Finite-Temperature and Excited-State Entanglement

Finite-temperature and excited-state entropies mix quantum entanglement with thermal and classical correlations. Clean statements compare matched states or use relative and excess quantities; thermal entropy density, entanglement entropy, and eigenstate entanglement coincide only in specific limits.

Required background. Use regulated subregion entropy. Helpful background. Mutual Information and Regulator-Independent Correlations and Entanglement Negativity in QFT separate total and quantum correlations.

Excess entropy and crossover

For a state $\rho$ relative to a reference $\sigma$, define a matched excess entropy

$$\Delta S_A=S(\rho_A)-S(\sigma_A).$$

Ultraviolet terms cancel when the states share the same local short-distance structure and regulator. Relative entropy sharpens the comparison:

$$D(\rho_A\Vert\sigma_A) =\Delta\langle K_{\sigma,A}\rangle-\Delta S_A\geq0.$$

For a small perturbation, the first law $\delta S_A=\delta\langle K_{\sigma,A}\rangle$ is the linear term, while relative entropy begins at quadratic order.

The structure figure separates state-dependent entropy from mixed-state and correlation measures.

A thermal reduced entropy contains both boundary entanglement and extensive thermal entropy. Mutual information and negativity probe different surviving correlations, so the measure must be chosen before interpreting a temperature crossover. Schematic.

In a two-dimensional CFT at inverse temperature $\beta$,

$$S_A(\beta)=\frac{c}{3} \log\!\left[ \frac{\beta}{\pi\epsilon} \sinh\!\left(\frac{\pi\ell}{\beta}\right) \right]+c_1.$$

For $\ell\ll\beta$, this approaches vacuum entanglement plus a small thermal correction. For $\ell\gg\beta$, it contains the thermal entropy density times $\ell$. The latter is not purely bipartite quantum entanglement.

The conformal-map derivation and both limits are given by Calabrese and Cardy 2004, § 3.

Excited states and ensembles

For a one-particle excitation in a free field, compare $S_A$, $\Delta S_A$, relative entropy, and mutual information at fixed volume and cutoff. Finite volume matters because a delocalized particle has a subsystem occupation probability that depends on $A/V$. Degeneracies and coherent superpositions can change the answer.

A high-energy eigenstate may have subsystem entropies close to a thermal ensemble under eigenstate-thermalization assumptions, but this is not a theorem for every QFT or every subsystem fraction. Integrable theories, conserved charges, scars, and finite-size effects require different ensembles or can violate the approximation.

Negativity can retain quantum entanglement after thermal entropy makes $S_A$ extensive, but adjacent-region ultraviolet terms and finite-temperature replica continuations must still be controlled.

Matched comparison

The validity map lists the choices that must not drift between vacuum, thermal, and excited calculations.

Excess and relative quantities cancel ultraviolet structure only for matched prescriptions. Changing volume, ensemble, energy window, boundary conditions, or regulator can leave a spurious contribution; ETH is an additional state-class assumption. Schematic.

Report temperature or energy, ensemble, total volume, subsystem fraction, boundary conditions, cutoff, and reference state. Separate the short-region, crossover, and volume-law regimes.

References

• Calabrese, Pasquale, and John Cardy. “Entanglement Entropy and Quantum Field Theory.” Journal of Statistical Mechanics (2004): P06002. DOI.

Further reading

• Alcaraz, Francisco C., Miguel Ibáñez Berganza, and Germán Sierra. “Entanglement of Low-Energy Excitations in Conformal Field Theory.” Physical Review Letters 106 (2011): 201601. DOI. Open preprint.