The Exercise

A qubit passes through the bit flip channel: with probability $p$ the Pauli matrix $X$, which exchanges the two basis states, acts on it, and otherwise it is left alone. A second process treats the qubit another way. With probability $1-2p$ it leaves the qubit alone, and with probability $2p$ it measures the observable $X$ projectively and throws the outcome away; for $1-2p$ to be a probability, $p$ is at most $1/2$.^✦ The exercise is to show that for $0 \le p \le 1/2$ the two processes are the same channel, taking every state of the qubit to the same state.^✦

A Quantum System

A quantum system $S$ is described by its state space, a finite dimensional complex Hilbert space.^✦ Any such space $H$ is the state space of a system^✦, and the qubit is the two level system whose state space is $H = \mathbb{C}^2$^✦. A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦

A unitary $U$ of $S$, a closed system evolution of $S$, is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$.^✦ Under a unitary $U$ of $S$, a state $\rho$ of $S$ evolves to $U \rho U^\dagger$.^✦

Rotations of the Qubit

The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(1) and $\vec\sigma = (X, Y, Z)$ is the Pauli vector. For a real vector $\vec n = (n_x, n_y, n_z)$ the matrix $\vec n \cdot \vec\sigma$ is $n_x X + n_y Y + n_z Z$.^✦ For such a vector and a real angle $\theta$, the axis rotation $R_{\vec n}(\theta)$ is the unitary of the qubit with matrix $$R_{\vec n}(\theta) = \exp\Bigl(-\frac{i\theta}{2}\, \vec n \cdot \vec\sigma\Bigr) ;$$(2)^✦ for a unit vector $\hat n$ it is the rotation $R_{\hat n}(\theta)$ through $\theta$ about the axis $\hat n$.

With $\hat x$, $\hat y$, $\hat z$ the unit vectors along the three axes, the Pauli conjugations of the qubit are the four unitaries $R_{\hat x}(0) = I$, the identity, and $R_{\hat x}(\pi)$, $R_{\hat y}(\pi)$ and $R_{\hat z}(\pi)$.^✦

The Bit Flip Channel

A random unitary channel of $S$ is a finite family of unitaries $U_k$ of $S$ together with probabilities $p_k$, nonnegative and summing to one.^✦ It acts on a state $\rho$ of $S$ as $$\rho \mapsto \sum_k p_k\, U_k \rho U_k^\dagger ,$$(3) The Pauli channel with weights $p_0, \dots, p_3$, nonnegative and summing to one, is the random unitary channel of the qubit that mixes the four Pauli conjugations with these weights, $$\rho \mapsto p_0\, \rho + p_1\, X \rho X + p_2\, Y \rho Y + p_3\, Z \rho Z .$$(4)^✦

For $0 \le q \le 1$ the bit flip channel $\mathcal{E}_q$ is the Pauli channel with weights $(q, 1-q, 0, 0)$: $$\mathcal{E}_q(\rho) = q\, \rho + (1-q)\, X \rho X .$$(5)^✦ The channel of the exercise is $\mathcal{E}_{1-p}$.

Measuring $X$

An observable $A$ on $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$.^✦ For a real vector $\vec n$ the Pauli observable $\vec n \cdot \vec\sigma$ is the observable of the qubit whose operator is the matrix $\vec n \cdot \vec\sigma$.^✦ The observable $X$ is $\hat x \cdot \vec\sigma$^✦, and it squares to the identity, $X^2 = 1$^✦.

The projectors $P_+$ and $P_-$ of an observable $A$ on $S$ are the operators $$P_\pm = \tfrac{1}{2}\,(1 \pm A),$$(6) which project onto the eigenspaces of $A$ for the eigenvalues $+1$ and $-1$ when $A^2 = 1$.

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 $i$, satisfying the completeness relation $\sum_i M_i^\dagger M_i = 1$.^✦ If the outcome is not read, the measurement takes a state $\rho$ of $S$ to the unread postmeasurement state $$\rho' = \sum_i M_i \rho M_i^\dagger ,$$(7)^✦

For an observable $A$ with $A^2 = 1$, the sign measurement of $A$ is the measurement with two outcomes whose operators are $M_0 = P_+$ and $M_1 = P_-$, the outcome $0$ standing for the eigenvalue $+1$ and the outcome $1$ for $-1$.^✦

The Measurement Model

The mixture of two states $\rho$ and $\sigma$ of $S$ with weight $\lambda$, a real number with $0 \le \lambda \le 1$, is the convex combination $\lambda \rho + (1-\lambda) \sigma$, prepared by choosing $\rho$ with probability $\lambda$ and $\sigma$ otherwise.^✦

For $0 \le p \le 1/2$ the measurement model $\mathcal{M}_p$ takes a state $\rho$ of the qubit to the mixture, with weight $1-2p$, of $\rho$ itself and its unread postmeasurement state under the sign measurement of $X$: $$\mathcal{M}_p(\rho) = (1-2p)\, \rho + 2p\, \bigl(P_+ \rho P_+ + P_- \rho P_-\bigr), \qquad P_\pm = \tfrac{1}{2}\,(1 \pm X).$$(8)^✦ It is the second process of the exercise.

The Claim

What Is Checked

The claim is stated over two maps on the states of the qubit: the bit flip channel (5) and the measurement model (8).

About these constructions the library proves that the bit flip weights $(q, 1-q, 0, 0)$ are nonnegative for $0 \le q \le 1$^✦ and sum to one^✦, so that $\mathcal{E}_q$ is a Pauli channel. It proves that conjugating a state by a unitary gives a state and that a random unitary channel takes states to states; that the unread postmeasurement state is a state, its trace being one by the completeness relation; and that a mixture of two states is a state. The matrix $\vec n \cdot \vec\sigma$ is Hermitian, so the Pauli observable is self adjoint, and for an observable with $A^2 = 1$ the projectors $P_\pm$ satisfy the completeness relation, so that the sign measurement of $X$ is a measurement.