Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 53 / 3

Let $c=⟨R_0|R_1⟩$. The joint state after the environment interaction is $\alpha |0⟩|R_0⟩+\beta |1⟩|R_1⟩$. Taking its partial trace over the environment gives the exact reduced density operatorThe conjugate in the upper off-diagonal entry follows from tracing $|R_0⟩⟨R_1|$. Normalization of the environment states and the Cauchy-Schwarz inequality imply $|c|\le 1$.

For comparison, a phase-flip channel sends $\rho$ to $(1-p)\rho +pZ\rho Z$, multiplying its off-diagonal entries by the real number $1-2p$. Consequently the claimed representation is valid, for a general input, provided the environment overlap is real, withThis is a genuine probability between zero and one. The environment maps printed in the question do not themselves imply that $c$ is real. For example, choose $|R_1⟩=i|R_0⟩$, so $c=i$, and take $\alpha =\beta =1/\sqrt{2}$. The qubit becomes $(|0⟩+i|1⟩)/\sqrt{2}$, whose imaginary off-diagonal entries cannot come from any probabilistic mixture of $\rho$ and $Z\rho Z$. These environment maps are a valid isometry and can be extended to a unitary interaction, so this is an actual counterexample to the unrestricted premise.

The general complex environment overlap in qubit dephasing has an exact corrected description. Write $c=r{e}^{i\phi }$, $0\le r\le 1$, and let $D_{\phi }=\mathrm{diag}(1,e^{i\phi })$. ThenAt $r=0$, choose any phase. The dephasing channel includes both random phase errors and, when the overlap is complex, a coherent diagonal rotation. The subsequent probability formulas use the intended real-overlap phase-flip model.

Subtract the two density operators. The difference is Hermitian with zero diagonal and off-diagonal entry $\alpha \overline{\beta }(1-\overline{c})$. A matrix $\left(\begin{array}{cc}0& a\\ \overline{a}& 0\end{array}\right)$ has eigenvalues $\pm |a|$, because its square is $|a{|}^{2}I$. Its two singular values are therefore both $|a|$. By the definition of the trace norm,For the intended real-overlap phase-flip channel, $1-c=2p$, givingThis vanishes for a computational basis state, since a phase flip then changes only an unobservable global phase. Its largest value at fixed $p$ is $p$, attained when $|\alpha |=|\beta |=1/\sqrt{2}$. The general formula remains correct even when the probabilistic premise of part 1 fails.

Assume the three physical phase-flip channels act independently, with the same probability $p$ on each qubit, and encoding, recovery and decoding are ideal. Zero or one physical error is corrected. HenceThe probability of an uncorrectable two- or three-site error is $p_L=3p^2(1-p)+p^3=3p^2-2p^3$.

One must also choose the logical convention consistently with the output phase error specified here. The usual phase-flip repetition code with codewords $|+++⟩$ and $|---⟩$ has residual operator $Z_1{Z}_2{Z}_3$ after two or three physical errors: their error syndrome equals that of the complementary zero- or one-error pattern. This operator exchanges those codewords, so plain inverse encoding would give logical $X$, not $Z$.

To obtain the stated logical phase-flip convention, use the logical Hadamard convention for a phase-flip repetition code. Put a logical Hadamard gate before that encoding and after its decoding. The encoded state iswhere the plus and minus states in these codewords are the real Hadamard basis states. The uncorrectable exchange is now conjugated to $HXH=Z$. With that convention the decoded density operator and its trace distance are exactlyFor $0<p<1/2$, $p-p_L=p(1-p)(1-2p)>0$, so coding reduces the trace distance. The displayed recovery probability means success of correction as a quantum operation on an arbitrary unknown input. If the particular input is a $Z$ eigenstate, even the logical phase-error branch represents the same physical state, and the probability of recovering that particular state is one.

Independence is an extra noise assumption needed for the numerical probability. If all three physical qubits flip together with probability $p$, each marginal still has the given phase-flip channel, but the logical failure probability is $p$, not $3p^2-2p^3$. More generally, for a classical joint phase-error distribution with a random number $K$ of flips, $P_{\mathrm{rec}}=\mathrm{Pr}(K\le 1)$ and $p_L=\mathrm{Pr}(K\ge 2)$; the same decoded mixture holds in the stated logical convention. Thus marginal error probabilities alone do not determine the requested recovery probability. A complex environment overlap likewise requires the coherent rotation to be retained or separately compensated before this stochastic phase-error analysis applies.