The Exercise

Alice prepares a qubit in one of six states. She picks one of the Pauli operators $X$, $Y$ and $Z$ and a bit $s$, and sends the eigenstate of that operator with eigenvalue $+1$ if $s = 0$ and $-1$ if $s = 1$. Bob measures the qubit in the eigenbasis of one of the three operators. When his choice agrees with Alice’s, he obtains her bit with certainty.^✦ When it does not, each of his outcomes is as likely for $s = 0$ as for $s = 1$, and his result tells him nothing about her bit.^✦

A Qubit and Its Observables

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 $\mathbb{C}^2$.^✦

A pure state $\psi$ of $S$ is a vector $|\psi\rangle$ of unit norm in the state space of $S$.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ Each pure state $\psi$ determines the state $|\psi\rangle\langle\psi|$ of $S$, the projector onto $|\psi\rangle$.^✦

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 $\vec{n} = (n_x, n_y, n_z)$ the matrix $\vec{n} \cdot \vec{\sigma}$ is $n_x X + n_y Y + n_z Z$.^✦

An observable $A$ on $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$.^✦ Since $\vec{n} \cdot \vec{\sigma}$ is Hermitian, the Pauli observable $\vec{n} \cdot \vec{\sigma}$ is the observable on the qubit with this matrix as its operator.^✦ The observables $X$, $Y$ and $Z$ are the Pauli observables for the unit vectors $\hat{n} = (1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$.

Label the three bases of the exercise by $i \in \{0, 1, 2\}$ and write $O_i$ for the observable of basis $i$, with $O_0 = X$, $O_1 = Y$ and $O_2 = Z$.^✦

Six States

The computational basis kets of $\mathbb{C}^2$ are the standard basis vectors $|0\rangle = (1, 0)$ and $|1\rangle = (0, 1)$^✦, and the basis states $|0\rangle$ and $|1\rangle$ of the qubit are the pure states with these vectors.^✦ With the normalization $1/\sqrt{2}$, four more pure states of the qubit, $|+\rangle$, $|-\rangle$, $|{+i}\rangle$ and $|{-i}\rangle$, have as their vectors the columns $$|\pm\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ \pm 1 \end{pmatrix} = \frac{|0\rangle \pm |1\rangle}{\sqrt{2}}, \qquad |{\pm i}\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ \pm i \end{pmatrix} = \frac{|0\rangle \pm i|1\rangle}{\sqrt{2}}.$$(2) For a basis $i$ and a bit $s \in \{0, 1\}$, Alice’s encoding state $\psi_{i,s}$ is the pure state of the qubit given by $$\psi_{0,0} = |+\rangle, \quad \psi_{0,1} = |-\rangle, \quad \psi_{1,0} = |{+i}\rangle, \quad \psi_{1,1} = |{-i}\rangle, \quad \psi_{2,0} = |0\rangle, \quad \psi_{2,1} = |1\rangle.$$(3)^✦

A pure state $\psi$ of $S$ is an eigenstate of an observable $A$ on $S$ with eigenvalue $\mu$, a real number, when $A|\psi\rangle = \mu|\psi\rangle$.^✦ To the bit $s$ belongs the sign $(-1)^s$.^✦

Measurements

A measurement on $S$ with a finite set of outcomes is a family of operators $M_o$ on the state space of $S$, one for each outcome $o$, satisfying the completeness relation $$\sum_o M_o^\dagger M_o = 1.$$(4)^✦ The positive operator valued measure of a measurement $\{M_o\}$ on $S$ consists of the positive operators $E_o = M_o^\dagger M_o$, the effects of the measurement.^✦ Measuring $\{M_o\}$ on a state $\rho$ of $S$ gives the outcome $o$ with a probability given by the Born rule, $$p(o) = \operatorname{Tr}(E_o \rho) = \operatorname{Tr}(M_o \rho M_o^\dagger).$$(5)^✦

Let $A$ be an observable with $A^2 = 1$. The projectors $P_\pm$ of $A$ onto its eigenspaces of eigenvalue $+1$ and $-1$ are $$P_\pm = \tfrac{1}{2}(1 \pm A),$$(6) The sign measurement of $A$ is the measurement on $S$ with two outcomes, $M_0 = P_+$ for the eigenvalue $+1$ and $M_1 = P_-$ for $-1$.^✦

Each $O_i$ squares to the identity, $$O_i^2 = 1.$$(7)^✦

Bob’s measurement in basis $i$ is the sign measurement $M^{(i)}$ of $O_i$.^✦ We write $p_i(o \mid \psi)$ for the probability (5) of the outcome $o$ when $M^{(i)}$ is performed on the state $|\psi\rangle\langle\psi|$ of a pure state $\psi$.

The Claim

What Is Checked

The claim is stated over the six states $\psi_{i,s}$ of (3), the observables $O_i$ and their eigenstates, and the measurements $M^{(i)}$ with the Born probabilities (5) of their outcomes on the states $|\psi_{i,s}\rangle\langle\psi_{i,s}|$.

About these constructions the library proves (7) for each $i$, so that the projectors (6) of $O_i$ satisfy the completeness relation (4) and $M^{(i)}$ is a measurement. It also proves that each of the six vectors in (3) has unit norm, and that the projector onto a unit vector is a density operator, so that each $|\psi_{i,s}\rangle\langle\psi_{i,s}|$ is a state.