# Superfidelity

Superfidelity is a measure of similarity between density operators. It is defined as

$G(\rho,\sigma) = \mathrm{tr}\rho\sigma + \sqrt{1-\mathrm{tr}(\rho^2)}\sqrt{1-\mathrm{tr}(\sigma^2)},$

where σ and ρ are density matrices.

Superfidelity was introduced inmiszczak09sub as an upper bound for fidelity.

### Properties

Super-fidelity has also properties which make it useful for quantifying distance between quantum states. In particular we have:

• Bounds: 0 ≤ G(ρ1, ρ2) ≤ 1.
• Symmetry:  G(ρ1, ρ2) = G(ρ2, ρ1).
• Unitary invariance: for any unitary operator U, we have  G(ρ1, ρ2) = G(Uρ1U † , Uρ2U † ).
• Concavity:
G(ρ1, αρ2 + (1 − α)ρ3) ≥ αG(ρ1, ρ2) + (1 − α)G(ρ1, ρ3)

for any ρ1, ρ2, ρ3 ∈ ΩN and α ∈ [0, 1].

• Supermultiplicativity: for ρ1, ρ2, ρ3, ρ4 ∈ ΩN we have
G(ρ1 ⊗ ρ2, ρ3 ⊗ ρ4) ≥ G(ρ1, ρ3)G(ρ2, ρ4).