The Exercise

A qubit is prepared in the state $|0\rangle$, and the observable $\hat n \cdot \vec\sigma$ is measured on it, for a unit vector $\hat n = (n_1, n_2, n_3)$. The exercise asks for the probability of the result $+1$, and for the state of the qubit after the measurement when $+1$ is obtained.

The library answers both questions in one theorem about the projective measurement of $\hat n \cdot \vec\sigma$, built on the definitions below.

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ The qubit is the two level system whose state space is $\mathbb{C}^2$, the standard two dimensional complex Euclidean space; its computational basis kets are the standard basis vectors $|0\rangle = (1, 0)$ and $|1\rangle = (0, 1)$.

A pure state $\psi$ of $S$ is a unit vector $|\psi\rangle$ in the state space of $S$, and a nonzero vector $v$ there determines the pure state $v / \lVert v \rVert$. A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$, a density operator.^✦ A pure state $\psi$ gives the state $|\psi\rangle\langle\psi|$, the projector onto its vector.^✦ The qubit state $|0\rangle$ is the pure state of the qubit whose vector is the basis ket $|0\rangle$.^✦

Observables and Measurements

An observable $A$ on $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$; its projectors and the Pauli observables are introduced next, each in its own paragraph.^✦

The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(1) and for a real vector $\vec n = (n_1, n_2, n_3)$ the matrix $\vec n \cdot \vec\sigma$ is the complex $2 \times 2$ matrix $$\vec n \cdot \vec\sigma = n_1 X + n_2 Y + n_3 Z .$$(2) For a real vector $\vec n$, the Hermitian matrix $\vec n \cdot \vec\sigma$, read as an operator on the state space of the qubit, is the Pauli observable of $\vec n$.^✦

The projectors $P_\pm$ of an observable $A$ on $S$ are the operators $$P_+ = \tfrac12 (1 + A), \qquad P_- = \tfrac12 (1 - A);$$(3) when $A^2 = 1$ they project onto the eigenspaces of $A$ for the eigenvalues $+1$ and $-1$.

A measurement on $S$ with a finite set of outcomes is a family of measurement operators $M_i$ on the state space of $S$, one for each outcome $i$, that satisfies the completeness relation $$\sum_i M_i^\dagger M_i = 1 .$$(4)^✦ The positive operator valued measure (POVM) of the measurement is the family of positive operators $E_i = M_i^\dagger M_i$.^✦ Performed on a state $\rho$ of $S$, the measurement gives the outcome $i$ with the Born probability $$p(i) = \operatorname{Tr}(E_i \rho) = \operatorname{Tr}(M_i \rho M_i^\dagger),$$(5) and when $p(i) \neq 0$ the state after the outcome $i$ is the renormalized collapse $$\rho_i = \frac{M_i \rho M_i^\dagger}{p(i)} .$$(6)

The sign measurement of an observable $A$ on $S$ with $A^2 = 1$ is the measurement on $S$ with the two outcomes $+1$ and $-1$ and the operators $M_{+1} = P_+$ and $M_{-1} = P_-$: the projective measurement of $A$.^✦ For a unit vector $\hat n$, the measurement of $\hat n \cdot \vec\sigma$ is the sign measurement of the Pauli observable of $\hat n$, with $$M_{+1} = \tfrac12 (1 + \hat n \cdot \vec\sigma), \qquad M_{-1} = \tfrac12 (1 - \hat n \cdot \vec\sigma).$$(7)^✦

The projected pure state $\psi_+$ of a pure state $\psi$ of $S$, for an observable $A$ on $S$ with $P_+|\psi\rangle \neq 0$, is the normalized projection $$|\psi_+\rangle = \frac{P_+ |\psi\rangle}{\lVert P_+ |\psi\rangle \rVert} .$$(8)^✦

The Claim

What Is Checked

The claim is stated over the measurement (7) of $\hat n \cdot \vec\sigma$ on the qubit state $|0\rangle\langle 0|$, with the Born probability (5) and the collapsed state (6) of its outcome $+1$, and over the projected pure state (8) of $|0\rangle$.

About these constructions the library proves that $(\hat n \cdot \vec\sigma)^2 = 1$ for every unit vector $\hat n$, the condition of the sign measurement, and that a nonzero probability of $+1$ in the sign measurement of an observable on a pure state $\psi$ forces $P_+|\psi\rangle \neq 0$, so that the right side of (10) is defined. The collapsed state (6) is a density operator whenever $p(i) \neq 0$, and so is $|\psi\rangle\langle\psi|$ for every pure state $\psi$.