First Law of Entanglement Entropy

• First Law of Entanglement Entropy is a universal linear relation that links changes in entanglement entropy to local energy variations in quantum field theories.
• Holographic methods like the Ryu–Takayanagi prescription yield explicit geometric scaling laws, defining an effective entanglement temperature based solely on subsystem size.
• The law bridges thermodynamic stability and quantum information, elucidating how strong subadditivity and energy changes underlie gravitational dynamics in holographic duals.

The First Law of Entanglement Entropy refers to a universal linear relation, applicable in quantum field theories (QFT) and their holographic duals, linking the change in entanglement entropy of a subsystem to the change in local energy under small excitations. For sufficiently small subsystems, this relation is closely analogous to the first law of thermodynamics and emerges as a fundamental property of ground and excited states in strongly coupled conformal field theories (CFTs), particularly those admitting an AdS/CFT dual description.

1. Holographic Entanglement and the First Law Analogy

In a quantum field theory, consider a spatial subsystem $A$ of characteristic size $l$, and let $S_A$ denote its entanglement entropy. Under perturbations exciting the system—such as local energy insertions, thermal excitations, or mass perturbations—the entanglement entropy deviates from its ground state value: $S_A = S_A^{(0)} + \Delta S_A,$ where $\Delta S_A$ quantifies the extra quantum information. The local energy in $A$ likewise shifts by $\Delta E_A$. For $l$ sufficiently small (so that $ml^d \ll 1$ and the associated minimal surface in the AdS bulk remains near the boundary), the paper (Bhattacharya et al., 2012) demonstrates: $T_{\text{ent}} \cdot \Delta S_A = \Delta E_A,$ where $T_{\text{ent}}$ is an effective "entanglement temperature" set by geometry alone: $$T_{\text{ent}} = \frac{c}{l}, \qquad c = \frac{d+1}{2\pi}$$ when $A$ is a ball of radius $l$ in $d$ spatial dimensions.

This "first law of entanglement entropy" expresses a universal proportionality between energy and quantum information in small subsystems, independent of the details of the excitation.

2. Explicit Holographic Expressions and Universality

Quantitative predictions for $\Delta S_A$ and $\Delta E_A$ can be obtained in holographic CFTs using the Ryu–Takayanagi prescription. For specific geometries:

• Strip region (width $l$):

$$\frac{\Delta S_A}{\Delta E_A} = (\text{const}_d) \cdot l$$

where the constant involves $d$-dimensional Gamma functions.

• Spherical region (radius $l$):

$$\frac{\Delta S_A}{\Delta E_A} = \frac{2\pi}{d+1} \, l \implies T_{\text{ent}} = \frac{d+1}{2\pi l}$$

which is a direct realization of the canonical entanglement temperature and first law relation.

These relationships are robust for arbitrary, small excitations (not only thermal) as long as the geometric scale $l$ remains the smallest physical scale.

3. Field Theory Confirmation and Universality

The universality of the first law is confirmed in two-dimensional CFTs by direct calculation. For example, at finite temperature $\beta$, the entanglement entropy for an interval of length $l$,

$$S_A = \frac{c}{3} \log\left(\frac{\beta}{\pi} \sinh \frac{\pi l}{\beta}\right),$$

yields, for $l \ll \beta$,

$$\Delta S_A \simeq \frac{\pi}{3} l \cdot \Delta E_A,$$

matching the holographic prediction with $T_{\text{ent}} = 3/(\pi l)$.

This identification demonstrates that the first law of entanglement entropy is a field-theoretic, not solely holographic, phenomenon, characteristic of theories with local modular Hamiltonians for small regions.

4. Thermodynamic Stability, Subadditivity, and Negative Specific Heat

The paper also identifies a deep connection between thermodynamic stability and quantum information inequalities. For certain holographic states (e.g., the D3-brane shell realizing negative specific heat), two branches of holographic minimal surfaces exist:

• A concave branch obeys strong subadditivity ($d^2S_A/dl^2 \leq 0$), as required for stable thermodynamics (positive specific heat).
• A non-concave branch violates strong subadditivity and signals thermodynamic instability (negative specific heat).

Formally, if the finite part of $S_A$ scales as $l^{q+1}$, strong subadditivity imposes $q \leq 0$. Interpreting the thermal entropy as $S_{\text{th}}\sim V T^{-q/z}$, with $T\sim 1/l^z$, reveals that positivity of specific heat corresponds precisely to the concavity constraint on $S_A$.

This correspondence establishes a concrete bridge between the second law of thermodynamics and constraints on entanglement in quantum information theory.

5. Entanglement Temperature: Geometry and Scaling

The effective entanglement temperature $T_{\text{ent}}$ is a property of the geometry of the subsystem, not the details of the excitation. For a ball of radius $l$ in $d$ spatial dimensions,

$T_{\text{ent}} = \frac{d+1}{2\pi l},$

highlighting that the energy scale for "entanglement thermodynamics" is set by the inverse size of the region.

The entanglement temperature is distinct from any physical temperature present in the system (e.g., due to thermal excitation) and controls the ratio of the entanglement entropy change to energy for infinitesimal local state variations.

6. Broader Implications and Foundations

The first law of entanglement entropy provides evidence for a universal relationship between quantum energy distribution and quantum information content. It underpins recent derivations of gravitational dynamics from entanglement principles, acting as a cornerstone in the holographic emergence of Einstein's equations from CFT (Mosk, 2016, Oh et al., 2017).

The interplay between strong subadditivity and thermodynamic stability suggests that fundamental information-theoretic inequalities directly govern the allowed dynamics of strongly coupled field theories and their gravitational duals.

The law's geometric universality, its confirmation across CFTs of various dimensions, and its role in connecting quantum information and gravitational dynamics mark it as a central organizing principle for holographic quantum systems.

7. Summary Table: Key Formulas and Correspondences

Relation	Formula	Interpretation
First Law (small $A$)	$T_{\text{ent}}\, \Delta S_A = \Delta E_A$	Entanglement thermodynamics
Entanglement Temperature (ball)	$T_{\text{ent}} = \frac{d+1}{2\pi l}$	Geometry sets energy scale
Strip: ratio of entropy/energy	$(\Delta S_A)/(\Delta E_A) = (\text{const}) \, l$	Universal for small $A$
Ball: ratio	$(\Delta S_A)/(\Delta E_A) = \frac{2\pi}{d+1} \, l$	Matches above
2d CFT small interval	$\Delta S_A \simeq (\pi/3)\, l \Delta E_A$	Field theory confirmation
Concavity and specific heat	$$[S_A]_{\text{finite}} \sim l^{q+1};~q\leq0\Leftrightarrow~C_v\geq0$$	Stability/inequality link

This synthesis establishes the first law of entanglement entropy as a precise, quantitative bridge between energy, geometry, and quantum information in quantum field theories, with profound consequences for the understanding of gravity–information dualities and thermodynamic–entanglement correspondences.