemble itself.

Quantum Cheshire-Cat Dynamics

Beyond the arrow of time, the paper examines a lattice interferometer conditioned only at its input and output ports. It finds that the quantum Cheshire-cat structure—where a particle and its polarization are assigned to different arms—emerges spontaneously without being imposed by hand. The particle and polarization obey separate continuity equations. A local field in the polarization-carrying arm rotates the polarization phase at twice the field strength, appearing as a rigid imaginary shift in the extracted generator, while the particle's weak density remains invariant.

Scaling and Practical Implications

The findings were verified in chaotic XXZ spin chains ranging from 2^8 to 2^20 Hilbert-space dimensions. The study demonstrates that self-averaging makes the results insensitive to the specific post-selection class, provided they share the same boundary modulation. This effectively removes the exponential scaling obstruction (the 2^-N/2 overlap problem) typically associated with independent post-selection in large quantum systems.