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...

For PSD A and p ≠ 0, A^{-p} * A^p = HermitianMat.supportProj A.

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...

For positive A ≤ 1 and p ≥ 1, Tr[A^p] ≤ Tr[A].

For $0<y\le1$ and $y/2\le y'\le1$, with both $y$ and $y'$ nonnegative, the negative-side Hilbert kernel satisfies the quantitative regularity estimate

$\lVert k(-y)-k(-y')\rVert \le 2^6\,\frac{1}{|y|}\,\frac{|y-y'|}{|y|}.$