The Exercise

Phase damping of a qubit, with a parameter $\lambda$ between $0$ and $1$, and the phase flip channel, which leaves the qubit alone with some probability and otherwise applies the Pauli operator $Z$, are each given by a pair of operation elements, and the two pairs look quite different. The exercise asks for a $2 \times 2$ unitary matrix $u$ that turns one pair into the other, the phase flip elements being linear combinations of the phase damping elements with coefficients from the rows of $u$, once the probability of no flip is chosen as a suitable function of $\lambda$.

A Qubit

The state space $H$ of the qubit is a two-dimensional complex Hilbert space with a fixed orthonormal basis $|0\rangle, |1\rangle$. For $i = 0, 1$ the projector $P_i$ on $H$ is the rank-one operator $P_i = |i\rangle\langle i|$ onto the basis vector $|i\rangle$.^✦

Two Pairs of Operation Elements

The phase damping elements with a real parameter $\lambda$ are the ordered pair of operators on $H$ $$(E_k)_{k=0,1} = (E_0, E_1),$$(1) where, with matrices in the basis $|0\rangle, |1\rangle$, $$E_0 = P_0 + \sqrt{1-\lambda}\,P_1 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\lambda} \end{pmatrix},$$(2) $$E_1 = \sqrt{\lambda}\,P_1 = \begin{pmatrix} 0 & 0 \\ 0 & \sqrt{\lambda} \end{pmatrix}.$$(3)

The no-flip probability $\alpha$ belonging to $\lambda$ is the real number $$\alpha = \frac{1 + \sqrt{1-\lambda}}{2}.$$(4)^✦

The phase flip elements belonging to $\lambda$ are the ordered pair of operators on $H$ $$(\tilde E_k)_{k=0,1} = (\tilde E_0, \tilde E_1),$$(5) where, with $\alpha$ the no-flip probability (4), $$\tilde E_0 = \sqrt{\alpha}\, I,$$(6) $$\tilde E_1 = \sqrt{1-\alpha}\,(P_0 - P_1) = \sqrt{1-\alpha}\, Z,$$(7) with $I$ the identity on $H$ and $Z = P_0 - P_1$ the Pauli operator in the basis $|0\rangle, |1\rangle$.

The mixing matrix $u$ belonging to $\lambda$ is the $2 \times 2$ complex matrix $$u = \begin{pmatrix} \sqrt{\alpha} & \sqrt{1-\alpha} \\ \sqrt{1-\alpha} & -\sqrt{\alpha} \end{pmatrix},$$(8)^✦ with $\alpha$ again given by (4), and with rows and columns indexed by $0$ and $1$, like the operation elements.

The Claim

The matrix $u$ answers the exercise.

What Is Checked

The theorem is stated over the phase damping elements (1) and the phase flip elements (5), two ordered pairs of operators on a qubit built from the projectors $P_0$ and $P_1$ of a fixed orthonormal basis, and over the mixing matrix $u$ of (8), whose entries depend on $\lambda$ through the no-flip probability $\alpha$. What the library proves about these constructions is the unitarity (9) of $u$: the matrix that carries the one pair into the other is a unitary.