Given a system $A$ and its complement $\bar{A}$, we know that the entanglement entropy is given by $$ S_A = - \text{Tr} ( \rho_A \log \rho_A ), $$ where $\rho_A$ is the reduced density matrix obtained by tracing out the degrees of freedom in $\bar{A}$.

It is possible to geometrize this quantity for a field theory by considering its dual spacetime and calculating the area of a minimal surface anchored on the boundary of the region $A$, as proposed by Ryu-Takayanagi.

WLOG, let us assume that the dual spacetime is Schwarzschild-AdS$_3$, such that the field theory is at finite temperature and minimal surfaces are simply geodesics. It is possible to show that these geodesics start wrapping around the event horizon as the size of the region $A$ increases. However, the length of a geodesic completely wrapped around the horizon corresponds to the Bekenstein-Hawking thermal entropy.

Therefore we know that the entanglement entropy contains information about the thermal state of the system, at least for large enough regions. What if the size of $A$ is such that the corresponding minimal surface does not go deep in the bulk? Is there any way of formalizing the distinction between thermal and quantum entropies for a subsystem $A$ of arbitrary size?

  • 2
    $\begingroup$ A quantum information person will tell you that entanglement entropy is not a good measure of quantum entanglement between A and its complement \bar A when the total system is not in a pure state, precisely because there can be this type of thermodynamic contribution that you describe. Different measures have been proposed to deal with this case, for example negativity. $\endgroup$ Commented Dec 15, 2016 at 0:46
  • $\begingroup$ It must goes deep to the center. Since RT relation asks for that. The minimal surface must be a continuous deformation of the boundary, so you can not pass by the central black hole. $\endgroup$
    – XXDD
    Commented Mar 21, 2018 at 6:17

1 Answer 1


I'm not sure whether it would be relevant to your question but either way, one good measure in this case seems to be the Mutual Information. As discussed in [1] , let's say that we have two disjoint subsystems (infinite rectangular strips) $A$ and $B$ in $CFT_d$ , each specified by $$x^1\in [-\frac{l}{2},\frac{l}{2}] \quad ,\quad x^i\in [-\frac{L}{2},\frac{L}{2}] \quad s.t. \quad i=2,3,...,d-2 \quad and \quad L\to \infty$$ which are separated by the distance $x$ along $x^1$ direction. The bulk geometry is $Schwarzschild-AdS_{d+1}$ of radius $R$ with the Hawking temperature $\,T=\frac{r_hd}{4\pi R}\,$ . The authors calculated the HEE for both low temperature $(lT \ll 1)$ and high temperature $(lT \gg 1)$ cases. By the definition of Mutual Information as $$I(A:B)=S(A)+S(B)-S(A\cup B)$$ and using their results for $\,S\,$, in the $\,\frac{1}{l}\ll T \ll \frac{1}{x}\,$ case in which one expects that extremal surfaces wrap a part of horizon with the area contribution of $$\mathcal{A}_{ext}\sim r_{h}^{d-1}lL^{d-2} \sim r_{h}^{d-1}\mathcal{V} \quad s.t. \,\, \mathcal{V}\equiv Vol(A)=Vol(B)$$ interestingly by taking the $\,x\to 0\,$ limit $($i.e. two subsystems approach each other$)$ , they found that $I(A:B)|_{x\to 0}$ subtracts out the thermal contribution $(S_{th}\sim T^{d-1}\mathcal{V})$ and contains two area law terms:
$i)$ a universal divergent term $I_{div}\sim S_{div}\sim \frac{L^{d-2}}{x^{d-2}}$ which is expected [2].

$ii)$ the finite term $S_{ent}\sim L^{d-2}T^{d-2} \sim T^{d-2}\mathcal{A} \quad s.t.\,\, \mathcal{A}\,$ is the area of subsystems.

Therefore $S_{ent}$ term would be the measure of actual quantum entanglement, hence $I(A:B)|_{x\to0}$ would give us a good probe of entangled regions. I hope it helps you.

[1] https://arxiv.org/abs/1212.4764
[2] https://arxiv.org/abs/1010.4038


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