The Exercise

A box applies one of three unitaries to a qubit, and we do not know which. Each is a reflection about a state of the qubit: about the basis state $|1\rangle$, about the eigenstate $|-\rangle$ of the Pauli operator $X$, or about the eigenstate $|-i\rangle$ of $Y$. The box is used once. Two qubits are prepared in the Bell state $|\Phi^+\rangle$, the first of them is sent through the box, and the pair is measured in the Bell basis; the outcome of that one measurement is to name the box.

A second question compares the reflections about two states $|\varphi\rangle$ and $|\psi\rangle$ of one system, when a unitary $U$ prepares $|\psi\rangle$ from $|\varphi\rangle$.

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ The composite system $S \otimes T$ of two systems $S$ and $T$ is the quantum system whose state space is the tensor product of their state spaces.^✦

A pure state $|\psi\rangle$ 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 $|\psi\rangle$ gives the state $|\psi\rangle\langle\psi|$, the projector onto its vector. For pure states $|\psi\rangle$ of $S$ and $|\varphi\rangle$ of $T$, the product $|\psi\rangle|\varphi\rangle$ is the pure state of $S \otimes T$ whose vector is $|\psi\rangle \otimes |\varphi\rangle$.^✦

The qubit is the quantum system whose state space is $\mathbb{C}^2$ itself.^✦ For a bit $i$, its basis state $|i\rangle$ is the pure state whose vector is the standard basis vector $e_i$ of $\mathbb{C}^2$.^✦ The states $|-\rangle$ and $|-i\rangle$ are the pure states of the qubit whose vectors are the columns $$|-\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \qquad |-i\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix}.$$(1)

The Bell Basis

For bits $x$ and $y$, the Bell state $|\beta_{xy}\rangle$ is the pure state of the composite system of two qubits whose vector is $$|\beta_{xy}\rangle = \frac{1}{\sqrt{2}} \bigl( |0\rangle|y\rangle + (-1)^x\, |1\rangle|\bar y\rangle \bigr),$$(2)^✦ where $\bar y$ is the flip of $y$ and $|0\rangle|y\rangle$ and $|1\rangle|\bar y\rangle$ are products of basis states. The four Bell states form the Bell basis, an orthonormal basis of the state space of two qubits indexed by the pairs $(x, y)$.^✦ The Bell state $|\Phi^+\rangle$ is the pure state of two qubits whose vector is $|0\rangle|0\rangle + |1\rangle|1\rangle$ divided by its norm $\sqrt{2}$.^✦

Unitaries and Reflections

A unitary $U$ of a system $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = I$, where $I$ is the identity operator of that space.^✦ A unitary $U$ of $S$ carries a pure state $|\psi\rangle$ of $S$ to the pure state $U|\psi\rangle$ of $S$.^✦ For unitaries $U$ and $V$ of $S$, the composite $UV$ is the unitary of $S$ that applies $V$ first and then $U$, and the adjoint $U^\dagger$ is the unitary of $S$ whose operator is the adjoint of that of $U$.

For a unitary $U$ of $S$ and a unitary $V$ of $T$, the product unitary $U \otimes V$ is the unitary of $S \otimes T$ whose operator is the tensor product of the two.^✦ For a unitary $U$ of $S$ and the identity $I$ of $T$, the gate $U \otimes I$ is the unitary of $S \otimes T$ that applies $U$ to the first system and leaves the second alone.^✦

For a pure state $|\psi\rangle$ of $S$, the reflection $U_\psi$ about $|\psi\rangle$ is the unitary of $S$ $$U_\psi = I - 2\,|\psi\rangle\langle\psi| .$$(3)^✦ The three oracles, numbered $0$, $1$ and $2$, are the reflections of the qubit about $|1\rangle$, $|-\rangle$ and $|-i\rangle$: $$O_0 = I - 2\,|1\rangle\langle 1|, \qquad O_1 = I - 2\,|-\rangle\langle -|, \qquad O_2 = I - 2\,|-i\rangle\langle -i| .$$(4)

For $j \in \{0, 1, 2\}$, the state $|\psi_j\rangle$ of two qubits after the oracle $O_j$ is the pure state obtained from $|\Phi^+\rangle$ by the gate $O_j \otimes I$ on the first qubit: $$|\psi_j\rangle = (O_j \otimes I)\,|\Phi^+\rangle .$$(5)^✦

The Bell Measurement

A measurement $M$ on a system $S$, 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 $\sum_i M_i^\dagger M_i = I$.^✦ The positive operator valued measure (POVM) of a measurement $M$ on $S$ is the family of effects $E_i = M_i^\dagger M_i$, one for each outcome $i$, and the probability of $i$ when $M$ is performed on a state $\rho$ of $S$ is the Born rule value $$p(i) = \operatorname{Tr}(E_i \rho) = \operatorname{Tr}(M_i \rho M_i^\dagger) .$$(6)

The projective measurement in an orthonormal basis $\{|b_i\rangle\}$ of $S$ is the measurement whose operator for the outcome $i$ is the projector $|b_i\rangle\langle b_i|$.^✦ The Bell measurement is the projective measurement of two qubits in the Bell basis: its outcomes are the pairs $(x, y)$ of bits, and the operator of $(x, y)$ is $|\beta_{xy}\rangle\langle\beta_{xy}|$.^✦

The Bell outcome of the oracle $O_j$ is the pair of bits $(x_j, y_j)$ given by $$(x_0, y_0) = (1, 0), \qquad (x_1, y_1) = (0, 1), \qquad (x_2, y_2) = (1, 1).$$(7)^✦

The Claim

The three outcomes (7) are distinct, so the one Bell measurement names the box.

What Is Checked

The first claim is stated over the Born probabilities (6) of the Bell measurement on the states (5), each the oracle (4) applied to the first qubit of $|\Phi^+\rangle$, at the outcomes (7). The second is stated over the reflection (3), the action of a unitary on a pure state, and the composite and the adjoint of unitaries.

Of these constructions the library proves that each Bell vector (2) has unit norm, that the four are orthonormal, and that their number equals the dimension $4$ of the state space of two qubits, so that they form a basis. The projectors onto an orthonormal basis satisfy the completeness relation, the effects $M_i^\dagger M_i$ are positive and sum to $I$, and $|\psi\rangle\langle\psi|$ is a density operator. The reflection $U_\psi$, the product $U \otimes V$, the composite $UV$ and the adjoint $U^\dagger$ are unitary, while $|\Phi^+\rangle$, the states (1), $U|\psi\rangle$ and the product $|\psi\rangle|\varphi\rangle$ are unit vectors.