Research proof index

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.

V_rho^H * V_rho simplifies to sandwiching the traceRight by the inverse square root.

V_rho is an isometry.