Lean, informalized

Nielsen & Chuang · Chapter 9, Distance measures for quantum information · p. 400

Exercise 9.3 - The fidelity of a binary and a ternary pair of probability distributions, in closed form

Classical fidelity of two pairs of distributions.

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The Exercise

The exercise asks for the fidelity of two pairs of probability distributions. In the first pair, an experiment with two outcomes gives the first one with certainty under px=(1,0)p_x = (1, 0) and either one with probability one half under qx=(12,12)q_x = (\frac12, \frac12): a coin that always lands heads set against a fair coin. In the second pair, an experiment with three outcomes has the probabilities px=(12,13,16)p_x = (\frac12, \frac13, \frac16) under one distribution and qx=(34,18,18)q_x = (\frac34, \frac18, \frac18) under the other.

The Classical Fidelity

The classical fidelity F(px,qx)F(p_x, q_x) of two real functions pxp_x and qxq_x on the same finite set of outcomes xx is the real number F(px,qx)=∑xpxqx.F(p_x, q_x) = \sum_x \sqrt{p_x q_x} .(1)✦

The Claim

Theorem 1. The fidelity of the certain distribution (1,0)(1, 0) and the uniform distribution (12,12)(\frac12, \frac12) on two outcomes is F((1,0),(12,12))=22,F\bigl((1, 0), (\tfrac12, \tfrac12)\bigr) = \frac{\sqrt{2}}{2} ,(2)✦ and the fidelity of the distributions (12,13,16)(\frac12, \frac13, \frac16) and (34,18,18)(\frac34, \frac18, \frac18) on three outcomes is F((12,13,16),(34,18,18))=46+312.F\bigl((\tfrac12, \tfrac13, \tfrac16), (\tfrac34, \tfrac18, \tfrac18)\bigr) = \frac{4\sqrt{6} + \sqrt{3}}{12} .(3)✦

What Is Checked

The claim is stated over the classical fidelity (1), evaluated on the four distributions exactly as the exercise gives them, two on two outcomes and two on three.