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Apply part (i) for every positive integer $K$ and discard the countable union of its exceptional null sets. A sample path bounded below never reaches $-K$ for some such $K$, and must therefore converge to a finite limit of a sequence.

Apply the same argument to the martingale $(-M_n)$. Outside one more null set, every sample path bounded above also converges finitely. Thus a sample path which does not converge finitely must be unbounded in both directions. Each finite initial segment has finite values, so removing any such segment preserves unboundedness. It follows that the limit superior is $+\infty$ and the limit inferior is $-\infty$. ConsequentlyThis proves the bounded-increment martingale convergence-or-oscillation dichotomy; in particular one-sided divergence to infinity is impossible with positive probability.

Suppose the events $(B_n)$ are mutually independent events. Then $B_k$ is independent of ${\mathcal{F}}_{k-1}$, first by factorization against finite intersections of previous events and then by part (a). ThereforeIf ${\sum }_k\mathbb{P}(B_k)=\infty$, the sum in part (ii) is infinite almost surely, and consequentlyThis is the Second Borel-Cantelli lemma. In fact part (ii) also yields probability zero in the convergent case for these independent events.

Both $X_n$ and $A_n$ are increasing, so each has a limit of a sequence in $[0,\infty ]$. Apply part (c)(ii) to the martingale $M_n=X_n-A_n$, whose bounded increments were verified in part (i).

On $\{X_{\infty }<\infty ,A_{\infty }=\infty \}$ we would have $M_n\to -\infty$. On $\{X_{\infty }=\infty ,A_{\infty }<\infty \}$ we would have $M_n\to +\infty$. Both events have probability zero by the convergence-or-oscillation conclusion. Hence the two divergence events agree up to a null set. Since $X_{\infty }=\infty$ means precisely that infinitely many $B_n$ occur,Taking probabilities proves the displayed equality in the question. This stronger agreement of events is the Conditional Borel-Cantelli lemma.

The floor function gives the deterministic boundfor every outcome, including $X=1$. Thus all three convergence assertions hold: there is pointwise, and hence almost sure convergence, as well as convergence in probability and convergence in L2. In detail, for $\epsilon >0$ the probability $\mathbb{P}(|X-X_n|>\epsilon )$ is zero as soon as $2^{-n}\le \epsilon$; alsoThe uniform distribution even gives the exact second moment of the error by integrating over the dyadic partition:

For $1\le j\le n$, repeated use of the floor function identity givesThus the filtration satisfies ${\mathcal{F}}_n=\sigma (X_n)$. Each positive-probability atom of a sigma-algebra here has the formwith probability $2^{-n}$; $\{X=1\}$ is a possible additional null set atom. Averaging over each dyadic interval gives a versionFor a globally defined version of the right side, assign any value, for example zero, on $\{X=1\}$.

Indeed, the displayed random variable is ${\mathcal{F}}_n$-measurable and bounded by $\sup |h|$. On each $A_{n,k}$ its integral isby the uniform distribution. Every member of ${\mathcal{F}}_n$ is a union of the displayed sets and the possible zero-probability atom. Summing these identities therefore verifies the defining conditional expectation property on all of ${\mathcal{F}}_n$.

Choose $Z_{\infty }$ as the pointwise limit of a sequence of $(Z_n)$ where it exists, and as zero otherwise. The convergence event and every $Z_n$ are $\sigma (X)$-measurable, so this is an exactly $\sigma (X)$-measurable version, bounded by $L$. Part (a)(ii), the Doob-Dynkin lemma, supplies a Borel measurable function $g$ on $[0,1]$ such that $Z_{\infty }=g(X)$. Applying scalar clipping to $g$ with bounds $[-L,L]$ preserves that equality and ensures $|g(u)|\le L$ everywhere.

Part (c)(ii) and the conditional expectation formula from part (b)(iv) now giveSince the indices $n$ form a countable family, these identities may be taken on one common full-probability event.

For fixed $n,k$, both the slope and the displayed average are constant on the atom $A_{n,k}$, whose probability is $2^{-n}>0$. An equality holding almost surely of these constants forces the deterministic equalitySum over $0\le k<j$. The telescoping sum givesThus the desired identity holds at every dyadic endpoint, including $1$, without evaluating the undefined slope at $X=1$.

