Entropy functions and determinant inequalities

In this paper, we show that the characterisation of all determinant inequalities for n × n positive definite matrices is equivalent to determining the smallest closed and convex cone containing all entropy functions induced by n scalar jointly Gaussian random variables. We have obtained inner and outer bounds on the cone by using representable functions and entropic functions. In particular, these bounds are tight and explicit for n ≤ 3, implying that determinant inequalities for 3 × 3 positive definite matrices are completely characterized by Shannon-type information inequalities.