Research proof index

The kernel condition σ.M.ker ≤ ρ.M.ker is preserved under partial trace. This follows because supp(ρ) ⊆ supp(σ) implies supp(Tr_B ρ) ⊆ supp(Tr_B σ): if v ∈ supp(Tr_B ρ), then ⟨v, (Tr_B ρ) v⟩ > 0, so for some basis vector e_b we have v ⊗ e_b ∈ supp(ρ) ⊆ supp(σ), hence ⟨v, (Tr_B σ) v⟩ ≥ ⟨v ⊗ e_b, σ (v ⊗ e_b)⟩ > 0.

A ket is a product state iff its components are cross-multiplicative.

If $X, Y, Z$ are independent $G$-valued random variables, then $D_{KL}(X+Z\Vert Y+Z) \leq D_{KL}(X\Vert Y).$

The operator norm of a matrix, with respect to the Euclidean norm (l2 norm) on the domain and codomain. -/ noncomputable def Matrix.opNorm [RCLike 𝕜] (A : Matrix m n 𝕜) : ℝ := ‖LinearMap.toContinuousLinearMap (Matrix.toEuclideanLin A)‖ /- An isometry preserves the Euclidean norm.

The operator norm of a matrix, with respect to the Euclidean norm (l2 norm) on the domain and codomain. -/ noncomputable def Matrix.opNorm [RCLike 𝕜] (A : Matrix m n 𝕜) : ℝ := ‖LinearMap.toContinuousLinearMap (Matrix.toEuclideanLin A)‖ /- An isometry preserves the Euclidean norm. -/ theorem Matrix.isometry_preserves_norm (A : Matrix n m 𝕜) (hA : A.Isom...

The operator norm of a matrix product is at most the product of the operator norms.