Don N. Page investigates the average entropy of a subsystem in a quantum system. He considers a quantum system with Hilbert space dimension mn in a random pure state and calculates the average entropy of a subsystem of dimension m ≤ n. He conjectures that the average entropy $ S_{m,n} $ is approximately $ \ln m - \frac{m}{2n} $ for $ 1 \ll m \leq n $. This result implies that, on average, less than one-half unit of information is in the smaller subsystem of a total system in a random pure state. Page explains that the entropy of a subsystem can be calculated by taking the partial trace of the total system's density matrix. For a pure state, the sum of the entropies of the subsystems is greater than the entropy of the total system. He calculates the average entropy $ S_{m,n} $ by considering the probability distribution of the eigenvalues of the subsystem's density matrix for random pure states. He finds that for $ m \ll n $, the typical entropy of the smaller subsystem is very nearly maximal. Page conjectures that the exact general formula for $ S_{m,n} $ is $ \sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n} $, which agrees with his calculations for m = 4 and m = 5. He also derives an asymptotic expansion for large m and n, which fits both his previous results and the conjecture. He shows that for large m and n, the average entropy $ S_{m,n} $ is approximately $ \ln m - \frac{m}{2n} $, which is consistent with the conjecture. Page concludes that for a typical pure quantum state of a large system, the smaller subsystem is very nearly maximally mixed, showing little signs that the total system is pure. The information in the total system is mainly in the correlations between the subsystems, with only a small amount in the correlations within the smaller subsystem. The average deviation or information in the smaller subsystem is $ \frac{m}{2n} $, which is always less than one half of a natural logarithmic unit.Don N. Page investigates the average entropy of a subsystem in a quantum system. He considers a quantum system with Hilbert space dimension mn in a random pure state and calculates the average entropy of a subsystem of dimension m ≤ n. He conjectures that the average entropy $ S_{m,n} $ is approximately $ \ln m - \frac{m}{2n} $ for $ 1 \ll m \leq n $. This result implies that, on average, less than one-half unit of information is in the smaller subsystem of a total system in a random pure state. Page explains that the entropy of a subsystem can be calculated by taking the partial trace of the total system's density matrix. For a pure state, the sum of the entropies of the subsystems is greater than the entropy of the total system. He calculates the average entropy $ S_{m,n} $ by considering the probability distribution of the eigenvalues of the subsystem's density matrix for random pure states. He finds that for $ m \ll n $, the typical entropy of the smaller subsystem is very nearly maximal. Page conjectures that the exact general formula for $ S_{m,n} $ is $ \sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n} $, which agrees with his calculations for m = 4 and m = 5. He also derives an asymptotic expansion for large m and n, which fits both his previous results and the conjecture. He shows that for large m and n, the average entropy $ S_{m,n} $ is approximately $ \ln m - \frac{m}{2n} $, which is consistent with the conjecture. Page concludes that for a typical pure quantum state of a large system, the smaller subsystem is very nearly maximally mixed, showing little signs that the total system is pure. The information in the total system is mainly in the correlations between the subsystems, with only a small amount in the correlations within the smaller subsystem. The average deviation or information in the smaller subsystem is $ \frac{m}{2n} $, which is always less than one half of a natural logarithmic unit.