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 ∈ Ω_N and α ∈ [0, 1].

• Supermultiplicativity: for ρ₁, ρ₂, ρ₃, ρ₄ ∈ Ω_N we have

G(ρ₁ ⊗ ρ₂, ρ₃ ⊗ ρ₄) ≥ G(ρ₁, ρ₃)G(ρ₂, ρ₄).

See also

• Fidelity
• Trace distance
• Trace norm

References

Category:Handbook of Quantum Information Category:Mathematical Structure Category:Linear Algebra