Fidelity of Quantum States - Motivation

Motivation

In probability theory, given two random variables p = (p₁...p_n) and q = (q₁...q_n) on the probability space X = {1,2...n}. The fidelity of p and q is defined to be the quantity

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In other words, the fidelity F(p,q) is the inner product of and viewed as vectors in Euclidean space. Notice that F(p,q) = 1 if and only if p = q. In general, . This measure is known as the Bhattacharyya coefficient.

Given a classical measure of the distinguishability of two probability distributions, one can motivate a measure of distinguishability of two quantum states as follows. If an experimenter is attempting to determine whether a quantum state is either of two possibilities or, the most general possible measurement he can make on the state is a POVM, which is described by a set of Hermitian positive semidefinite operators . If the state given to the experimenter is, he will witness outcome with probability, and likewise with probability for . His ability to distinguish between the quantum states and is then equivalent to his ability to distinguish between the classical probability distributions and . Naturally, the experimenter will choose the best POVM he can find, so this motivates defining the quantum fidelity as the Bhattacharyya coefficient when extremized over all possible POVMs :

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It was shown by Fuchs and Caves that this manifestly symmetric definition is equivalent to the simple asymmetric formula given in the next section.