The Exercise

The spins of a nuclear magnetic resonance (NMR) sample at room temperature are in a state very close to the maximally mixed one, and each computational basis state of the $n$ spins of a molecule has a population close to $2^{-n}$ . When the spins all have the same Larmor frequency, the population of a basis state departs from $2^{-n}$ by an amount that depends only on how many of its spins are in $|1\rangle$ .

A set of basis states in which one state’s population stands apart, while all the others share a common value, carries an effective pure state: restricted to the subspace the set spans, the state of the ensemble is a multiple of the identity plus a multiple of one pure state. How many basis states can such a set hold, and is the largest size reached?

Basis States and Their Magnetization

The computational basis states of $n$ spins are labelled by sets of spins: for a set $A$ of positions among $1, \dots, n$ , the state $|A\rangle$ has the spins in $A$ in $|1\rangle$ and the others in $|0\rangle$ . The magnetization $m(A)$ of the basis state $|A\rangle$ , its signal population, is the integer $m(A) = n - 2\,|A|,$ (1)^✦ where $|A|$ is the number of spins in $A$ . The magnetization $m(A)$ is the eigenvalue of $Z_1 + \cdots + Z_n$ on $|A\rangle$ , where $Z_i$ is the Pauli $Z$ operator of spin $i$ .

Effective Pure Manifolds

An effective pure manifold of $n$ spins is a set $S$ of computational basis states that contains a state $|g\rangle$ of larger magnetization than every other state of $S$ , the other states of $S$ all having one and the same magnetization: $m(A) < m(g) \quad\text{and}\quad m(A) = m(B) \qquad \text{for all } |A\rangle, |B\rangle \in S \text{ other than } |g\rangle.$ (2)^✦ The number of states in $S$ is written $|S|$ .

The maximal effective pure dimension $D(n)$ of $n$ spins is the integer $D(n) = \binom{n}{\lfloor n/2 \rfloor} + 1.$ (3)^✦

The Claim

Theorem 1. Let $n$ be a number of spins. Every effective pure manifold $S$ of $n$ spins has at most $D(n)$ states: $|S| \le D(n).$ (4)^✦ If there is at least one spin, $n \ge 1$ , then some effective pure manifold $S$ of $n$ spins has exactly $D(n)$ states: $|S| = D(n).$ (5)^✦

What Is Checked

The claim is stated over effective pure manifolds of $n$ spins, sets of computational basis states compared through their magnetization (1) as in (2), and over the maximal effective pure dimension $D(n)$ of (3).