The Exercise

A state $\rho$ of $n$ spins is a density operator on $(\mathbb{C}^2)^{\otimes n}$, and it is a combination of the $4^n$ Pauli products, $$\rho = \frac{1}{2^n} \sum_{g} \mathrm{tr}(\rho\,\sigma_g)\,\sigma_g , \qquad \sigma_g = \sigma_{g_1} \otimes \sigma_{g_2} \otimes \cdots \otimes \sigma_{g_n} ,$$(1) where a Pauli string $g = (g_1, \dots, g_n)$ has $g_i \in \{0,1,2,3\}$, with $\sigma_0 = I$, $\sigma_1 = X$, $\sigma_2 = Y$, $\sigma_3 = Z$.

In nuclear magnetic resonance (NMR) tomography, every spin is measured along one of the axes $x$, $y$, $z$, chosen spin by spin. The setting of an experiment is a string $s = (s_1, \dots, s_n)$ in $\{1,2,3\}^n$, with $s_i$ the axis of spin $i$, so there are $3^n$ settings, and one experiment gives some of the expectation values $\mathrm{tr}(\rho\,\sigma_g)$, not all of them. The exercise is to count the experiments that tomography of $n$ spins needs: $3^n$, which is $27$ for three spins.

Settings and Completeness

A setting $s$ observes a Pauli string $g$ when the factor of $\sigma_g$ on every spin is the identity or the Pauli matrix of the axis that $s$ chooses for that spin: $$g_i = 0 \quad \text{or} \quad g_i = s_i \qquad (i = 1, \dots, n) .$$(2)^✦ The experiment with setting $s$ gives $\mathrm{tr}(\rho\,\sigma_g)$ for every string $g$ that $s$ observes.

A set $T$ of settings is tomographically complete when every Pauli string other than the identity string is observed by some setting in $T$: $$g \neq (0, \dots, 0) \;\Longrightarrow\; \text{there is } s \in T \text{ with } g_i \in \{0, s_i\} \text{ for all } i .$$(3)^✦ The experiments of a complete set together give every coefficient of (1), and with them $\rho$.

The number $N(n)$ of NMR tomography experiments on $n$ spins is the number of settings, $$N(n) = 3^n .$$(4)^✦

The Claim

For $n \ge 1$, a setting $s$ read as a Pauli string, with $\sigma_{s_i}$ on every spin, is not the identity string, and by (2) the only setting that observes it is $s$ itself.

What Is Checked

The claim is stated over the settings $\{1,2,3\}^n$ and the Pauli strings $\{0,1,2,3\}^n$ of $n$ spins, through the relation (2) of a setting observing a string, the completeness (3) of a set of settings and the count (4).