# Entanglement velocity The entanglement entropy for a large region $R$ at some $t$ following a uniform [[0558 Quantum quench|quench]] at $t=0$ displays a linear growth at late time:$S_R(t)-S_R(t=0)\sim v_E \,s_{\mathrm{th}} \operatorname{Area}[\partial R] \,t,$which defines $v_E$, the entanglement velocity. ## Refs - 2d CFT: [[2005#Calabrese, Cardy]] - higher dimensions: [[2013#Liu, Suh (May)]] and [[2013#Liu, Suh (Nov)]] ## Bounds - $v_{E} \leqslant v_{E}^{\mathrm{Sch}}=\frac{\sqrt{d}(d-1)^{\frac{1}{2}-\frac{1}{d}}}{[2(d-1)]^{1-\frac{1}{d}}}$ - conjectured in [[2013#Liu, Suh (May)]] and [[2013#Liu, Suh (Nov)]] - $v_E<c$ - proof using positivity of [[0300 Mutual information|mutual information]] ([[2015#Casini, Liu, Mezei]]) - proof using inequalities involving [[0199 Relative entropy|relative entropy]] ([[HartmanAfkhami-Jeddi2015]]) - $v_E<v_B$ - [[2016#Mezei, Stanford]] ## Temperature dependence - "Just like in the case of $v_B$, the entanglement velocity in thermal CFTs does not depend on the temperature. But $v_E$ acquires a temperature dependence if we deform the CFT and move along the corresponding RG flow. In these cases, $v_E$ violate the above bound, but it remains bounded by its corresponding value at the IR fixed point, never surpassing the speed of light" - see [[Jahnke2017]] and [[AvilaJahnkePatino2018]]