The Exercise

A source emits qubits one at a time, each in one of two pure states: $|0\rangle$, or the state obtained from $|0\rangle$ by a rotation through $\pi/2$ about the $\hat y$ axis.^✦ A block of $n$ emissions is a product state of $n$ qubits, labelled by the string $k$ of $n$ bits that records which of the two states each qubit received. Rotating $|0\rangle$ and $|1\rangle$ through $\pi/4$ about the same axis gives a second basis of the qubit, and its products, labelled by strings $x$ of $n$ bits, are product states of the $n$ qubits as well. The exercise asks for the overlap of a block of the source with one of these products, and for the fidelity between the two. The library states both for every length $n$ and every pair of labels.

States of 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 system whose state space is $\mathbb{C}^2$ itself^✦, and its computational basis states $|0\rangle$ and $|1\rangle$ are the standard basis vectors of $\mathbb{C}^2$.^✦

A pure state $|\psi\rangle$ of $S$ is a unit vector in the state space of $S$, fixed physically up to a global phase.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ Each pure state $|\psi\rangle$ gives the state $|\psi\rangle\langle\psi|$, the projector onto its vector.^✦

The overlap $\langle\varphi|\psi\rangle$ of two pure states $|\varphi\rangle$ and $|\psi\rangle$ of $S$ is the complex inner product of their vectors, conjugate linear in $|\varphi\rangle$.^✦ The fidelity of two states $\rho$ and $\sigma$ of $S$ is the nonnegative number $$F(\rho,\sigma) = \operatorname{tr}\sqrt{\sqrt{\rho}\,\sigma\sqrt{\rho}},$$(1)^✦ where the square root of a positive operator takes the square root of each eigenvalue.

The composite $\bigotimes_i S_i$ of a finite family of systems $S_i$ is the system whose state space is the tensor product of the state spaces of the $S_i$.^✦ Given a pure state $|\chi_i\rangle$ of each $S_i$, their product is the pure state of the composite with vector $\bigotimes_i |\chi_i\rangle$.^✦

Rotations of a Qubit

A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$.^✦ It carries a pure state $|\psi\rangle$ of $S$ to the pure state $U|\psi\rangle$.^✦

On the qubit, the Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(2) 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 rotation about $\vec n$ through a real angle $\theta$ is $$R_{\vec n}(\theta) = \exp\Bigl(-\frac{i\theta}{2}\,\vec n \cdot \vec\sigma\Bigr),$$(3)^✦ written $R_{\hat n}(\theta)$ when the axis is a unit vector $\hat n$. About the axis $\hat y = (0, 1, 0)$ this is $$R_y(\theta) = R_{\hat y}(\theta) = e^{-i\theta Y/2} = \begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\[2pt] \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}.$$(4)^✦ For a real angle $\theta$, the $y$ rotation gate $R_y(\theta)$ of the qubit is the unitary of the qubit whose matrix is the one displayed in (4).

The Source and the Rotated Basis

The two states of the source are $$|\psi_0\rangle = |0\rangle, \qquad |\psi_1\rangle = R_y(\pi/2)\,|0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt 2}.$$(5)^✦ The rotated basis has one state for each bit $b$, $$|\phi_b\rangle = R_y(\pi/4)\,|b\rangle, \qquad |\phi_0\rangle = \cos\tfrac{\pi}{8}\,|0\rangle + \sin\tfrac{\pi}{8}\,|1\rangle, \quad |\phi_1\rangle = -\sin\tfrac{\pi}{8}\,|0\rangle + \cos\tfrac{\pi}{8}\,|1\rangle.$$(6)^✦

For strings of bits $x = x_1 \cdots x_n$ and $k = k_1 \cdots k_n$, the blocks are the product states of the composite of $n$ qubits, with state space $(\mathbb{C}^2)^{\otimes n}$, $$|\Phi_x\rangle = |\phi_{x_1}\rangle \otimes \cdots \otimes |\phi_{x_n}\rangle, \qquad |\Psi_k\rangle = |\psi_{k_1}\rangle \otimes \cdots \otimes |\psi_{k_n}\rangle.$$(7)

The Claim

What Is Checked

The claim is stated over the blocks (7), products of the source states (5) and of the rotated basis states (6) as pure states of the composite of $n$ qubits.

The library proves that each of these is a state of its system. A tensor product of unit vectors is a unit vector, so each block is a pure state of the composite, whose state space is finite dimensional because the products of basis vectors of the factors form a basis of it.