Finally, given $x\in [0,1]$, take dyadic endpoints $x_n\to x$. The Lipschitz condition implies $|f(x_n)-f(x)|\le L|x_n-x|$, and the bound on the Borel measurable function $g$ givesPassing to the limit of a sequence provesThis is a martingale proof of the integral representation of a Lipschitz function. It constructs a bounded Lebesgue integrable function representing the increments of $f$, and in particular proves absolute continuity of $f$. No prior assumption that $f$ has a derivative is needed.

Each $Z_n$ is a bounded Borel measurable function of $X_n$, including the specified value at $X_n=1$. Therefore $(Z_n)$ is an adapted process for $({\mathcal{F}}_n)$ and consists of integrable random variables.

Fix an atom of a sigma-algebra in ${\mathcal{F}}_n$ with positive probability, corresponding to the dyadic interval $[u,u+\delta )$, with $\delta =2^{-n}$. By the uniform distribution, each half has conditional probability $1/2$. The conditional expectation of $Z_{n+1}$ on that atom is thereforeThe middle values cancel. On the zero-probability atom the conditional expectation can be chosen as zero, agreeing with our version of $Z_n$. Henceso $(Z_n)$ is a martingale. This is the dyadic slope martingale associated with the Lipschitz function $f$.

After $\tau$, the token product remains zero, whereas the deterministic term $nN/3$ continues to increase. Combining this with the pre-elimination calculation in part (ii),The stochastic process is adapted and integrable at each fixed time, and the displayed conditional expectation is nonnegative. Thus $(M_n)$ is a submartingale.

For $N>0$ it is neither a martingale nor a supermartingale. If $a,b,c>0$, there is positive probability that the first player transfers a token to the second in each of the next $a$ rounds. Until the last such round all three remain active, so this event has probability $6^{-a}>0$ and entails $\tau \le a$. At time $n=a$ the displayed conditional expectation is therefore strictly positive on an event of positive probability, ruling out both the zero drift required of a martingale and the nonpositive drift required of a supermartingale. If an initial holding is zero, $\tau =0$ and the same conclusion follows immediately. The degenerate case $N=0$ gives a constant martingale.

Put $P_n=X_n{Y}_n{Z}_n$. Before $\tau$, all three players are active. Each unordered pair is selected with probability $1/3$, and each direction of transfer then has conditional probability $1/2$. Given the current holdings $(x,y,z)$, a transfer within the first pair has mean productFor transfers within the other two pairs the analogous values are $xyz-y$ and $xyz-x$. Averaging over pairs givesAt and after elimination the product is zero, since an eliminated player never re-enters. In particular the stopped process has incrementThe indicator function is ${\mathcal{F}}_n$-measurable, and the expression in parentheses has zero conditional expectation on its support. The stopped process is adapted and integrable: at fixed $n$, its absolute value is bounded by $N^3+nN/3$. HenceThis is the product martingale for three-player token exchange.

Children possess structured expectations about living things, but knowledge of biological words, causal mechanisms and illness risk develops unevenly. Those expectations can shape pandemic explanations and emotions without determining one uniform reaction. Developmental findings are useful for generating predictions about children's interpretations; direct pandemic data are needed before those predictions are treated as demonstrated responses to COVID-19.

Intuitive biology includes expectations about growth, internal properties, needs and the persistence of kinds despite changes in appearance. Susan A. Gelman and colleagues' experiments showed that children can attend to nonobvious internal properties rather than reasoning only from what an object looks like. This challenges a picture of young children as entirely restricted to surface appearance. It does not establish a complete adult theory of physiology or a correct understanding of every invisible cause. An experiment on familiar animals is also not a direct test of understanding a novel infectious disease.^[ref]

Children's explanations can nevertheless extend beyond adult biological boundaries. Deborah Kelemen compared preferences for purpose-based and physical explanations of natural properties. Children accepted teleological explanations more broadly than adults, including for nonliving features. The tasks suggest a readiness to interpret nature in terms of what things are for. Their forced alternatives and wording constrain what follows: the result does not show that every child believes all natural events are intentionally designed, or that a child cannot understand a mechanistic explanation when it is made available.^[ref]

Applied to a pandemic, teleological reasoning could encourage a question about what a virus or restriction is “for,” rather than what caused it. A virus need not be understood as a tiny agent with human intentions for the child to ask that question. Adults can help distinguish the reason for a public-health decision from the biological cause of an illness. This is an educational implication of the developmental account, not an empirical claim that the laboratory study measured children's pandemic beliefs.

