Monotonicity of Fisher Discord under Partial Trace

Establish whether Fisher discord C(ρ12, H ⊗ 1) = I_F(ρ12, H ⊗ 1) − I_W(ρ12, H ⊗ 1), defined as the difference between the symmetric-logarithmic-derivative quantum Fisher information I_F and the Wigner–Yanase skew information I_W for a bipartite quantum state ρ12 and an observable H on subsystem 1, is nonincreasing under the partial trace over subsystem 2; equivalently, determine whether C(ρ12, H ⊗ 1) ≥ C(ρ1, H) holds for all bipartite states ρ12 with ρ1 = tr_2(ρ12) and all observables H on subsystem 1.

Background

The paper defines Fisher discord C(ρ, H) as the difference between two prominent versions of quantum Fisher information: the symmetric-logarithmic-derivative quantum Fisher information I_F(ρ, H) and the Wigner–Yanase skew information I_W(ρ, H). Both I_F and I_W are known to be nonincreasing under partial trace, i.e., I_F(ρ12, H ⊗ 1) ≥ I_F(ρ1, H) and I_W(ρ12, H ⊗ 1) ≥ I_W(ρ1, H) for bipartite states ρ12 and observables H acting on subsystem 1.

Whether the difference C(ρ, H) inherits this monotonicity property under partial trace is explicitly posed as an open question. An affirmative result would enable using the difference C(ρ12, H ⊗ 1) − C(ρ1, H) to probe correlations in bipartite quantum systems, thereby linking the proposed complexity quantifier to correlation measures.