Research proof index

The operator norm of the conjugate transpose is equal to the operator norm. -/ theorem Matrix.opNorm_conjTranspose_eq_opNorm {m n : Type*} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] (A : Matrix m n 𝕜) : Matrix.opNorm Aᴴ = Matrix.opNorm A := by unfold Matrix.opNorm rw [← ContinuousLinearMap.adjoint.norm_map (toEuclideanLin A).toContinuousLine...

The convex roof extension of g : KetUpToPhase d → ℝ≥0 applied to a pure state ψ is g (KetUpToPhase.mk ψ).

The identity channel on D dimensional space achieves a rate of log2(D).

A channel cannot achieve a rate greater than log2(D), where D is the output dimension.

Doubly stochastic Hölder inequality: for nonneg a, b, doubly stochastic w, and conjugate p, q > 1: ∑{ij} a_i * b_j * w{ij} ≤ (∑ a_i^p)^{1/p} * (∑ b_j^q)^{1/q}.

The average of f : MState d → T on an ensemble that mixes to a pure state ψ is f (pure ψ)