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

For a positive contraction $Q$ on $\ell^2(E)$ with $E$ finite, let $P_Q$ be the determinantal probability measure on $2^E$ and $\operatorname{Ent}(Q)$ its Shannon entropy. Lyons conjectured in 2003 that $\operatorname{Ent}$ is concave in $Q$. We prove the conjecture when $|E|=2$. For general $E$ we write the second derivative of $\operatorname{Ent}$ along a line as a negative quadratic form in the derivatives of the principal minors of $Q$ plus a sum, over subsets $T$ with $|T|\ge2$, of the second derivatives of the minors $\det Q_T$ weighted by Möbius sums of $\log P_Q$. 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 $Q=Q_1\oplus Q_2$ the second variation vanishes in every off-block direction, and the entropy decreases there at fourth order, by the $\chi^2$ 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 $Q=\frac12I$; 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 $|E|\le5$, for real and complex kernels: the largest eigenvalue of the Hessian of $\operatorname{Ent}$ is negative at random kernels, and its supremum, zero, is approached at reducible kernels.