Concavity of the entropy of determinantal measures on two points, and the second variation in general

For a positive contraction Q on ²(E) with E finite, let PQ be the determinantal probability measure on 2E and Ent(Q) its Shannon entropy. Lyons conjectured in 2003 that Ent is concave in Q. We prove the conjecture when $|E|=2$. For general E we write the second derivative of Ent along a line as a negative quadratic form in the derivatives of the principal minors of Q plus a sum, over subsets T with |T|≥2, of the second derivatives of the minors QT weighted by Möbius sums of log PQ. For two-element T these weights are nonnegative by the Hadamard–Fischer inequality, which is what makes the case $|E|=2$ work. At a reducible kernel Q=Q₁⊕ Q₂ the second variation vanishes in every off-block direction, and the entropy decreases there at fourth order, by the χ² term of the mutual information between the two blocks. For blocks of sizes k and $n-k$ and \|D\|F=1 the fourth-order coefficient is at least $2/(k(n-k))$, with equality at Q=12I; the proof is a Cramér–Rao bound for a centred pair statistic combined with the nonnegativity of the inverse of the indicator covariance. We also report numerical evidence for |E|≤5, for real and complex kernels: the largest eigenvalue of the Hessian of Ent is negative at random kernels, and its supremum, zero, is approached at reducible kernels.