Definition
Given two density matrices ρ and σ, the fidelity is defined by
By M½ of a positive semidefinite matrix M, we mean its unique positive square root given by the spectral theorem. The Euclidean inner product from the classical definition is replaced by the Hilbert-Schmidt inner product. When the states are classical, i.e. when ρ and σ commute, the definition coincides with that for probability distributions.
An equivalent definition is given by
where the norm is the trace norm (sum of the singular values). This definition has the advantage that it clearly shows that the fidelity is symmetric in its two arguments.
Notice by definition F is non-negative, and F(ρ,ρ) = 1. In the following section it will be shown that it can be no larger than 1.
In the original 1994 paper of Jozsa the name 'fidelity' was used for the quantity and this convention is often used in the literature. According to this convention 'fidelity' has a meaning of probability.
Read more about this topic: Fidelity Of Quantum States
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