Extensions of the Quantum Fano Inequality

The system Q undergoes a completely positive trace-preserving transformation or quantum operation E and R is assumed to be isolated and its state remains the same. This quantum operation is also represented by IR ⊗ E , where IR is the identity superoperator on R. We add subscript ‘1’ to denote the state of the system (joint or otherwise) after this quantum operation. So the state of the joint system is denoted by ρ11. Note that ρ1 = E(ρ) and ρ1 = ρ. The entanglement fidelity is defined by Schumacher [1] as F (ρ, E) = 〈ψ|ρ11|ψ〉 (3) and the entropy exchange as S(ρ, E) = S(ρ11) (4) where S(ρ11) is the von-Neumann entropy of ρ11. The QFI upper bounds S(ρ, E) by a function of the entanglement fidelity as [1] S(ρ, E) ≤ H(F (ρ, E)) + (1− F (ρ, E)) log(d − 1) (5)