# Average entropy of a subsystem

In this paper by Don Page, https://arxiv.org/pdf/gr-qc/9305007.pdf, He conjectures average entropy of a substem of dimension m with Hilbert space dimension mn, $$m \leq n$$. to be : $$S_{mn} = \sum_{n+1}^{mn} \dfrac{1}{k} - \dfrac{m-1}{2n}$$. Suppose I have a spin chain with 8 spins. Then the dimension of the Hilbert space would be $$2^8$$. If I divide this into equal subsystems, then the split up would be $$m= 2^4, n=2^4$$, Hence I would have to calculate $$S_{16,16} = \sum_{17}^{256} \dfrac{1}{k} - \dfrac{15}{32}$$. Unfortunately, this computation of the summation is giving different results than what was expected from numerical calculations. I think I am making some mistake with the dimension number or some silly mistake with the series. Can someone please point out what mistake I am making?

• Equation (7) in that paper says $\langle \text{trace}(\rho_A^2)\rangle=(m+n)/(mn+1)$, where $\rho_A$ is the reduced density matrix for one subsystem. Have you tried computing $\langle \text{trace}(\rho_A^2)\rangle$ numerically to see if it agrees with equation (7)? – Chiral Anomaly Nov 12 '18 at 11:43

Let me illustrate/confirm Page's prediction for an $$8$$-site spin chain, where we bipartition it into a region A and B, each containing $$4$$ sites. There is no universal result for the entanglement entropy $$S_A(|\psi\rangle)$$ of a single random state $$|\psi\rangle$$, but Page predicts that if we average $$S_A$$ over random states (he notes he uses the Haar measure in his paper), then $$\langle S_A \rangle$$ should obey the formula you quoted. So I went ahead and did this for the $$8$$-site chain, obtaining:
More precisely, I average over only 2000 states (whose entropies are shown in the blue-ish histogram), yet there is already a nice approximate agreement between Page's prediction and the center-of-mass of this histogram: $$S_\textrm{Page} \approx 2.275 \qquad \textrm{and} \qquad \langle S_A \rangle \approx 2.276$$