The Exercise

In nuclear magnetic resonance the qubits are nuclear spins in a strong static magnetic field, at thermal equilibrium at inverse temperature $\beta$. A single spin has the Hamiltonian $\hbar\omega Z$, where $Z$ is the Pauli matrix $\operatorname{diag}(1,-1)$ and $\omega$ the precession frequency of the spin; two spins $A$ and $B$ in the same field have $H = \hbar\omega_A Z_A + \hbar\omega_B Z_B$, with $Z_A$ and $Z_B$ the matrix $Z$ acting on spin $A$ and on spin $B$. At the temperatures of the experiment $\beta\hbar\omega$ is small, and the equilibrium state $e^{-\beta H}/\operatorname{tr} e^{-\beta H}$ is replaced by its expansion to first order in $\beta$.

The exercise computes this state in two cases: for one spin, as a point of the Bloch ball, the closed unit ball of vectors $\vec r$ that label the states of a qubit; for two spins whose frequencies stand in the ratio $\omega_A = 4\omega_B$, as four populations on the diagonal of the computational basis. The library gives both answers as equalities of states, each under a bound on the product $\beta\hbar\omega$.

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ 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 $A = A^\dagger$ on the state space of $S$, and a Hamiltonian is an observable.^✦

The qudit with $d$ levels is the quantum system whose state space is $\mathbb{C}^d$ with its standard inner product.^✦ The qubit is the qudit with two levels and holds one spin; a register of two qubits is the qudit with four levels and holds two. Matrices on the two spins are written in the computational basis in the order $|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle$, the first label belonging to spin $A$.^✦

On the qubit, in the basis $|0\rangle$, $|1\rangle$, the Pauli matrices are $$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1) For a real vector $\vec n = (n_x, n_y, n_z)$ the matrix $\vec n \cdot \vec\sigma$ is $n_x X + n_y Y + n_z Z$, and the Pauli observable $\vec n \cdot \vec\sigma$ is the observable on the qubit with this matrix. For a real vector $\vec r$ in the closed unit ball, $|\vec r| \le 1$, the Bloch state $\rho(\vec r)$ is the state of the qubit whose matrix is the Bloch matrix $$\rho(\vec r) = \tfrac12 \left( I + \vec r \cdot \vec\sigma \right).$$(2)^✦

Spins in a Field

The Hamiltonian of one spin is $\hbar\omega Z$, the Pauli observable of the vector $(0, 0, \hbar\omega)$.^✦ The Hamiltonian of two spins is the observable on the four level system whose matrix is $$H = \hbar\omega_A \, Z \otimes I + \hbar\omega_B \, I \otimes Z,$$(3)^✦ diagonal with entries $\hbar(\omega_A + \omega_B)$, $\hbar(\omega_A - \omega_B)$, $\hbar(\omega_B - \omega_A)$ and $-\hbar(\omega_A + \omega_B)$.

For an observable $H$ on a system $S$ whose state space has dimension $d$, and a real number $\beta$ for which the operator $(I - \beta H)/d$ is positive with unit trace, the high temperature thermal state of $S$ at inverse temperature $\beta$ is the state $$\rho_\beta(H) = \frac{1}{d} \left( I - \beta H \right).$$(4)^✦

The populations of two spins with frequencies $\omega_A = 4\omega_B$ and $\omega_B$ at inverse temperature $\beta$ are the four numbers $$p_{00}, \, p_{11} = \tfrac14 \left( 1 \mp 5\beta\hbar\omega_B \right), \qquad p_{01}, \, p_{10} = \tfrac14 \left( 1 \mp 3\beta\hbar\omega_B \right),$$(5) and when $5|\beta\hbar\omega_B| \le 1$ the state $\sigma_\beta$ of the populations is the state of the four level system whose matrix is diagonal with entries $p_{00}$, $p_{01}$, $p_{10}$, $p_{11}$ in that order.

The Claim

The vector $\vec r$ of the Bloch state $\rho(\vec r)$ is its Bloch vector. For positive $\beta$ and $\hbar\omega$ the Bloch vector in (6) points toward $|1\rangle$, the lower level of $\hbar\omega Z$, and its length $\beta\hbar\omega$ is the polarization of the spin.

What Is Checked

The first claim is stated over the qubit, the Pauli observable $\hbar\omega Z$ built from the matrices (1), the high temperature thermal state (4) with $d = 2$, and the Bloch state (2). The second is stated over the register of two qubits as the four level system, the Hamiltonian (3) written in the order $|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle$, the same thermal state with $d = 4$, and the state $\sigma_\beta$ of the populations (5).

The matrices $\hbar\omega Z$ and (3) are Hermitian, so $\hbar\omega Z$ and $H$ are observables. When $|\beta\hbar\omega| \le 1$, the library shows that the operator $\tfrac12 (I - \beta\hbar\omega Z)$ is positive with unit trace and that the vector $(0, 0, -\beta\hbar\omega)$ lies in the Bloch ball. When $5|\beta\hbar\omega_B| \le 1$ and $\omega_A = 4\omega_B$, it shows that $\tfrac14 (I - \beta H)$ is positive with unit trace, and that the populations are nonnegative and sum to one. The Bloch matrix $\tfrac12 (I + \vec r \cdot \vec\sigma)$ is positive exactly when $|\vec r| \le 1$, and its trace is one.