Knowledge about germs is a particularly important case of facts versus a causal system. Gregg E. A. Solomon and colleagues presented stories in which similar symptoms arose through different causes and asked about contagion. Across five studies, preschoolers' answers often failed to integrate germs into a consistent biological causal account. Some responses were more compatible with treating particular symptoms as contagious than with tracking the cause. Additional tasks questioned whether children understood germs as distinctively biological agents. Knowing the word “germ” therefore should not be equated with understanding how illness can spread.^[ref]

That distinction yields concrete predictions. A child might overgeneralize danger from any visible sign of illness, fail to understand why someone who appears well can still matter in a disease account, or apply a rule correctly without being able to explain it. Conversely, prior discussion, schooling and family experience can support more coherent reasoning. These possibilities require direct assessment rather than inference from age alone. Children can also learn a practical action through trusted instruction before they possess an adult mechanism; practical participation and conceptual completeness are different outcomes.

Psychological essentialism can support expectations of stable kinds, but it can also encourage an overly fixed distinction between “safe people” and “sick people.” Infection status and a person's moral worth are not the same property. A useful educational account can explain change and recovery without assigning blame to a person or community. This is a reason to examine the interpretation of biological categories, not to attribute prejudice automatically to every child's causal error. Context and what adults say matter alongside the child's prior concepts.

There is direct evidence that school-age children actively sought to understand the pandemic. Lucy Bray and colleagues surveyed 390 children aged seven to twelve and caregivers in six countries during the first wave. The online survey combined closed and open questions about information, knowledge and needs. Parents were a major information source, with school more prominent in Sweden; adults often filtered information, yet children still reported awareness of serious consequences. The study shows active interpretation and unmet information needs rather than simple ignorance protected by adult silence.^[ref]

Its method also limits the conclusion. An online volunteer sample is not representative of all children, countries experienced different conditions, and a factual statement in a survey is not a detailed causal model. Older school-age participants should not be treated as interchangeable with preschoolers in the germ studies. Nor does the survey establish that a particular explanation reduced anxiety or changed behavior: that would require a comparison or intervention design. Cross-study differences in age, questions and context can explain apparent inconsistencies without requiring that one result be dismissed.

Emotional reactions can be understandable even with accurate biological knowledge. Concern about relatives, changes to routines and loss of ordinary contact are not simply failures to understand germs. Information can also interact with perceived control and the consistency of adults' explanations. An implication is to ask what the child thinks and fears before selecting an explanation. Developmental knowledge should help make a conversation interpretable, not be used to assume that a child is incapable of understanding or ought to be unconcerned.

An educational approach can distinguish agents, causes and protective decisions using accessible explanations, ask children to apply an idea to a new case and correct a misunderstanding without shame. Checking transfer is more informative than asking whether the child can repeat a phrase. Explanations can acknowledge uncertainty and distinguish a possibility from its probability. This remains a general communication principle; it is not a substitute for appropriate, time-specific health advice, and the examination does not provide data for choosing a clinical or public-health protocol.

A rigorous follow-up would measure prior biological reasoning and language, elicit pandemic explanations in the child's own words, and separately assess factual knowledge, emotion and action. An age-appropriate explanatory intervention could then be compared with equally supportive information, using delayed tests and unfamiliar scenarios. Family circumstances and educational access would need measurement. Such a design would test whether changing understanding contributes to a better outcome, rather than assuming that knowledge and reassurance are equivalent.

The research predicts a mixture of genuine biological insight, incomplete mechanisms and active meaning-making. It supports listening to children's explanations, differentiating factual labels from causal understanding and testing communication rather than relying on a simple age stereotype. Pandemic reactions arise within families, schools and changing circumstances as well as within the child's developing concepts.

