Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 5

The normalized input density operator isA quantum measurement in the computational basis has projectors $P_j=|j⟩⟨j|$. Discarding its recorded outcome gives the dephasing channelThe original off-diagonal coherence disappears. If the outcome is retained instead, the conditional state is $|0⟩$ or $|1⟩$ and the record carries information about which occurred. Thus discarding the record is an essential part of this channel specification.

A random Pauli Z gate with equal application and nonapplication probabilities givesThis is exactly the same dephasing channel as the unrecorded measurement, for every input density operator, including one entangled with an auxiliary system. Individual trials differ: conditioned on the classical random choice, the state is the pure state $|\psi ⟩$ or $Z|\psi ⟩$, generally neither a computational basis vector. Knowledge of that choice permits recovery by the same Pauli Z gate; a measurement outcome does not restore an unknown destroyed superposition.

Use the rotation convention $R_z(\varphi )=e^{-i\varphi Z/2}=\mathrm{diag}(e^{-i\varphi /2},e^{i\varphi /2})$. For uniform $\varphi$ modulo $2\pi$,because the off-diagonal entry acquires $e^{-i\varphi }$ with zero average. Each trial retains a pure state with a randomly rotated relative phase, reversible if $\varphi$ is known. All three procedures therefore give identical statistics for every measurement on the unconditioned system, but their records and conditional ensembles differ. A density operator does not identify its preparation ensemble uniquely; access to a record changes which information is available.

For short times, survival under unitary time evolution is $1-(\Delta H{)}^2{t}^2/{\hslash }^2+O(t^3)$. Dividing a fixed time into $N$ intervals and projecting back after each gives survival approaching one as $N\to \infty$. This suppression of departure by repeated projective measurements is the quantum Zeno effect.

For measurement-guided Zeno transfer, rotate the measured basis gradually to transport a known state. LetMeasure in these orthonormal bases successively for $j=1,\dots ,N$. Starting in $|0⟩=|b_0⟩$, the probability to follow the $b_j$ outcome at every step isIts final state is $|b_N⟩=|1⟩$. If every outcome is retained without postselection, the final probability for $|1⟩$ is $[1+{\cos }^N(\pi /N)]/2$, also approaching one; histories with two switches can return to that branch. These two finite-$N$ probabilities should not be confused.

For a general unknown input, the same slowly turning measurements first resolve its amplitudes into the two branches, then drag each branch separately. In the large-$N$ limit, after discarding the records,The first basis approaches the computational one as $N$ grows. Inserting an initial computational measurement makes this branch decomposition exact at the outset, without changing the limiting result. Relative phase is lost, so this procedure transports a known basis state with high probability rather than implementing a coherent Pauli X gate on an arbitrary qubit.

Quantum error correction detects an error syndrome and actively restores a logical state within a correctable error set. A decoherence-free subspace instead encodes into a subspace on which every allowed noise operator acts as the same scalar; no relative logical phase or mixing is produced by that noise. It relies on the specified correlation structure, and does not protect against arbitrary independent errors.

For collective rotations $R_z(\varphi )\otimes R_z(\varphi )$,Thus every encoded superposition $\alpha |01⟩+\beta |10⟩$ is invariant. Equivalently it lies in the zero-eigenvalue subspace of $Z\otimes I+I\otimes Z$.

By contrast,The relative phase changes by $2\varphi$, although $Z\otimes Z$ acts as $+I$ on both basis states. Invariance under one discrete operation does not establish invariance under its continuous rotation family. A concrete counterexample is $\varphi =\pi /2$: the equally weighted encoded plus state becomes the encoded minus state, up to a global phase, and is orthogonal to its original state.

Let $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$ be the Pauli X eigenstates. EncodeBoth have zero total $X$ eigenvalue, so $R_x(\varphi )\otimes R_x(\varphi )$ is the identity on their span. This is the decoherence-free subspace obtained from the previous code by applying a Hadamard gate to each physical qubit.

Since $Z|+⟩=|-⟩$ and $Z|-⟩=|+⟩$, a correlated $Z\otimes Z$ error exchanges the codewords. It acts as logical $X_L$, not as harmless identity. The Pauli Y gate obeys $Y|+⟩=-i|-⟩$ and $Y|-⟩=i|+⟩$, soThe phases cancel within each codeword. Thus either unknown correlated gate gives a logical bit error on a general encoded state, though a logical $X_L$ eigenstate is unchanged as a physical ray. The code prevents collective $x$ rotations, not every possible correlated error.