Rényi Entropies and Spectral Information

The von Neumann entropy compresses an entire radiation spectrum into one number. Rényi entropies retain a one-parameter family of spectral moments, separating sensitivity to the rank, typical eigenvalues, and largest eigenvalue. In replica calculations they are also the quantities directly available at positive integer index before any continuation to the von Neumann limit.

Required background. Quantum Information supplies density-matrix spectra, Rényi entropy, trace moments, and continuity used below.

Helpful background. Semiclassical Gravity supplies the generalized-entropy setting; Quantum Fields in Curved Spacetime supplies regulated radiation states; Black-Hole Information supplies the Page-curve and purity questions being diagnosed.

Rényi family and its limits

For a normalized finite-dimensional radiation state $\rho_R$ with eigenvalues $\lambda_i$, define for $n\gt0$, $n\neq1$,

$$S_n(R)=\frac{1}{1-n}\ln\operatorname{Tr}\rho_R^n =\frac{1}{1-n}\ln\!\left(\sum_i\lambda_i^n\right).$$

Natural logarithms are used. Three limits have distinct meanings:

$$S_0=\ln\operatorname{rank}\rho_R, \qquad \lim_{n\to1}S_n=-\operatorname{Tr}(\rho_R\ln\rho_R), \qquad S_\infty=-\ln\lambda_{\max}.$$

$S_2=-\ln\operatorname{Tr}\rho_R^2$ is the collision entropy and is determined by purity. For a fixed state, $S_n$ is nonincreasing with $n$: large index emphasizes the largest eigenvalues, while small positive index emphasizes the spectral tail. Every $S_n$ equals $\ln d$ for $I_d/d$, and every $S_n$ vanishes for a pure state. These limiting checks fix the sign and normalization.

The definitions assume a trace-class density operator. Continuum local algebras require a regulator and subtraction or an algebraic replacement. A subtraction that renders $S_1$ finite does not automatically define every $S_n$ in a common scheme.

From moments to a spectrum

At integer $n\geq2$, the quantity

$$Z_n=\operatorname{Tr}\rho_R^n=\sum_i\lambda_i^n$$

is a power sum of the eigenvalues. In exact dimension $d$, the moments $Z_1,\ldots,Z_d$ determine the elementary symmetric polynomials through Newton identities and hence determine the characteristic polynomial of $\rho_R$. Thus the complete finite sequence contains the spectrum up to ordering.

This reconstruction is poorly conditioned when moments are noisy or $d$ is very large. Small eigenvalues contribute negligibly at large $n$, so quite different spectral tails can share the same accurately known low moments. An entropy curve at one index therefore cannot establish spectral agreement at all indices.

For a concrete independent check, compare

$$\rho_a=\operatorname{diag}\!\left(\tfrac12,\tfrac12,0\right)$$

with

$$\rho_b=\operatorname{diag}\!\left(x,\tfrac{1-x}{2},\tfrac{1-x}{2}\right), \qquad x\simeq0.7729.$$

Both have $S_1\simeq\ln2$, but their purities are respectively $1/2$ and approximately $0.6232$. Hence $S_2$ distinguishes states that the von Neumann entropy alone does not. Equality of one Page curve is weaker than equality of the radiation spectrum.

Replica access and continuation

For positive integer $n$, a replica construction represents $Z_n$ by $n$ copies cyclically joined along the radiation region. The von Neumann entropy is formally

$$S(R)=-\left.\frac{\partial}{\partial n}\ln Z_n\right|_{n=1},$$

because $Z_1=1$. Harlow reviews the finite-dimensional identities and replica bookkeeping Harlow 2016, Appendix C; the gravitational use and its saddle structure are developed in Almheiri et al. 2021, § 4 and Appendix A.

There are two different continuation questions. For an actual finite-dimensional density operator, sufficiently many exact moments determine its spectrum as above, so the physical function $\sum_i\lambda_i^n$ and its entropy are fixed. More generally, the complete exact sequence is constrained by positivity: $Z_{k+1}=\int_0^1x^k\,d\mu(x)$ for the probability measure $\mu=\sum_i\lambda_i\delta_{\lambda_i}$, and all moments on the compact interval determine that measure.

A finite set of moments, or only their leading saddle approximations, does not provide this complete information. Arbitrary analytic functions agreeing at the sampled integers need not agree near one, and an arbitrary interpolation need not represent a positive density operator. In gravitational path integrals there can also be multiple saddles, changes of dominance, replica-symmetry questions, and contour choices. A calculation must state whether it computes exact moments, an asymptotic saddle sequence, or a controlled continuation whose derivative at one gives the entropy.

Spectral Page diagnostics

In a unitary finite-dimensional model, every Rényi entropy of the complete radiation returns to zero at an endpoint with no remnant. Their turnover times and finite-size corrections need not coincide. A spectrum with one anomalously large eigenvalue affects $S_\infty$ well before it strongly changes $S_1$; a long tail affects $S_0$ while barely moving large-index entropies.

These comparisons can test a proposed Page curve. Trace normalization requires $Z_1=1$, positivity requires all reconstructed eigenvalues to be nonnegative, and purity bounds give $1/d\leq Z_2\leq1$. Violating any check signals a regulator, continuation, or numerical error rather than exotic information recovery.

The family still does not specify a microscopic decoding map or prove that the radiation algebra is complete. Generic spectral theory belongs to Quantum Information, fixed-background mode regularization to Quantum Fields in Curved Spacetime, and explicit replica-gravity mechanisms to later chapters. String or holographic constructions are not assumed here.

Exercise

A two-level radiation state has purity $P=\operatorname{Tr}\rho^2$. Reconstruct its two eigenvalues and state the allowed range of $P$.

Solution

Write the eigenvalues as $\lambda$ and $1-\lambda$. Then $P=\lambda^2+(1-\lambda)^2$, so

$$\lambda=\frac{1\pm\sqrt{2P-1}}{2}.$$

Positivity requires $1/2\leq P\leq1$. The sign only exchanges the eigenvalue labels. Thus $S_2=-\ln P$ determines the complete two-level spectrum, whereas in dimension three or larger a single purity leaves additional spectral freedom.

References

• Almheiri, Ahmed, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini. 2021. “The Entropy of Hawking Radiation.” Reviews of Modern Physics 93: 035002. DOI.
• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88: 015002. Open text.