Use an orthonormal basis of stationary states and project the Time-dependent Schrödinger equation onto state 2 after removing its free-evolution phase. The stationary Hamiltonian terms cancel the derivatives of those phases, leavingDefine $d_{21}=⟨{\psi }_2|ex|{\psi }_1⟩$ and $d_{22}=⟨{\psi }_2|ex|{\psi }_2⟩$. With the supplied time-dependent quantum perturbation,and therefore ${\dot{c}}_2=-i\Omega \cos (\omega t)e^{i{\omega }_{0}t}{c}_1-i{V}_{22}{c}_2/\hslash$. The state phases may be chosen so $d_{21}$ and $\Omega$ are real for this single driven pair; otherwise probabilities depend on $|\Omega {|}^2$ and the conjugate matrix element appears in the equation for $c_1$. Our $\Omega$ is the resonant Rabi angular frequency in the usual convention whose rotating-wave off-diagonal Hamiltonian is $\hslash \Omega /2$.

For a field-free atom with symmetry under spatial reflection, each selected nondegenerate stationary orbital can be chosen with definite parity. Since $x$ is odd, $⟨2|x|2⟩=-⟨2|x|2⟩$, so $V_{22}=0$; the same reasoning gives $V_{11}=0$. Being an energy eigenstate alone is not sufficient. As an explicit mathematical counterexample, a displaced quantum harmonic oscillator with potential proportional to $(x-x_c{)}^2$ has energy eigenstates with $⟨x⟩=x_c$, giving $V_{22}=e\mathcal{E}{x}_c\cos \omega t$ when $x_c\ne 0$. The vanishing used in the atomic model rests on its parity assumption, not merely on stationarity.

At first order in time-dependent perturbation theory set $c_1\simeq 1$ and use the parity result $V_{22}=0$. Decomposing the cosine into its two exponentials givesBoth exponentials contribute to the exact first-order result:Each apparent zero denominator has a continuous limit. In particular the near-resonant fraction tends to $it$ as $\omega \to {\omega }_0$, while the counter-rotating fraction remains bounded on the optical timescale. One may equivalently write the expression asThis form makes $c_2(0)=0$ explicit and avoids cancellation at resonance. At extremely short time both sinc functions tend to one, so $c_2\simeq -i\Omega t$, agreeing with the initial derivative of the original cosine-driven equation. No rotating-wave approximation has yet been made. The separate assumption of weak excitation remains necessary: the first-order expression must not be extended until it predicts probabilities comparable to or larger than one.

Resolve the weak broadband radiation into narrow frequency intervals whose relative phases are uncorrelated. The transition probabilities, rather than the coherent amplitudes, can then be added after ensemble or time averaging. For the x-polarized field, the energy density in one interval is $\rho (\omega )d\omega ={ϵ}_0{\mathcal{E}}_{\omega }^2/2$, so its squared coupling is $|d_{21}{|}^2{\mathcal{E}}_{\omega }^2/{\hslash }^2$. Inserting this into the weak rotating-wave result givesFor times long compared with the optical coherence time of the field, the narrow resonant kernel samples a nearly constant $\rho ({\omega }_0)$. Provided the resonance is well away from the zero-frequency boundary, extend the Rabi detuning integral to the full real line. Its value is $2\pi t$, yieldingThis is the stimulated absorption rate for the specified polarization and spectral radiation energy density per angular-frequency interval. The averaging window should also satisfy $Wt\ll 1$ while the lower state is treated as undepleted. A deterministic phase-locked broadband pulse would not in general permit adding these probabilities without its interference terms.

For isotropic unpolarized radiation with total spectral radiation energy density ${\rho }_{\mathrm{tot}}$, orientational averaging instead gives $W=\pi |{\mathbf{d}}_{21}{|}^2{\rho }_{\mathrm{tot}}({\omega }_0)/(3{ϵ}_0{\hslash }^2)$. The factor one third is not inserted into the original x-polarized formula. If ordinary frequency is used, ${\rho }_{\nu }=2\pi {\rho }_{\omega }$ and the isotropic coefficient becomes $|{\mathbf{d}}_{21}{|}^2/(6{ϵ}_0{\hslash }^2)$ multiplying ${\rho }_{\nu }({\nu }_0)$. The spectral convention and polarization must be stated together to avoid spurious factors of three or $2\pi$.

