Research proof index

Two maps are equal if they agree on all positive inputs with trace one

Shannon entropy of a distribution is at most ln d.

A symmetry identity in the $\tau$-minimizer endgame. Let $X_1',X_2'$ be independent copies of $X_1,X_2$, and set $U=X_1+X_2$, $V=X_1'+X_2$, $W=X_1'+X_1$, and $S=X_1+X_2+X_1'+X_2'$. Then the conditional mutual informations agree: $I[V:W\mid S]=I[U:W\mid S]$.

The partition function Z for an ideal gas.

For Antitone functions, the InfRegularized is the supremum of values.

Under the support condition σ.M.ker ≤ ρ.M.ker (i.e., supp(ρ) ⊆ supp(σ)), conjugation by σ^γ and σ^{-γ} preserves the inner product: ⟪ρ.M, H⟫ = ⟪σ^γ ρ σ^γ, σ^{-γ} H σ^{-γ}⟫. This holds because the kernel condition ensures ρ is supported on supp(σ), where σ^γ σ^{-γ} acts as the identity.