Fidelity

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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 \vert\langle\phi,\psi\rangle\vert^2. For a pure state (vector ψ) and a mixed state (density matrix ρ) this generalizes to \langle\psi,\rho\psi\rangle, 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 in<bibref>Jozsa94</bibref>, where the term fidelity appears to have been used first.

However, one can also start from \vert\langle\phi,\psi\rangle\vert, leading to the alternative

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

used in<bibref>NielsenChuang</bibref>. 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 \rho=|\psi\rangle\langle\psi| is pure, then F(\rho,\sigma)=\langle\psi|\sigma|\psi\rangle and if both states are pure i.e. \rho=|\psi\rangle\langle\psi| and \sigma=|\phi\rangle\langle\phi|, then F(\rho,\sigma)=|\langle\psi|\phi\rangle|^2.

Other properties:

  1. 0\leq F(\rho,\sigma)\leq 1
  2. F(\rho,\sigma)=F(\sigma,\rho)\,
  3. F(\rho_1\otimes\rho_2,\sigma_1\otimes\sigma_2)=F(\rho_1,\sigma_1)F(\rho_2,\sigma_2)
  4. F(U \rho U^\dagger,U\sigma U^\dagger)=F(\rho,\sigma)
  5. F(\rho,\alpha\sigma_1+(1-\alpha)\sigma_2)\geq \alpha F(\rho,\sigma_1)+(1-\alpha)F(\rho,\sigma_2),\ \alpha\in[0,1]

Bures distance

Fidelity can be used to define metric on the set of quantum states, so called Bures distance<bibref>fuchs96phd</bibref> DB

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

and the angle<bibref>NielsenChuang</bibref>

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}.

References

<bibreferences/>

See also