== Quantum Fidelity ==

Fidelity is a popular measure of distance between density operators. It is not a metric, but has some useful properties and it can be used to defined a metric on this space of density matrices, known as Bures metric.

Fidelity as a distance measure between pure states used to be called "transition probability". For two states given by unit vectors ϕ, ψ it is |⟨ϕ, ψ⟩|². For a pure state (vector ψ) and a mixed state (density matrix ρ) this generalizes to ⟨ψ, ρψ⟩, and for two density matrices ρ, σ it is generalized as the largest fidelity between any two purifications of the given states. According to a theorem by Uhlmann, this leads to the expression

$$F(\rho,\sigma)=\left(\textrm{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^2$$ This is precisely the expression used by Richard Jozsa inJozsa94, where the term fidelity appears to have been used first.

However, one can also start from |⟨ϕ, ψ⟩|, leading to the alternative

$$F'(\rho,\sigma)=\textrm{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$$

used inNielsenChuang. This second quantity is sometimes denoted as $\sqrt{F}$ and called square root fidelity. It has no interpretation as a probability, but appears in some estimates in a simpler way.

Basic properties

If ρ = |ψ⟩⟨ψ| is pure, then F(ρ, σ) = ⟨ψ|σ|ψ⟩ and if both states are pure i.e. ρ = |ψ⟩⟨ψ| and σ = |ϕ⟩⟨ϕ|, then F(ρ, σ) = |⟨ψ|ϕ⟩|².

Other properties:

1. 0 ≤ F(ρ, σ) ≤ 1
2. F(ρ, σ) = F(σ, ρ) 
3. F(ρ₁ ⊗ ρ₂, σ₁ ⊗ σ₂) = F(ρ₁, σ₁)F(ρ₂, σ₂)
4. F(UρU^†, UσU^†) = F(ρ, σ)
5. F(ρ, ασ₁ + (1 − α)σ₂) ≥ αF(ρ, σ₁) + (1 − α)F(ρ, σ₂), α ∈ [0, 1]

Bures distance

Fidelity can be used to define metric on the set of quantum states, so called Bures distancefuchs96phd D_B

$$D_B(\rho,\sigma) = \sqrt{2-2\sqrt{F(\rho,\sigma)}}$$

and the angleNielsenChuang

$$D_A(\rho,\sigma) = \arccos\sqrt{F(\rho,\sigma)}.$$

The quantity D_B(ρ, σ) is the minimal distance between purifications of ρ and σ using a common environment.

Classical fidelity

Fidelity is also defined for classical probability distributions. Let {p_i} and {q_i} where i = 1, 2, ..., n be probability distributions. The fidelity between p and q is defined as $F'(\{p_i\},\{q_i\})=\sum_{i=1}^n\sqrt{p_i,q_i}.$

References

See also

• Trace distance
• Trace norm
• Superfidelity

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