The Exercise

Two qubits are prepared in the Bell state $|\Phi^+\rangle$, and an observable is measured on the first of them: either $Z$, in the computational basis $|0\rangle, |1\rangle$, or $X$, in the basis $|+\rangle, |-\rangle$. The second qubit is left alone. The question is what state the pair is in after each outcome of either measurement.

The library answers it in four statements, one for each observable and outcome, over its definitions of states and measurements.

Two Qubits

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ Any such space $H$ is the state space of a system, the system modelled on $H$^✦; the qubit is the system modelled on $\mathbb{C}^2$.^✦ Two systems $S$ and $T$ form the composite system $S \otimes T$, whose state space is the tensor product of their state spaces.^✦

A pure state $\psi$ of $S$ is a unit vector in the state space of $S$.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$^✦, and a pure state $\psi$ gives the state $|\psi\rangle\langle\psi|$, the projector onto its vector.^✦ Pure states $\psi$ of $S$ and $\varphi$ of $T$ give the product pure state $\psi \otimes \varphi$ of $S \otimes T$, whose vector is the tensor product of their vectors.^✦

The basis states $|0\rangle$ and $|1\rangle$ of the qubit are the pure states whose vectors are the standard basis vectors of $\mathbb{C}^2$.^✦ The states $|+\rangle$ and $|-\rangle$ are the pure states whose vectors have the coordinates $(1/\sqrt{2}, 1/\sqrt{2})$ and $(1/\sqrt{2}, -1/\sqrt{2})$: $$|\pm\rangle = \frac{|0\rangle \pm |1\rangle}{\sqrt{2}}.$$(1) On two qubits, $|00\rangle$ stands for the product $|0\rangle \otimes |0\rangle$, and likewise $|11\rangle$, $|{+}{+}\rangle$ and $|{-}{-}\rangle$. The Bell state $|\Phi^+\rangle$ is the pure state of two qubits whose vector is $|00\rangle + |11\rangle$ divided by its norm $\sqrt{2}$: $$|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}.$$(2)^✦

Measurements

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 $i$, satisfying the completeness relation, in which $I$ is the identity operator on the state space of $S$, $$\sum_i M_i^\dagger M_i = I.$$(3) The positive operator valued measure (POVM) of a measurement $\{M_i\}$ on $S$ is the family of effects $E_i = M_i^\dagger M_i$, one for each outcome $i$; the effects are positive. Performed on a state $\rho$ of $S$, the measurement gives the outcome $i$ with the probability of the Born rule, $$p(i) = \operatorname{Tr}(E_i \rho) = \operatorname{Tr}(M_i \rho M_i^\dagger),$$(4)^✦ and when $p(i) \neq 0$ the state after the outcome is $\rho$ collapsed and renormalised, $$\rho_i = \frac{M_i \rho M_i^\dagger}{p(i)}.$$(5)^✦

An observable $A$ on $S$ is a Hermitian operator on the state space of $S$.^✦ The identity observable $I$ on $S$ is the observable of $S$ whose operator is the identity on the state space of $S$.^✦ Observables $A$ of $S$ and $B$ of $T$ give the observable $A \otimes B$ of $S \otimes T$, the tensor product of the two operators, which measures $A$ on the first factor and $B$ on the second.^✦ The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(6) and for a real vector $\vec{n}$ in three dimensions the Pauli observable on the qubit is the Hermitian matrix $\vec{n} \cdot \vec{\sigma} = n_x X + n_y Y + n_z Z$ acting on $\mathbb{C}^2$.^✦ The observables $X$ and $Z$ are the Pauli observables for $\hat{n} = (1, 0, 0)$ and $\hat{n} = (0, 0, 1)$.

The projectors $P_+$ and $P_-$ of an observable $A$ on $S$ are the operators $$P_\pm = \tfrac{1}{2}(I \pm A),$$(7) which for $A^2 = I$ project onto the eigenspaces of $A$ for $+1$ and $-1$. The sign measurement of an observable $A$ on $S$ with $A^2 = I$ is the measurement $\{P_+, P_-\}$ on $S$ with two outcomes, $+1$ with the operator $P_+$ and $-1$ with the operator $P_-$: the projective measurement of $A$.^✦

The measurement of $Z$ on the first of two qubits is the sign measurement of the observable $Z \otimes I$, which squares to the identity^✦; the measurement of $X$ on the first qubit is the sign measurement of $X \otimes I$.^✦ Their projectors are written $P^Z_\pm$ and $P^X_\pm$.

The Claim

What Is Checked

The claim is stated over the Bell state $|\Phi^+\rangle$ of two qubits, taken as the state $|\Phi^+\rangle\langle\Phi^+|$; the measurements of $Z$ and of $X$ on the first qubit, as the sign measurements of $Z \otimes I$ and $X \otimes I$; the state after an outcome, with the probability of the outcome by the Born rule; and the product states $|00\rangle$, $|11\rangle$, $|{+}{+}\rangle$ and $|{-}{-}\rangle$.

Of these constructions the library proves that the vectors of $|\Phi^+\rangle$ and of $|\pm\rangle$ have unit norm, as does the product of any two pure states, and that the projector onto a unit vector is a density operator. It proves that $\vec{n} \cdot \vec{\sigma}$ and a tensor product of observables are Hermitian, that $Z \otimes I$ and $X \otimes I$ square to the identity, so that their projectors satisfy the completeness relation (3), and that the state after an outcome of nonzero probability is again a density operator.