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 \phi,\psi it is \vert\langle\phi,\psi\rangle\vert^2. For a pure state (vector \psi) and a mixed state (density matrix \rho) this generalizes to \langle\psi,\rho\psi\rangle, and for two density matrices \rho,\sigma 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 \vert\langle\phi,\psi\rangle\vert, 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 \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: # 0\leq F(\rho,\sigma)\leq 1 # F(\rho,\sigma)=F(\sigma,\rho)\, # F(\rho_1\otimes\rho_2,\sigma_1\otimes\sigma_2)=F(\rho_1,\sigma_1)F(\rho_2,\sigma_2) # F(U \rho U^\dagger,U\sigma U^\dagger)=F(\rho,\sigma) # 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''fuchs96phd D_B :D_B(\rho,\sigma) = \sqrt{2-2\sqrt{F(\rho,\sigma)}} and the ''angle''NielsenChuang :D_A(\rho,\sigma) = \arccos\sqrt{F(\rho,\sigma)}. The quantity D_B(\rho,\sigma) is the minimal distance between purifications of \rho and \sigma 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

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Monday, October 26, 2015 - 17:56