Entanglement, Correlation, and Hydrodynamic Fronts

A “front” is an arrival contour of a declared observable at a declared threshold. Entanglement, mutual information, connected correlators, conserved density, and operator growth can produce different contours and different broadening. In relativistic QFT, microcausality fixes the causal boundary; it does not require every diagnostic to propagate at that speed.

Required background. Operator spreading and scrambling defines operator fronts, microcausality supplies the exact relativistic support condition, entanglement growth supplies entropy data, and mutual information supplies the correlation measure.

Helpful background. Membrane entanglement supplies one coarse-grained definition of $v_E$.

Direct front extraction

For a space–time diagnostic $D(x,t)$, define the threshold arrival time

$$t_\eta(x)=\inf\{t:D(x,t)-D_{\rm base}(x)>\eta\},$$

and fit $x$ as a function of $t_\eta$ only where the signal exceeds numerical and experimental error. Repeat over a threshold interval $\eta\in[\eta_{\min},\eta_{\max}]$. Threshold drift indicates front broadening or a poorly isolated asymptotic regime.

For a ballistic broadened front one might test

$$D(x,t)\simeq F\!\left(\frac{x-v_ft}{t^\alpha}\right),$$

where $v_f$ is the contour speed and $\alpha$ its width exponent. Diffusive transport has a length scale $\sqrt{Dt}$ rather than a nonzero ballistic $v_f$. A long-range initial state can give nonzero correlations at arbitrary separation at $t=0$ even though changes caused by a compact local intervention remain causal.

Which velocity?

• $c$ is the relativistic causal speed in the chosen units.
• A lattice Lieb–Robinson velocity bounds commutator tails and depends on the microscopic norm bound.
• A group velocity $v_g(k)=\partial_k\omega_k$ follows the dispersion of a mode.
• $v_E$ is an entropy-production coefficient in a coarse-grained growth law.
• $v_B$ is extracted from an operator-growth or OTOC front.
• A correlation-front velocity depends on the chosen operators, state, and threshold.

No equality among these follows from notation. The lattice commutator estimate of Lieb and Robinson 1972, Theorem 1, pp. 253–255 is a bound rather than a measured signal velocity. Mezei and Stanford 2017, §§ 2–4 give useful relations among chaotic-system velocities in stated regimes, not a universal identification across QFTs.

Fronts belong to the direct-diagnostic layer. A velocity obtains meaning from its observable, threshold, and scaling form, not from the visual slope of an effective picture. The map is schematic.

Matched contour test

In one regulated quench, extract contours for $\Delta S_A(t)$, $I(A:B;t)$, a connected two-point function, and a conserved density. Use the same spatial and time interpolation. Fit ballistic, diffusive, and broadened-ballistic models over nested windows, and compare predictive residuals on withheld times. Then vary operator choice and threshold.

Microcausality is checked through an intervention contrast or commutator, not the raw vacuum correlator. On a lattice, compare the observed contour with the lattice dispersion and the Lieb–Robinson bound; the latter is usually not a measured propagation speed.

The central failure is to label one fitted slope “the information velocity.” Changing observable, threshold, resolution, or time window can reveal a different front or no ballistic front at all. The map is schematic.

Evidence boundary

This comparison reflects results available through 10 August 2026. Exact causal support in relativistic QFT, lattice commutator bounds, and model-specific entanglement or operator fronts are well established. A universal equality between their velocities is not.

References

• Lieb, Elliott H., and Derek W. Robinson. “The Finite Group Velocity of Quantum Spin Systems.” Communications in Mathematical Physics 28 (1972): 251–257. DOI.
• Mezei, Márk, and Douglas Stanford. “On Entanglement Spreading in Chaotic Systems.” Journal of High Energy Physics 2017, no. 5 (2017): 065. DOI.