Wigner-entropy lower bound (Wigner entropy conjecture)

Prove that for any subsystem A consisting of M canonical modes (here, the + relative-mode phase space of a subset of wells in the spin-1 Bose–Einstein condensate), the differential entropy S(W_+^A) of its Wigner function satisfies the entropic uncertainty lower bound S(W_+^A) ≥ M (1 + ln π), with equality if and only if W_+^A is a product of pure Gaussian states.

Background

In the paper’s discussion of classical entropies for phase-space distributions, the authors define the Wigner entropy S(W_+^A) for a subsystem A and note it is well-defined when the Wigner distribution is non-negative. They then cite a conjectured lower bound, often referred to as the Wigner entropy conjecture, which encodes an entropic uncertainty principle.

The conjectured inequality takes the form S(W_+^A) ≥ M (1 + ln π), where M is the number of modes in A, and is stated to be tight exactly for products of pure Gaussian states. Establishing (or refuting) this bound would clarify the fundamental entropic uncertainty structure associated with Wigner functions and link classical phase-space entropies to quantum uncertainty principles.