The Exercise

Let $|\psi\rangle$ be a normalized state vector, and let $|x\rangle$ be $N$ orthonormal vectors of the same state space, the label $x$ running over $N$ values. Each $|x\rangle$ lies at some distance from $|\psi\rangle$ . The exercise bounds the sum of the squares of these $N$ distances from below by a quantity that depends on $N$ alone, whatever the state $|\psi\rangle$ and whatever the orthonormal vectors.

Unit and Orthonormal Vectors

The vectors lie in an inner product space $\mathcal{H}$ . The length of a vector $|\phi\rangle$ of $\mathcal{H}$ is its norm, $\big\| |\phi\rangle \big\| = \sqrt{\langle \phi | \phi \rangle},$ (1) and the distance between two vectors is the norm of their difference. A unit vector $|\psi\rangle$ of $\mathcal{H}$ is a vector of norm one.

A family of vectors $|x\rangle$ of $\mathcal{H}$ , indexed by a finite set of $N$ labels, is orthonormal when $\langle x | y \rangle = \delta_{xy}$ (2) for all labels $x$ and $y$ .

The Claim

Theorem 1. Let $|x\rangle$ , for $x$ in a finite set of $N$ labels, be an orthonormal family of vectors of an inner product space $\mathcal{H}$ , and let $|\psi\rangle$ be a unit vector of $\mathcal{H}$ . Then the squared distances from $|\psi\rangle$ to the $|x\rangle$ satisfy $\sum_x \big\| |\psi\rangle - |x\rangle \big\|^2 \;\ge\; 2N - 2\sqrt{N}.$ (3)^✦

What Is Checked

The claim is stated over an inner product space $\mathcal{H}$ whose scalars are the real or the complex numbers, a unit vector $|\psi\rangle$ in it, and an orthonormal family $|x\rangle$ indexed by any finite set, with $N$ the number of its labels. The family need not span $\mathcal{H}$ .

About these objects the library proves the inequality (3), for every unit vector and every finite orthonormal family of $\mathcal{H}$ .