# Fidelity

### 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) = ⟨ψσψ⟩ and if both states are pure i.e. ρ = ∣ψ⟩⟨ψ∣ and σ = ∣ϕ⟩⟨ϕ∣, then F(ρ, σ) = ∣⟨ψϕ⟩∣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 DB

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

and the angleNielsenChuang

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

The quantity DB(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 {pi} and {qi} where i = 1, 2, ..., n be probability distributions. The fidelity between p and q is defined as

$F'(\{p_i\},\{q_i\})=\sum_{i=1}^n\sqrt{p_i,q_i}.$