Finite-Temperature and Excited-State Entanglement

A subregion entropy at finite temperature contains boundary entanglement, classical uncertainty, and thermodynamic entropy. It is not, by itself, a mixed-state entanglement measure. Clean comparisons keep the region algebra and ultraviolet prescription fixed, state the ensemble and total volume, and use entropy, mutual information, relative entropy, or negativity only for the question each quantity answers.

Required background. Use regulated subregion entropy to fix the region algebra, cutoff, and subtraction before comparing states. Helpful background. Mutual Information and Regulator-Independent Correlations supplies a total-correlation diagnostic, while Entanglement Negativity in QFT supplies the partial-transpose test for non-PPT entanglement.

The chapter’s structure map separates state-dependent entropy from correlation and entanglement constructions. Its comparison table gives their operational domains, and its validity map identifies the choices that a vacuum–thermal or vacuum–excited subtraction must match.

Matched entropy and relative entropy

In a type-I realization, choose states $\rho_A$ and $\sigma_A$ on the same Hilbert-space factor. If $\operatorname{supp}\rho_A\subseteq\operatorname{supp}\sigma_A$, define

$$\Delta S_A=S(\rho_A)-S(\sigma_A), \qquad D(\rho_A\Vert\sigma_A) =\operatorname{Tr}\rho_A(\log\rho_A-\log\sigma_A).$$

With $K_{\sigma,A}=-\log\sigma_A$,

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

If the support condition fails, the relative entropy is $+\infty$. For a differentiable family $\rho_A(t)$ through $\sigma_A$, the linear term gives the first law

$$\delta S_A=\delta\langle K_{\sigma,A}\rangle,$$

while relative entropy begins at quadratic order. In an intrinsic continuum treatment, these statements use two normal states on one local von Neumann algebra and Araki’s relative modular operator, not a sharp-region density matrix Araki 1976, pp. 809–817 and Eqs. (1.1)–(1.2).

Ultraviolet cancellation in $\Delta S_A$ is a property of a matched family, not of the minus sign alone. The theory, region, algebra and center, regulator, boundary condition, and local short-distance state class must agree.

Worked thermal crossover in a 1+1-dimensional CFT

Let $A$ be one interval of length $\ell$ on the infinite spatial line, let the global state be the Gibbs state of a $1+1$-dimensional CFT at inverse temperature $\beta$, take the thermodynamic limit, set the signal velocity to one, and use the same short-distance cutoff $\epsilon$ as in the vacuum. Then

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

as derived in Calabrese and Cardy 2004, §III.A, Eq. (18). This is not the formula for a finite circle, a boundary interval, a microcanonical ensemble, or a generic massive theory.

Subtracting the vacuum entropy $S_A(\infty)=(c/3)\log(\ell/\epsilon)+c_1$ gives the cutoff-independent function

$$\Delta S_A(\beta) =\frac{c}{3}\log\frac{\sinh x}{x}, \qquad x=\frac{\pi\ell}{\beta}.$$

The two useful limits follow directly:

$$\Delta S_A =\frac{c}{18}x^2-\frac{c}{540}x^4+O(x^6), \qquad x\ll1,$$

and

$$S_A(\beta) =\frac{\pi c}{3\beta}\ell +\frac{c}{3}\log\frac{\beta}{2\pi\epsilon} +c_1+O(e^{-2\pi\ell/\beta}), \qquad x\gg1.$$

The coefficient $s_{\mathrm{th}}=\pi c/(3\beta)$ is the Gibbs entropy density. Thus the extensive large-interval term is thermal entropy, not bipartite quantum entanglement. Some representative exact values are

The universal excess-entropy crossover $3\Delta S_A/c=\log(\sinh x/x)$.

$x=\pi\ell/\beta$	$3\Delta S_A/c$	Regime
$0.25$	$0.01040$	vacuum-dominated
$1$	$0.16144$	crossover
$3$	$1.20576$	approaching the thermal volume term

Application: vacuum, Gibbs, and one-particle states

Fix one finite-volume regulator and a complementary bipartition $A\cup B$ before making this comparison:

What entropy, mutual information, and negativity conclude for three matched state classes.

