Continuum hydrodynamic description of local-information dynamics

Determine whether the dynamics of local quantum information on the information lattice admits a hydrodynamic description in the continuum limit, including a formulation that correctly captures the non-local current dependence, packet broadening, and stochastic subleading fluctuations beyond the singular leading-order evolution.

Background

The paper derives a formal continuum continuity equation for the local-information density on the information lattice. In this formulation, the currents depend non-locally on integrated information along diagonal regions of the lattice, reflecting the non-local constraints inherited from quantum entanglement inequalities.

At leading order, the resulting Euler-type equation admits singular, delta-function-like solutions corresponding to ballistic propagation of information packets. The authors note that a satisfactory theory would likely require higher-derivative corrections to generate regular solutions and stochastic noise consistent with the information constraints in order to reproduce the observed subleading broadening and Tracy–Widom-like fluctuations. A complete continuum theory incorporating these effects is left unresolved.

References

Given that the analyses presented in the previous sections present a compelling picture for the universal properties of the dynamics of local information in terms of their flow on the information lattice, a natural question that arises is does it admit a hydrodynamic description in the continuum limit.

— Dynamics of local quantum information in random unitary circuits  (2608.27440 - Thakur et al., 27 Aug 2026) in Section 6, “Towards a hydrodynamic theory”