**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*(*ρ*_{1},*ρ*_{2}) ≤ 1. - Symmetry:
*G*(*ρ*_{1},*ρ*_{2}) =*G*(*ρ*_{2},*ρ*_{1}). - Unitary invariance: for any unitary operator
*U*, we have*G*(*ρ*_{1},*ρ*_{2}) =*G*(*U**ρ*_{1}*U*^{ † },*U**ρ*_{2}*U*^{ † }). - Concavity:

*G*(*ρ*_{1},*α**ρ*_{2}+ (1 −*α*)*ρ*_{3}) ≥*α**G*(*ρ*_{1},*ρ*_{2}) + (1 −*α*)*G*(*ρ*_{1},*ρ*_{3})

for any *ρ*_{1}, *ρ*_{2}, *ρ*_{3} ∈ Ω_{N} and *α* ∈ [0, 1].

- Supermultiplicativity: for
*ρ*_{1},*ρ*_{2},*ρ*_{3},*ρ*_{4}∈ Ω_{N}we have

*G*(*ρ*_{1}⊗*ρ*_{2},*ρ*_{3}⊗*ρ*_{4}) ≥*G*(*ρ*_{1},*ρ*_{3})*G*(*ρ*_{2},*ρ*_{4}).

### See also

### References

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

## Last modified:

Monday, October 26, 2015 - 17:56