Global state	$S_A$	$I(A{:}B)$	logarithmic negativity $\mathcal E$
Vacuum, pure	boundary entanglement plus regulator terms	$2S_A$ for a complementary pure bipartition	$S_{1/2}(A)$ for that pure bipartition
Gibbs state, mixed	boundary contribution plus an extensive thermal term	total classical and quantum correlation; extensive bulk terms cancel in a matched combination	certifies only NPT entanglement when nonzero; it is not obtained from $S_A$ by substitution
One delocalized particle above the vacuum, pure	$\Delta S_A=H_2(r)$ in the large-volume regime	$\Delta I=2H_2(r)$ for complementary regions	$\Delta\mathcal E=2\log(\sqrt r+\sqrt{1-r})$

For the last row, consider a one-particle state in a massive free theory on a circle of total length $V$, with both $V$ and the region length large at fixed fraction $r=\ell/V$. The particle is in $A$ with probability $r$ and in $B$ with probability $1-r$, so

$$\Delta S_A^{(1)} =H_2(r) =-r\log r-(1-r)\log(1-r).$$

This excitation-sector “qubit” result and its Rényi generalization are Castro-Alvaredo et al. 2018, §2.1, Eqs. (2.1)–(2.3). Because both the global one-particle state and the vacuum are pure, $\Delta I(A{:}B)=2\Delta S_A$. In the same large-volume scaling limit, the excitation sector has coefficients $\sqrt r$ and $\sqrt{1-r}$, so its contribution to the partial-transpose trace norm gives

$$\Delta\mathcal E^{(1)} =2\log(\sqrt r+\sqrt{1-r}).$$

At $r=1/2$, the three excess quantities are respectively $\log2$, $2\log2$, and $\log2$. Equal numerical values at this symmetric point do not make their definitions interchangeable.

For a thermal mixed state, mutual information measures total correlation. Logarithmic negativity can remain nonzero after $S_A$ becomes extensive, but there is no universal persistence theorem. For a finite interval at finite temperature, the naïve two-point twist-field continuation is wrong; the correct construction uses a four-point function and depends on the full operator content Calabrese, Cardy, and Tonni 2015, §§3–4. Adjacent-region negativity also retains ultraviolet contact terms.

Adversarial test: ensemble and volume

An energy eigenstate is globally pure, whereas a canonical Gibbs state is mixed. Even when local observables thermalize, their reduced states need only agree in a declared subsystem regime. Subsystem ETH explicitly distinguishes a subsystem much smaller than the total system from one comparable with its complement Dymarsky, Lashkari, and Liu 2018, §§II–III. In large-$c$ CFT, relative-entropy calculations likewise show that matching energy density does not make the reduced states exactly equal He, Lin, and Zhang 2017, §§2 and 4.

At fixed energy density, repeat the calculation for total sizes $V,2V,4V$ and for several fractions $r=\ell/V$. The leading thermodynamic expectations illustrate the failure injection:

Large-volume entropy densities used to expose an ensemble or finite-size artifact.

$r$	canonical Gibbs $S_A/V$	pure thermalizing eigenstate $S_A/V$
$0.10$	$0.10s_{\mathrm{th}}$	approximately $0.10s_{\mathrm{th}}$
$0.25$	$0.25s_{\mathrm{th}}$	approximately $0.25s_{\mathrm{th}}$
$0.50$	$0.50s_{\mathrm{th}}$	turnover region; finite-size corrections are largest
$0.75$	$0.75s_{\mathrm{th}}$	approximately $0.25s_{\mathrm{th}}$ by $S_A=S_{A^c}$

The eigenstate column is a leading chaotic-system expectation, not a theorem for every QFT. Its exact constraint $S_A=S_{A^c}$ follows from global purity. Therefore an apparent canonical volume law inferred only from $r<1/2$ cannot be extrapolated to all subsystem fractions. Acceptance requires convergence with $V$, agreement for genuinely local or small-fraction observables, and an explicit trace-distance or relative-entropy test. Integrable theories require conserved charges and usually a generalized Gibbs ensemble; scars and other nonthermal states fail still earlier.

Report the Hamiltonian, energy or temperature, ensemble and energy window, total volume, subsystem fraction, boundary conditions, algebra and center, regulator, reference state, and uncertainty. Keep short-interval, crossover, thermodynamic, and Page-turnover regimes separate.

Exercises

1. Derive both thermal limits

Starting from $\Delta S_A=(c/3)\log(\sinh x/x)$, derive the first two nonzero terms for $x\ll1$ and the leading large-$x$ entropy.

Solution

For small $x$,

$$\frac{\sinh x}{x}=1+\frac{x^2}{6}+\frac{x^4}{120}+O(x^6).$$

Using $\log(1+y)=y-y^2/2+\cdots$ gives

$$\log\frac{\sinh x}{x} =\frac{x^2}{6}-\frac{x^4}{180}+O(x^6),$$

and hence

$$\Delta S_A=\frac{c}{18}x^2-\frac{c}{540}x^4+O(x^6).$$

For large $x$, $\sinh x=(e^x/2)(1-e^{-2x})$. Substitution into the full entropy gives

$$S_A=\frac{\pi c}{3\beta}\ell +\frac{c}{3}\log\frac{\beta}{2\pi\epsilon} +c_1+O(e^{-2x}).$$

The linear coefficient is the thermal entropy density.

