Physlib

QuantumInfo.Entropy.Relative

To do relative entropies, we start with the _sandwiched Renyi Relative Entropy_ which is a nice general form. Then instead of proving many theorems (like DPI, relabelling, additivity, etc.) several times, we just prove it for this one quantity, then it follows for other quantities (like the relative entropy) as a special case.

2 declarations

theorem

D(ρ∥σ)≠∞  ⟺  ker⁡σ⊆ker⁡ρD(\rho \parallel \sigma) \neq \infty \iff \ker \sigma \subseteq \ker \rho

Let ρ\rho and σ\sigma be quantum states (mixed states) of dimension dd. The quantum relative entropy D(ρ∥σ)D(\rho \parallel \sigma) is finite (i.e., not equal to ∞\infty) if and only if the kernel of the density matrix σ\sigma is a subspace of the kernel of the density matrix ρ\rho, denoted as ker⁡σ⊆ker⁡ρ\ker \sigma \subseteq \ker \rho.

theorem

D(ρ∥σ)=∞  ⟺  ker⁡σ⊈ker⁡ρD(\rho \parallel \sigma) = \infty \iff \ker \sigma \not\subseteq \ker \rho

For any two quantum states ρ\rho and σ\sigma of dimension dd, the quantum relative entropy D(ρ∥σ)D(\rho \parallel \sigma) is equal to ∞\infty if and only if the kernel of the matrix σ\sigma is not contained within the kernel of the matrix ρ\rho (i.e., ker⁡σ⊈ker⁡ρ\ker \sigma \not\subseteq \ker \rho). This condition is equivalent to the failure of the support condition supp(ρ)⊆supp(σ)\text{supp}(\rho) \subseteq \text{supp}(\sigma).