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At the last step, $|{\psi }_{0,n}⟩=|1⟩$ and $|{\psi }_{1,n}⟩=|0⟩$. Ignoring the recorded outcome, the final density matrix isFor large $n$, the Taylor expansion $\log \cos x=-x^2/2+O(x^4)$ givesThus $q_n\to 1$, and the final state tends to $|1⟩⟨1|$, or $|1⟩$ as a pure-state representative. In fact $1-q_n\sim {\pi }^2/(4n)$, and its trace distance from $|1⟩⟨1|$ is exactly $1-q_n$. With records retained, the final conditional state is $|1⟩$ with probability tending to one. These are compatible descriptions, not a coherent sum of the two outcome branches. This quantum Zeno effect guides a known preparation through slowly rotating measurement bases; it is not a unitary Pauli X gate acting coherently on an arbitrary unknown input.

In the ideal state-selective fluorescence detection model, driving the $a\leftrightarrow c$ optical transition produces a bright record only for $a$, while $b$ is dark. A sufficiently discriminating pulse therefore measures the basis $\{|a⟩,|b⟩\}$. The optical cycle is assumed to return population to the appropriate logical state without leakage. Ignoring the recorded outcome removes the off-diagonal quantum coherence between $a$ and $b$.

Take $n$ equally spaced measurements at $t_j=j{T}_{\pi }/n$, including the final time. Between checks the resonant drive rotates through angle $\Omega T_{\pi }/n=\pi /n$. The probability of changing basis state in one interval isAfter each ideal projective measurement, the populations therefore follow the symmetric two-state Markov chainThe population difference is multiplied by $1-2p=\cos (\pi /n)$ at each step. Starting in $a$, the nonselective two-level Zeno dynamics givesFor $n=64$, the exact final population is $P_b\simeq 0.03712$, while the requested large-$n$ estimate is ${\pi }^2/256\simeq 0.03855$, about four percent. Thus the transition that an uninterrupted pi pulse would complete is strongly inhibited.

If “leaving” instead means departing from $a$ at least once in the complete measurement record, use the repeated projective-measurement survival probability. The probability that every check returns $a$ isIt has the same leading estimate ${\pi }^2/(4n)$, but is not exactly the final $b$ population: a trajectory can leave $a$ and subsequently return. Keeping these events separate also distinguishes postselected records from an experiment that measures only final populations.

The mechanism is the quadratic short-time quantum survival probability: each short interval has transition probability of order $(T_{\pi }/n{)}^2$, so $n$ checks leave only an order-$1/n$ total transition probability. The quantum Zeno effect here requires effective, sufficiently fast measurements; a finite-duration optical probe is a physical interaction whose imperfections must be included when comparing with data.

The beryllium-ion quantum Zeno experiment of 1990 used the same ground-state transition and a finite-duration optical probing sequence, followed by a calibrated ensemble fluorescence readout. Its measured final populations motivate the nonselective model; the instantaneous projections here idealize those physical pulses.

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.

From b, the excited-state probability isFor a single short measurement interval $\Delta t$ beginning in $|e,0⟩$, requiring $P_e>0.9$ before the first departure from unity givesA zero-photon outcome resets the conditional state to $|e,0⟩$. After $n=T/\Delta t$ such tests, the probability of the all-zero-photon record isThus a 90 percent survival requirement over a specified total duration $T$ is stronger than the single-interval condition: approximately $\Delta t<-\ln (0.9)/(g^{2}T)$ in the short-interval limit. The limiting preservation as $\Delta t\to 0$ is the quantum Zeno effect.

The measurement instrument matters. If photons are absorbed and their first detection removes the excitation, this survival probability also equals the unconditional excited population. For ideal nondestructive photon-number measurements, excitation can transfer back after a one-photon result; the unconditional result after $n$ equal intervals is instead $P_e^{(n)}=[1+{\cos }^n(2g\Delta t)]/2$. It also tends to one for fixed $T$ as the intervals shrink. Neither interpretation guarantees indefinite 90 percent survival merely by imposing a 90 percent bound on each separate interval.

Continue with the initial quantum state ${\psi }_1$ from part (ii), and let $\Delta t=T/N$. Conditional on every preceding outcome being $+\hslash /2$, the post-measurement state after each ideal projective measurement is again ${\psi }_1$, up to global phase. Part (ii) therefore gives the same conditional probability ${\cos }^2(\omega \Delta t)$ for each successive outcome. Multiplying these conditional probabilities yields the repeated projective-measurement survival probabilityThis multiplication uses the quantum state reset at each successful projective measurement; the outcomes need not be assumed statistically independent.

For fixed $T$, put $z=\omega T$. The Taylor expansion ${\cos }^{2}u=1-u^2+O(u^4)$ gives the quadratic short-time quantum survival probability. For sufficiently large $N$, taking logarithms givesHere the remainder uses Big O notation. Exponentiating provesThis is the quantum Zeno effect: increasingly frequent ideal projective measurements inhibit the transition away from the initial quantum state. The departure probability in a short interval is quadratic in its duration, so $N$ intervals of duration $T/N$ produce a total leading departure probability proportional to $T^2/N$. Without intervening measurements, the return probability at time $T$ would be ${\cos }^2(\omega T)$. The result concerns obtaining $+\hslash /2$ at every measurement in the ideal limit of instantaneous measurements with unitary time evolution between them.

Let ${\theta }_j=j\pi /N$ and retain the positive spin beam at every stage. The overlap of successive positive eigenstates isThus the unconditional survival probability, relative to the original intensity, isThe last retained quantum state is $|{\psi }_{\pi }^{+}⟩=|\downarrow ⟩$, so its final negative-$z$ result is certain conditional on survival and adds no further factor. In particular $p_2=1/4$, $p_{10}\simeq 0.7805$, and $p_{100}\simeq 0.9756$. Since $\log \cos x=-x^2/2+O(x^4)$,This is a changing-measurement-axis form of the quantum Zeno effect: frequent selections steer the surviving spin through a continuous succession of nearby eigenstates.

For a normalized quantum state $\psi$ and a self-adjoint operator $H$ on a finite-dimensional Hilbert space, expand its unitary time evolution:Multiplying by its complex conjugate gives the survival probabilityThe function is even in $t$, so its odd coefficients vanish. The coefficient $(\Delta H{)}^2=⟨H^2⟩-⟨H{⟩}^2$ is the variance of the energy. This quadratic departure yields the quantum Zeno effect when ideal projective measurements repeatedly reset the quantum state.

For a normalized quantum state $\psi$ and a time-independent Hamiltonian operator $H$, consider ideal projective measurements of the orthogonal projection $P=|\psi ⟩⟨\psi |$ at intervals $\Delta t=T/N$. Write $U=e^{-iH\Delta t/\hslash }$. A successful check leaves the post-measurement state $\psi$ up to global phase. Thus each successive success, conditional on earlier successes, has conditional probability $|⟨\psi ,U\psi ⟩{|}^2$, and multiplication gives the displayed formula. Equivalently,This is the probability of the entire sequence of successful checks. It underlies the quantum Zeno effect.