Research proof index

Polynomial Freiman-Ruzsa theorem for bounded-torsion groups. Let $G$ be a finite abelian group in which $mx=0$ for every $x$, with $m\ge2$. If $A\subseteq G$ is nonempty and $|A+A|\le K|A|$, then there are a subgroup $H\le G$ and a set $c$ such that $A\subseteq c+H$, $|H|\le|A|$, and $|c|<mK^{256m^3+1}$.

Step 1 (Variational formula): For α > 1, the trace functional equals the supremum of f_α over all PSD H: Q̃_α(ρ‖σ) = ⨆ (H : HermitianMat d ℂ) (_ : 0 ≤ H), f_alpha α H ρ σ. The optimizer is H_hat = σ^γ (σ^γ ρ σ^γ)^{α−1} σ^γ.

Entanglement of Formation of bipartite systems. It is the convex roof extension of the von Neumann entropy of one of the subsystems (here chosen to be the left one, but see Entropy.Sᵥₙ_of_partial_eq). The function Sᵥₙ ∘ traceRight ∘ pure is phase-invariant because pure maps phase-equivalent kets to the same mixed state, so it descends to `KetUpToPha...

The trace functional is invariant under 1_A ⊗ V conjugation. -/ theorem sandwichedTraceFunctional_conj_tensorUnitary (ρ σ : MState (dA × dB)) (V : Matrix.unitaryGroup dB ℂ) : Q̃_ α(ρ.conjTensorUnitary V‖σ.conjTensorUnitary V) = Q̃_ α(ρ‖σ) := by exact sandwichedTraceFunctional_conj_unitary_MState _ ρ σ section twirling variable {dA dB : Type*} variable [...

The trace functional is invariant under 1_A ⊗ V conjugation. -/ theorem sandwichedTraceFunctional_conj_tensorUnitary (ρ σ : MState (dA × dB)) (V : Matrix.unitaryGroup dB ℂ) : Q̃_ α(ρ.conjTensorUnitary V‖σ.conjTensorUnitary V) = Q̃_ α(ρ‖σ) := by exact sandwichedTraceFunctional_conj_unitary_MState _ ρ σ section twirling variable {dA dB : Type*} variable [...

In a FreeStateTheory, we have free states of full rank, therefore the minimum relative entropy of any state ρ to a free state is finite.