The Exercise

A single photon is shared between two optical modes. With the photon in the second mode it encodes the logical state $|0_L\rangle$, with the photon in the first mode the logical state $|1_L\rangle$, and a superposition $c_0|0_L\rangle + c_1|1_L\rangle$ is a qubit carried by one photon. A beamsplitter of angle $\theta$ mixes the two modes. Send the photon through a beamsplitter of angle $\pi/4$, shift the phase of the second mode by $\varphi$, and recombine the modes at a beamsplitter of angle $-\pi/4$: this is the interferometer of Mach and Zehnder.^✦ The exercise is to show that a beamsplitter of angle $-\theta$ undoes one of angle $\theta$, and to compute the output of the interferometer for the input $c_0|0_L\rangle + c_1|1_L\rangle$. On the qubit the photon carries, the interferometer is the rotation $R_x(\varphi)$ about the $x$ axis, up to the global phase $e^{i\varphi/2}$.^✦

A Photon in Two Modes

A quantum system $S$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ The composite system $S \otimes T$ of two quantum systems $S$ and $T$ is the quantum system whose state space is the tensor product of their state spaces.^✦ The qubit is the two level system with state space $\mathbb{C}^2$; the standard basis vectors of $\mathbb{C}^2$ are its computational basis states $|0\rangle$ and $|1\rangle$.

A pure state $|\psi\rangle$ of a system $S$ is a vector of unit norm in the state space of $S$, determined physically up to a global phase.^✦ For pure states $|\psi\rangle$ of $S$ and $|\chi\rangle$ of $T$, the product $|\psi\rangle \otimes |\chi\rangle$ is the pure state of $S \otimes T$ whose vector is the tensor product of the two vectors.^✦

The photon lives in the composite system of two qubits, one qubit for each mode. The logical zero $|0_L\rangle$ is the pure state $|01\rangle = |0\rangle \otimes |1\rangle$ of two qubits; the logical one $|1_L\rangle$ is the pure state $|10\rangle = |1\rangle \otimes |0\rangle$. For complex numbers $c_0$ and $c_1$, the dual rail superposition is the vector $$c_0\,|0_L\rangle + c_1\,|1_L\rangle$$(1)^✦ in the state space of the two qubits. When $|c_0|^2 + |c_1|^2 = 1$, the dual rail state with amplitudes $c_0$ and $c_1$ is the pure state with this vector, the general state of a single photon in the two modes.^✦

Beamsplitters and the Interferometer

A unitary $U$ of a quantum system $S$ is a closed system evolution of $S$: 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 $U|\psi\rangle$.^✦ For two unitaries $U$ and $V$ of $S$, the unitary $U$ after $V$ is the composite $UV$, first $V$ and then $U$, and the identity evolution of $S$ is the unitary $1$. For unitaries $U$ of $S$ and $V$ of $T$, the product unitary $U \otimes V$ is the unitary of $S \otimes T$ whose operator is the tensor product of the two operators.^✦

The projector $\Pi$ onto the dual rail plane, spanned by $|0_L\rangle$ and $|1_L\rangle$, and the generator $G$ of rotations in that plane are the operators on two qubits $$\Pi = |0_L\rangle\langle 0_L| + |1_L\rangle\langle 1_L| , \qquad G = |1_L\rangle\langle 0_L| - |0_L\rangle\langle 1_L| .$$(2) For a real angle $\theta$, the dual rail beamsplitter $B(\theta)$ is the unitary of two qubits with operator $$B(\theta) = 1 + (\cos\theta - 1)\,\Pi + \sin\theta\, G .$$(3)^✦

The phase shift gate $P(\alpha)$ of angle $\alpha$ is the unitary of the qubit diagonal in the computational basis, $P(\alpha) = \mathrm{diag}(1, e^{i\alpha})$.^✦ On two qubits it acts on the second qubit, as the product unitary $1 \otimes P(\alpha)$ of the identity evolution of the first qubit with $P(\alpha)$ on the second.^✦ This gate multiplies $|0_L\rangle = |01\rangle$ by $e^{i\alpha}$ and leaves $|1_L\rangle = |10\rangle$ alone, a phase shift of the second mode.

Two beamsplitters in series, of angle $\theta$ and then $-\theta$, form the unitary $$C(\theta) = B(-\theta)\,B(\theta)$$(4)^✦ of two qubits. The interferometer of Mach and Zehnder with phase $\varphi$ is the unitary $$M(\varphi) = B(-\pi/4)\,\bigl(1 \otimes P(\varphi)\bigr)\,B(\pi/4) .$$(5)^✦

The Claim

What Is Checked

The claim is stated over the composite system of two qubits and its dual rail states (1), and over the beamsplitter (3) built from the projector and the generator of (2). The two circuits it is about are $C(\theta)$ of (4) and $M(\varphi)$ of (5).

About these constructions the library proves that $B(\theta)$ is unitary for every real $\theta$, and that the dual rail superposition has unit norm whenever $|c_0|^2 + |c_1|^2 = 1$. It proves the same of both outputs: the amplitudes in (7) and in (8) have squared moduli summing to one, so both are dual rail states. The phase shift gate is unitary, its diagonal entries having unit modulus. Products and tensor products of unitaries are unitary, and a unitary preserves norms, so it carries a pure state to a pure state.