The Exercise

Two spins one half, two qubits, are prepared in an unknown state $\rho$. In one experiment each spin is measured along an axis of its own, $x$, $y$ or $z$, which measures the Pauli observable $X$, $Y$ or $Z$ of that spin with outcomes $\pm 1$, and the experiment is repeated on many copies of $\rho$. The average outcome of each spin gives the expectation value of its observable, and the average product of the two outcomes gives that of the product: the setting $(x, z)$ yields the expectation values of $X \otimes I$, $I \otimes Z$ and $X \otimes Z$. With three axes for each spin there are nine settings. The exercise is to show that the nine experiments determine $\rho$: two states of the two spins with the same expectation values in all nine settings are the same state.^✦

Two Qubits

A quantum system $S$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ The system modelled on a finite dimensional complex Hilbert space $H$ is the quantum system whose state space is $H$ itself.^✦ The qudit with $d$ levels is the system modelled on $\mathbb{C}^d$ with its standard inner product; the qubit is the qudit with two levels, and the qudit with $2^n$ levels is the register of $n$ qubits.^✦ The two spins are the register with $n = 2$, on $\mathbb{C}^4$.

A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ An observable $A$ on $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$, whose real spectrum is the set of its possible measured values.^✦ The expectation value of $A$ in the state $\rho$ is the real number $$\langle A \rangle_\rho = \operatorname{tr}(\rho A).$$(1)^✦

Pauli Strings

The Pauli matrices are the four complex $2 \times 2$ matrices $\sigma_0 = I$ and $$\sigma_1 = X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma_2 = Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma_3 = Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(2)^✦ with rows and columns indexed by $\{0, 1\}$.

For a choice $g = (g_1, \dots, g_n)$ of an index in $\{0, 1, 2, 3\}$ for each of $n$ qubits, the Pauli string $P_g = \sigma_{g_1} \otimes \cdots \otimes \sigma_{g_n}$ is the operator on the $n$ qubits whose matrix, with rows labelled by strings of bits $i = (i_1, \dots, i_n)$ and columns by strings of bits $j = (j_1, \dots, j_n)$, has the entries $$(P_g)_{ij} = \prod_{k=1}^{n} (\sigma_{g_k})_{i_k j_k}.$$(3) For a real coefficient $c$, the Pauli string term $c\,P_g$ is the observable on the register that acts on state vectors as $c$ times this matrix.^✦ On the register of $n$ qubits, whose levels are numbered $0, \dots, 2^n - 1$, the operator $P_g$ is the matrix whose entry in row $a$ and column $b$ is the entry of $P_g$ at the bit strings obtained by writing $a$ and $b$ as binary numbers.^✦

Measurement Settings

A setting $s = (s_1, \dots, s_n)$ on $n$ qubits chooses for each qubit $k$ one of the matrices $\sigma_1, \sigma_2, \sigma_3$, by its index $s_k \in \{1, 2, 3\}$: the axis along which qubit $k$ is measured. The setting $s$ observes the Pauli string $P_g$ when each factor of $P_g$ is either the identity or the Pauli matrix the setting chooses on that qubit: $$g_k = 0 \quad \text{or} \quad g_k = s_k \qquad \text{for every qubit } k.$$(4)^✦

The Claim

What Is Checked

The claim is stated over states of the qudit with four levels and over the expectation values (1) in them of the Pauli string terms $P_g$ with coefficient one, for the strings a setting observes in the sense of (4).