Research proof index

Hecke operators are preserved under the identification 𝒮²(U, χ; M ⊗ N) ≃ M ⊗ 𝒮²(U, χ; N).

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...

If a Turing machine halts, the time step is uniquely determined.

This lemma translates between the relational notion and the iterated step notion. The latter can be more convenient especially for deterministic machines as we have here.

Total-variation distance is convex over a shared bind: if tvDist (f a) (g a) ≤ c for every a ∈ support mx, then tvDist (mx >>= f) (mx >>= g) ≤ c. The real-valued root of the const bound, with ℝ≥0∞ companion ofReal_tvDist_bind_left_le_const.