Research proof index

The trace functional is invariant under joint unitary conjugation of MStates. -/ theorem sandwichedTraceFunctional_conj_unitary_MState (U : Matrix.unitaryGroup d ℂ) (ρ σ : MState d) : Q̃_ α(ρ.U_conj U‖σ.U_conj U) = Q̃_ α(ρ‖σ) := by unfold sandwichedTraceFunctional MState.U_conj exact sandwichedTraceFunctional_conj_unitary_hermitian U ρ.M σ.M /-! ## Joint...

Convexity of trace functions: if g is convex on ℝ₊, then A ↦ Tr[g(A)] is convex on PSD matrices. This is Theorem 2.10 of Carlen.

The trace functional is invariant under joint unitary conjugation of MStates. -/ theorem sandwichedTraceFunctional_conj_unitary_MState (U : Matrix.unitaryGroup d ℂ) (ρ σ : MState d) : Q̃_ α(ρ.U_conj U‖σ.U_conj U) = Q̃_ α(ρ‖σ) := by unfold sandwichedTraceFunctional MState.U_conj exact sandwichedTraceFunctional_conj_unitary_hermitian U ρ.M σ.M /-! ## Joint...

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.

Sub-lemma for Step 1b: the conj of H_hat by σ^{−γ} simplifies to (ρ.M.conj (σ^γ).mat)^{α−1}. This uses σ^{−γ} · σ^γ = identity (on support) to cancel the outer σ^γ factors. -/ theorem H_hat_conj_sigma (hα : 1 < α) (ρ σ : MState d) : let γ := (1 - α) / (2 * α) (H_hat α ρ σ).conj (σ.M ^ (-γ)).mat = (ρ.M.conj (σ.M ^ γ).mat) ^ (α - 1) := by intro γ have hγ :...

Step 3 (Convexity in σ): For fixed H ≥ 0 and ρ, and α > 1, the map σ ↦ f_alpha α H ρ σ is convex. The key is that for p = α/(α−1) > 1: • A ↦ Tr[A^p] is convex on PSD matrices (trace function convexity, Theorem 2.10 of Carlen), • σ ↦ σ^{−γ} H σ^{−γ} is concave in σ by Lieb concavity (since −γ = (α−1)/(2α) ∈ (0,½)), • The composition...