Characterization of matrix classes satisfying the Bapat–Sunder inequalities

Characterize the restricted classes of positive semidefinite matrices that satisfy either the Bapat–Sunder Hadamard-product permanent inequality or the Bapat–Sunder eigenvalue inequality, despite the failure of both conjectures in general, with potential applications to quantum interferometry.

Background

The paper studies two conjectures of Bapat and Sunder concerning permanents of positive semidefinite Hermitian matrices. The first asserts an upper bound for the permanent of a Hadamard product, while the second bounds the largest eigenvalue of the matrix formed from the first-order permanent derivatives of a positive semidefinite matrix.

Although Drury had produced counterexamples to both conjectures, the paper emphasizes that determining which restricted matrix classes continue to satisfy one or both inequalities remains unresolved. The paper records several such classes, including rank-one matrices and positive semidefinite matrices with nonnegative entries, and establishes a logical implication from the first inequality to the second for any class satisfying the first inequality universally. The characterization problem is motivated in part by applications to quantum interferometry.