2. Check the entanglement first law

Let $\sigma=\operatorname{diag}(p,1-p)$ and $\rho=\operatorname{diag}(p+\delta,1-p-\delta)$, with $0<p<1$ and small $\delta$. Expand $\Delta S$, $\Delta\langle K_\sigma\rangle$, and $D(\rho\Vert\sigma)$ through second order.

Solution

For the binary entropy $H_2(p)$,

$$H_2'(p)=\log\frac{1-p}{p}, \qquad H_2''(p)=-\frac{1}{p(1-p)}.$$

Therefore

$$\Delta S =\delta\log\frac{1-p}{p} -\frac{\delta^2}{2p(1-p)}+O(\delta^3).$$

Since $K_\sigma=\operatorname{diag}(-\log p,-\log(1-p))$,

$$\Delta\langle K_\sigma\rangle =\delta\log\frac{1-p}{p}.$$

The linear terms agree, while

$$D(\rho\Vert\sigma) =\frac{\delta^2}{2p(1-p)}+O(\delta^3)\geq0.$$

This finite-dimensional calculation displays both the first law and the positive quadratic relative entropy.

3. Compare one-particle measures

For a delocalized one-particle state with probability $r$ in $A$, derive $\Delta S_A$, $\Delta I(A{:}B)$, and $\Delta\mathcal E(A{:}B)$. Evaluate them at $r=1/2$.

Solution

In the stated large-volume limit, the excitation sector has Schmidt coefficients $\sqrt r$ and $\sqrt{1-r}$. Its reduced probabilities are $r$ and $1-r$, so

$$\Delta S_A=H_2(r).$$

The global state is pure, hence $S_A=S_B$ and $S_{AB}=0$, giving

$$\Delta I(A{:}B)=2H_2(r).$$

For a pure state, the trace norm of the partial transpose is the square of the sum of Schmidt coefficients:

$$\left\lVert\rho^{T_B}\right\rVert_1 =(\sqrt r+\sqrt{1-r})^2.$$

Thus $\Delta\mathcal E=2\log(\sqrt r+\sqrt{1-r})$. At $r=1/2$, the results are $\log2$, $2\log2$, and $\log2$, respectively.

4. Reject an all-fraction thermal fit

A numerical study of a pure eigenstate finds $S_A\simeq s_{\mathrm{th}}\ell$ for $\ell/V=0.1$ and $0.25$ and claims the same formula at $\ell/V=0.75$. Give a model-independent objection and a finite-size test.

Solution

For every pure global state,

$$S_A=S_{A^c}.$$

At $\ell/V=0.75$, the complement has fraction $0.25$, so the proposed value $0.75s_{\mathrm{th}}V$ contradicts the value near $0.25s_{\mathrm{th}}V$ required by purity. The study has confused a small-subsystem thermal approximation with a global canonical state.

Repeat the calculation at fixed energy density for $V,2V,4V$ and fractions on both sides of $1/2$. Verify the exact reflection symmetry, locate the turnover, and compare reduced eigenstate and ensemble states using relative entropy or trace distance for fixed small fractions. A stable local limit may support subsystem ETH; it does not support the rejected all-fraction formula.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI and open article.
• Calabrese, Pasquale, and John Cardy. “Entanglement Entropy and Quantum Field Theory.” Journal of Statistical Mechanics: Theory and Experiment (2004): P06002. DOI. Open preprint.
• Calabrese, Pasquale, John Cardy, and Erik Tonni. “Finite Temperature Entanglement Negativity in Conformal Field Theory.” Journal of Physics A: Mathematical and Theoretical 48 (2015): 015006. DOI. Open preprint.
• Castro-Alvaredo, Olalla A., Cecilia De Fazio, Benjamin Doyon, and István M. Szécsényi. “Entanglement Content of Quantum Particle Excitations. Part I. Free Field Theory.” Journal of High Energy Physics 10 (2018): 039. DOI. Open preprint.
• Dymarsky, Anatoly, Nima Lashkari, and Hong Liu. “Subsystem Eigenstate Thermalization Hypothesis.” Physical Review E 97 (2018): 012140. DOI. Open preprint.
• He, Song, Feng-Li Lin, and Jia-ju Zhang. “Subsystem Eigenstate Thermalization Hypothesis for Entanglement Entropy in CFT.” Journal of High Energy Physics 08 (2017): 126. DOI. Open preprint.