First Law of Entanglement and Quadratic Corrections

The first law of entanglement is the linearization of an exact relative-entropy identity. For a normalized state perturbation on a fixed algebra and about a fixed reference state, the entropy variation equals the variation of the reference modular energy. Relative entropy has no linear term at its minimum, and its positive quadratic term measures the first distinguishable departure from the reference.

Required background. Relative entropy in QFT supplies the finite continuum comparison, and modular Hamiltonian definitions supply the generator and domain conventions.

Exact identity at fixed reference

Let $\sigma$ be a faithful reference density matrix and $K_\sigma=-\log\sigma$. For any normalized $\rho$ with compatible support,

$$\begin{aligned} S(\rho\Vert\sigma) &=\operatorname{Tr}\rho(\log\rho-\log\sigma)\\ &=\Delta\langle K_\sigma\rangle-\Delta S, \end{aligned}$$

where

$$\Delta\langle K_\sigma\rangle =\operatorname{Tr}[(\rho-\sigma)K_\sigma], \qquad \Delta S=S(\rho)-S(\sigma).$$

This equation is exact; Blanco, Casini, Hung, and Myers 2013, §2 develops the same identity and its positivity consequences. The modular Hamiltonian is held fixed at the reference state; $\Delta\langle K_\sigma\rangle$ is not the change of $\langle K_\rho\rangle_\rho$.

In an algebraic QFT formulation the same relation uses Araki relative entropy and the relative modular operator. The separate terms can require a regulator, but their difference is intrinsic when the relative entropy is finite.

The structural map places First Law of Entanglement and Quadratic Corrections along the perturbative sequence from normalized state or shape variations to information metrics and modular transport.

At fixed comparison algebra, normalization removes the linear term in relative entropy. The second response supports several inequivalent constructions: state and shape tangents, stress-tensor kernels, monotone metrics, and zero-mode-projected transport. The diagram is schematic and not to scale.

First variation

Take a differentiable normalized family

$$\rho_\lambda=\sigma+\lambda X+O(\lambda^2), \qquad \operatorname{Tr}X=0.$$

Since $S(\rho\Vert\sigma)\ge0$ and vanishes at $\rho=\sigma$, its first derivative at an interior faithful reference is zero. Differentiating the exact identity gives

$$\delta S=\delta\langle K_\sigma\rangle =\operatorname{Tr}(XK_\sigma).$$

One can verify this directly:

$$\delta S =-\operatorname{Tr}\bigl[X(\log\sigma+\mathbf1)\bigr] =-\operatorname{Tr}(X\log\sigma),$$

because normalization removes $\operatorname{Tr}X$. This shows exactly where the assumption enters. If the family is not normalized, an extra trace term remains.

The first law is kinematic. It does not require that $\rho_\lambda$ solve equations of motion, that $K_\sigma$ be local, or that the perturbation be thermal. Dynamics enters only when $X$ is related to sources or time evolution.

Positive quadratic correction

The Fréchet derivative of the logarithm gives the inverse Kubo–Mori map

$$\mathcal T_\sigma(X) =\int_0^\infty d\beta\; (\sigma+\beta)^{-1}X(\sigma+\beta)^{-1}.$$

For a linear family at a faithful finite-dimensional reference,

$$S(\rho_\lambda\Vert\sigma) =\frac{\lambda^2}{2}\, \operatorname{Tr}\!\left[X\,\mathcal T_\sigma(X)\right] +O(\lambda^3).$$

The quadratic form

$$\chi_{\rm rel}(X,X) =\operatorname{Tr}[X\mathcal T_\sigma(X)]$$

is nonnegative and vanishes only for the zero tangent on the support. In the eigenbasis $\sigma=\sum_n p_n\lvert n\rangle\langle n\rvert$,

$$\chi_{\rm rel}(X,X) =\sum_{m,n} \frac{\log p_m-\log p_n}{p_m-p_n} \lvert X_{mn}\rvert^2,$$

with the diagonal limit $1/p_n$. Positivity follows because $\log x$ is increasing.

Combining with the exact identity yields

$$\Delta S =\Delta\langle K_\sigma\rangle -\frac{\lambda^2}{2}\chi_{\rm rel} +O(\lambda^3).$$

Thus relative entropy is the positive deficit from saturating the linear first-law relation.

Remainders and directions at the support boundary

An $O(\lambda^3)$ symbol is meaningful only for a family that remains faithful and sufficiently differentiable. A useful finite-dimensional control is to write

$$\rho_\lambda =\sigma^{1/2}(\mathbf1+\lambda Y+\cdots)\sigma^{1/2}$$

and require $\lVert\lambda Y\rVert<1$. The nearest loss of positivity limits the Taylor expansion. If an eigenvalue of $\sigma$ is zero, generic tangents can change support and produce nonanalytic terms or infinite relative entropy; the simple Hessian formula must then be restricted to the supported face.

In QFT there may be no useful operator-norm ball. One instead controls a family of normal states on a fixed local algebra, an energy-bounded set of insertions, or a regulated sequence whose relative entropy and quadratic form converge. The stated topology should match the claimed remainder.

Fixed algebra is essential

If the state and region vary simultaneously, both $\rho_\lambda$ and the algebra of observables change. Identifying the moving algebras requires a specified pullback, and the derivative includes shape, displacement, and contact terms. Applying the fixed-algebra first law while silently changing the region mixes two different tangents.

The safe decomposition is:

1. vary the state on the reference algebra;
2. vary the region with the state held fixed and an explicit algebra identification;
3. compute mixed terms only after both conventions are fixed.

Shape-deformation theory handles the second step.

Common pitfalls

Varying the modular Hamiltonian in the first-law term. The identity uses $K_\sigma$ fixed at the reference. A variation of $K_\rho$ belongs to higher-order response.

Inferring dynamics from the first law. The equality follows from normalization and differentiability. Field equations or gravitational equations require additional input.

Claiming a cubic remainder at a support change. The logarithm is singular at zero eigenvalues. Check faithfulness and the radius to the positivity boundary.

Before interpreting this response coefficient, use the validity map to check normalization, tangent domains, contact and regulator terms, metric choice, and analyticity independently.

Normalization, domain, contact-term, metric, and analyticity checks are independent. A failed gate narrows or withdraws the physical claim; it is not a change of notation. The decision map is schematic and not to scale.

References

• Blanco, David D., Horacio Casini, Ling-Yan Hung, and Robert C. Myers. “Relative Entropy and Holography.” Journal of High Energy Physics 2013, no. 8 (2013): 060. DOI; arXiv.

Further reading

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.