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Put $\delta =\pi /(2n)$. Between successive measurement bases, the squared overlap is $c={\cos }^2\delta$ for retaining the same label and $s={\sin }^2\delta$ for changing label. The first step obeys the same rule if we regard the initial state as label 0 at angle zero. Thus the recorded labels form a Markov chain with transition matrix $\left(\begin{array}{cc}c& s\\ s& c\end{array}\right)$.

Let $q_k$ be the probability of label 0 after step $k$. Then $q_0=1$ andSince $c-s=\cos (\pi /n)$, iteration yieldsThis is the total final-outcome probability, including histories that change label an even number of times. The smaller quantity ${\cos }^{2n}(\pi /(2n))$ counts only the history with label 0 at every step and is not the requested total. For example, at $n=3$ the total is $9/16$, whereas the uninterrupted history has probability $27/64$. At $n=1$, the final basis is $\{|1⟩,|0⟩\}$ up to signs, so the total probability of label 0 is zero, as the formula also gives. This distinguishes nonselective measurement-guided Zeno transfer from postselection on one record.

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.