Research proof index

If $X, Y$ are $G$-valued random variables, and $Z$ is another random variable defined on the same sample space as $X$, then $D_{KL}((X|Z)\Vert Y) = D_{KL}(X\Vert Y) + \bbH[X] - \bbH[X|Z].$

Conditional multidistance is unchanged when both the random variables and their conditioning variables are reindexed along an equality $m'=m$. As with ordinary multidistance, the value does not depend on the chosen equal presentation of the finite index type.

A lower bound forced by $\tau$-minimality. If $(X_1,X_2)$ minimizes the source's $\tau$ functional, then for measurable $X_1',X_2'$ and conditioning variables $Z,W$, $d[X_1'\mid Z;X_2'\mid W]\ge d[X_1;X_2]-\eta\bigl(d[X_1^0;X_1'\mid Z]-d[X_1^0;X_1]\bigr)-\eta\bigl(d[X_2^0;X_2'\mid W]-d[X_2^0;X_2]\bigr).$

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...

A preliminary endgame bound. In the $\tau$-minimizer setup, let $k=d[X_1;X_2]$ and let measurable $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$. Put $\delta=I[T_1:T_2]+I[T_2:T_3]+I[T_3:T_1]$ and $c[T_1\#T_2]=d[X_1^0;T_1]-d[X_1^0;X_1]+d[X_2^0;T_2]-d[X_2^0;X_2].$ Then $k\le\delta+\eta c[T_1\#T_2]+\frac{\eta}{2}\bigl(I[T_1:T_3]+I[T_2:T_3]\bigr).$

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