Research proof index

"Ordinary" subadditivity of von Neumann entropy

Weak monotonicity, version with partial traces.

A pair of measures minimizing $\tau$ exists.

For p.η ≤ 1/8, there exist τ-minimizers X₁, X₂ at zero Rusza distance. For p.η < 1/8, all minimizers are fine, by tau_strictly_decreases'. For p.η = 1/8, we use a limit of minimizers for η < 1/8, which exists by compactness.

If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that $\tau[X'_1;X'_2] < \tau[X_1;X_2]$. Phrased in the contrapositive form for convenience of proof.

If G is torsion-free and X, Y are G-valued random variables then d[X; 2Y] ≤ 5d[X; Y].