Replica Construction of Rényi Entropy

The $n$th moment of a reduced density matrix is computed by cyclically gluing $n$ copies along the retained region while gluing each copy to itself along the traced region. This construction is exact at positive integer $n$. The von Neumann entropy follows only after normalization and a justified analytic continuation to $n=1$.

Required background. Quantum Information supplies reduced density operators, traces, Rényi entropy, and spectral decompositions.

Helpful background. Semiclassical Gravity supplies saddle-point gravitational path integrals; Quantum Fields in Curved Spacetime supplies Euclidean state preparation and regulated regional fields; Black-Hole Information supplies the radiation subsystem and Page problem.

Cyclic gluing of the retained region

Let a normalized state on $\mathcal H_R\otimes\mathcal H_C$ have density operator $\rho_{RC}$. Tracing the complement $C$ gives

$$\rho_R(r_+,r_-) = \int dc\, \rho_{RC}(r_+,c;r_-,c).$$

The equal $c$ data implement gluing of the upper and lower banks of the cut within each copy. Multiplying $n$ kernels and taking the trace gives

$$\operatorname{Tr}\rho_R^n = \int \prod_{k=1}^{n}dr_k\, \rho_R(r_1,r_2)\rho_R(r_2,r_3)\cdots \rho_R(r_n,r_1).$$

Thus the lower bank of the cut along $R$ in replica $k$ is glued to the upper bank in replica $k+1$, with $n+1\equiv1$. Along $C$, upper and lower banks are glued within the same replica. That is the boundary gluing asked for by the replica construction. The corresponding branch geometry for QFT entanglement is reviewed in Calabrese and Cardy 2004, §§2–3.

Equivalently, if $\tau_n$ cyclically permutes the $R$ factors of $n$ copies, then

$$\operatorname{Tr}\rho_R^n = \operatorname{Tr}\bigl(\rho_{RC}^{\otimes n}\tau_n\bigr).$$

This operator form makes clear that the moment is a nonlinear, multi-copy observable even though $\tau_n$ is linear on the replicated Hilbert space.

The orientation of the cut encodes operator order. Reversing the cyclic gluing replaces $\tau_n$ by $\tau_n^{-1}$; for a Hermitian positive density operator the trace is unchanged, but insertions on the sheets are reordered. Gluing every bank within its own copy would instead produce $(\operatorname{Tr}\rho_R)^n=1$ and erase the moment entirely. Gluing the complement cyclically computes the complementary reduced-state moment, which agrees with the $R$ moment only when the regulated global state is pure. These alternatives provide a direct topology-to-algebra check before gravity is introduced.

Normalized moments and entropy

A path integral often prepares an unnormalized operator $\widetilde\rho_R$ with one-copy partition function $Z_1=\operatorname{Tr}\widetilde\rho_R$. If $Z_n[R]$ denotes the cyclically glued path integral, the density-matrix moment is

$$Z_n^{\mathrm{norm}}[R] = \frac{Z_n[R]}{Z_1^n} =\operatorname{Tr}\rho_R^n.$$

The Rényi entropy and von Neumann limit are

$$S_n(R) =\frac{1}{1-n}\log Z_n^{\mathrm{norm}}[R], \qquad S(R) =-\left.\partial_n\log Z_n^{\mathrm{norm}}[R]\right|_{n=1}.$$

The normalization check is $Z_1^{\mathrm{norm}}=1$. Differentiating the spectral identity $Z_n^{\mathrm{norm}}=\sum_i p_i^n$ gives $-\sum_i p_i\log p_i$, so the sign and logarithm convention agree with von Neumann entropy. Equivalently, if $I_n:=-\log Z_n$, the contribution of a consistently normalized saddle family is

$$S(R)=\left.\partial_n I_n\right|_{n=1}-I_1.$$

The $-I_1$ term removes the normalization of state preparation. It cannot be discarded merely because the replicated geometry is connected.

In a continuum field theory, the cut introduces ultraviolet divergences localized near its endpoint or entangling surface. A regulator and subtraction must be held fixed as $n$ changes. The result here is the fine-grained Rényi entropy of the declared algebra, not a thermodynamic coarse graining.

Adding a gravitational region

When part of the path integral is gravitational, the boundary cyclic gluing is held fixed but the bulk metric is also integrated over:

$$Z_n[R] \simeq \sum_{g_n\,\mathrm{saddles}} e^{-I_{\mathrm{grav}}[g_n]} Z_{\mathrm{matter}}[g_n;R].$$

The notation is a semiclassical saddle expansion, not a nonperturbative definition. Different replicas need not remain disconnected in the gravitating region. Under replica symmetry, quotienting a smooth $n$-fold saddle produces a conical fixed locus; its continuation leads toward a cosmic-brane or QES condition. The integer construction and quotient logic are derived in Lewkowycz and Maldacena 2013, §§2–4.

Boundary gluing alone neither selects a bulk topology nor proves that a formal saddle lies on the integration contour. Those are gravitational questions addressed on the next pages.

For integer $n$, the cyclic group acts by relabeling sheets and the boundary conditions are unambiguous. The quotient step is an additional assumption about a dominant replica-symmetric bulk saddle. If several saddles contribute at comparable action, the logarithm must be taken after summing their amplitudes; differentiating one preferred action first can miss a dominance exchange. This ordering becomes especially important near a Page transition.

Checks and analytic-continuation limits

Pure-state check. For a regulated pure state on $R\otimes C$, the nonzero eigenvalues of $\rho_R$ and $\rho_C$ agree. Therefore $\operatorname{Tr}\rho_R^n=\operatorname{Tr}\rho_C^n$ for every integer $n$, and their Rényi entropies match.

Maximally mixed check. If $\rho_R=I_d/d$, then $Z_n^{\mathrm{norm}}=d^{1-n}$ and $S_n=\log d$ for every $n$. An expression with $d^n$ instead has missed normalization.

Continuation limitation. A bare list of integer values does not select an arbitrary analytic function. Exact moments of a positive density matrix also obey spectral constraints, whereas a truncated gravitational saddle calculation may not retain the information needed to impose them. Replica symmetry can break, saddle dominance can change with $n$, and a saddle may not lie on the chosen contour. The entropy review describes the gravitational replica construction in Almheiri et al. 2021, §10. The continuation page explains the distinction between spectral uniqueness and ambiguity of an approximate saddle continuation.

Exercises

For $\rho_R=\operatorname{diag}(1/2,1/3,1/6)$, compute $Z_2^{\mathrm{norm}}$ and $S_2$. Verify the $n=1$ normalization and explain what data would still be missing for a gravitational calculation.

Solution

$Z_2^{\mathrm{norm}}=1/4+1/9+1/36=7/18$, so $S_2=-\log(7/18)=\log(18/7)$. At $n=1$, the probabilities sum to one. A gravitational calculation would additionally need the state-preparing boundary conditions, the allowed bulk topologies, the action and counterterms, an integration contour, and a prescription for continuing the saddle sum away from positive integers.

References

• Almheiri et al. 2021, Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini, “The Entropy of Hawking Radiation,” Reviews of Modern Physics 93 (2021) 035002.
• Calabrese and Cardy 2004, Pasquale Calabrese and John Cardy, “Entanglement Entropy and Quantum Field Theory,” Journal of Statistical Mechanics: Theory and Experiment 2004 (2004) P06002.
• Lewkowycz and Maldacena 2013, Aitor Lewkowycz and Juan Maldacena, “Generalized Gravitational Entropy,” Journal of High Energy Physics 08 (2013) 090.