Thermodynamical Property of Entanglement Entropy for Excited States

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We will point out that there is a universal thermodynamical property of entanglement entropy for excited states.  We will derive this by using the AdS/CFT correspondence in any dimension. We will also directly confirm this property from direct field theoretic calculations in two dimensions. We will define a new quantity called entanglement density by taking derivatives of entanglement entropy with respect to the shape of subsystem. We will show that this quantity coincides with the energy density by taking the small subsystem limit and show that this is another equivalent statement of the thermodynamical property.