Optimal mode determining Sl± monotones in multi‑mode Gaussian states

Identify and characterize, for general multi‑mode Gaussian quantum states specified by second moments (μ, χ), the single mode (i.e., the optimizing direction/vector) that achieves the supremum in the definitions of the asymmetry monotones Sl±(μ, χ) = σ1[(μ* ± I/2)^(−1/2) χ (μ ± I/2)^(−1/2)]. Develop a method or closed‑form characterization to compute this optimal mode directly from (μ, χ) and elucidate its physical interpretation.

Background

The paper introduces two monotones, Sl±, that quantify type‑2 (second‑moment) asymmetry under Gaussian covariant operations. These monotones are finite, faithful, monotonic under GCOs, completely non‑extensive, and remain monotonic even under correlated catalysis. In single‑mode scenarios, Sl± are complete for state transformations.

While Definition 1 shows Sl± can be expressed as an optimization over a single effective mode (direction) in phase space, the identity and characterization of this optimal mode for general multi‑mode Gaussian states is not known. Clarifying this would likely reveal deeper structural properties of Gaussian states and streamline practical computation of Sl±.