Thermodynamical Property of Entanglement Entropy for Excited States

TL;DR

It is argued that the entanglement entropy for a very small subsystem obeys a property which is analogous to the first law of thermodynamics when the authors excite the system, and this provides a universal relationship between the energy and the amount of quantum information.

Abstract

We argue that the entanglement entropy for a very small subsystem obeys a property which is analogous to the first law of thermodynamics when we excite the system. In relativistic setups, its effective temperature is proportional to the inverse of the subsystem size. This provides a universal relationship between the energy and the amount of quantum information. We derive the results using holography and confirm them in two dimensional field theories. We will also comment on an example with negative specific heat and suggest a connection between the second law of thermodynamics and the strong subadditivity of entanglement entropy.

Figures (2)

• Figure 1: The effective temperature $T_{ent}={d(\Delta E_A)\over d(\Delta S_A)}|_{l=fixed}$ as a function of $l$ based on the holographic calculation using the AdS$_4$ black hole at temperature $T$. We set $T=1,2,3$.
• Figure 2: The plot describes the regularize minimal surface area (divided by $\pi^3R^8L^2$) as a function of the width $l$ in the D3-brane shell. We choose $z_0=1$. The two thick curves correspond to the connected minimal surfaces, where upper one is not concave and thus is not physical. The horizontal dashed line represents the disconnected surface in the D3-brane shell. The dotted curve is the result for the pure AdS.