For one coordinate with Hamiltonian $H={\hat{p}}^2/(2m)+V(\hat{x})$, the propagator is $K(x_f,t_f;x_i,t_i)=⟨x_f|e^{-iHT/\hslash }|x_i⟩$, where $T=t_f-t_i>0$. Divide $T$ into $M$ intervals $\delta t$, use the Trotter product formula to separate kinetic and potential evolution, and insert complete position and momentum bases at each slice. The momentum integral is Gaussian:Multiplying the short-time kernels gives the time-sliced configuration-space path integral,with $x_0=x_i$, $x_M=x_f$. Continuum notation abbreviates this limit as ${\int }_{x_i}^{x_f}\mathcal{D}x{e}^{iS[x]/\hslash }$, where $S=\int dt[m{\dot{x}}^2/2-V(x)]$. The measure includes the displayed normalization; it is not an unweighted count of paths. A small damping or analytic-continuation prescription defines the oscillatory integrals. The paths contribute complex amplitudes, so alternative histories interfere before probabilities are calculated. This is the central difference from a classical statistical mixture of trajectories.^[ref]

When actions vary rapidly relative to $\hslash$, stationary phase selects classical paths. Put $x=x_{\mathrm{cl}}+\eta$, with $\eta$ zero at the endpoints. Vanishing first variation gives $m{\ddot{x}}_{\mathrm{cl}}+V^{\prime }(x_{\mathrm{cl}})=0$. The quadratic term isThe resulting Gaussian fluctuation determinant is the semiclassical propagator prefactor. In one dimension, away from caustics,The branch convention makes $\nu =0$ for a direct free-particle path before any caustic. The Maslov index tracks the phase changes as fluctuations pass through a focal singularity. In several dimensions replace the endpoint derivative by the Van Vleck determinant. The approximation sums all relevant classical paths; real-time action need only be stationary, not a minimum. It fails in this simple form at a caustic or coalescing saddles, where a uniform approximation is needed. Quadratic actions have no higher fluctuation terms, making their Gaussian evaluation exact. For example $S_{\mathrm{cl}}=m(x_f-x_i{)}^2/(2T)$ gives the exact free-particle kernel $\sqrt{m/(2\pi i\hslash T)}{e}^{i{S}_{\mathrm{cl}}/\hslash }$. Tunnelling involves classically forbidden regions and can be described by imaginary-time saddles rather than real classical trajectories.

Quantum statistical mechanics follows by Wick rotation $t=-i\tau$. The imaginary-time evolution operator is $e^{-\tau H/\hslash }$, with Euclidean actionTaking a thermal trace identifies and integrates the endpoint positions:This thermal trace as a closed Euclidean path integral explains the periodic imaginary-time direction. A density-matrix element instead retains open endpoints. For many identical particles the trace also sums over endpoint permutations, with a plus sign for bosons and permutation parity for fermions. In coherent-state language this becomes periodic bosonic fields and antiperiodic Grassmann fields, with corresponding Matsubara frequencies.

The Euclidean weight resembles a classical Boltzmann weight. A time-sliced local quantum field in $d$ spatial dimensions becomes a classical-looking field model in $d+1$ dimensions, with anisotropic couplings and extra extent $\beta \hslash$. Thus zero-temperature quantum criticality can be treated using an infinitely extended imaginary-time direction, while nonzero temperature imposes a finite thickness. The dynamical scaling of time need not equal the spatial scaling: the dynamical critical exponent specifies their relationship. This quantum-to-classical imaginary-time mapping requires qualification when weights have signs or imaginary terms; such a functional integral is not automatically an ordinary positive classical probability distribution. Even with a positive weight, the mapping concerns equilibrium statistics, not literal dissipative real-time trajectories. For a free particle the Euclidean kernel is a diffusion kernel with diffusion coefficient $\hslash /(2m)$, illustrating the mathematical correspondence.

The classical high-temperature limit is a further approximation, rather than simply another name for imaginary time. If quantum level spacings and exchange effects are negligible, a one-particle partition function in $d$ dimensions becomesFor indistinguishable particles there is a factor $1/N!$ in the dilute classical gas, and exchange is negligible when the number density times the thermal de Broglie wavelength to power $d$ is small. For a harmonic oscillator, the exact quantum result $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}$ tends to $1/(\beta \hslash \omega )$ as $\beta \hslash \omega \to 0$, recovering the classical phase-space integral. stationary phase explains classical mechanics; the short imaginary-time, weak-exchange limit explains classical statistical mechanics. These are related but distinct limits of the same quantum formulation.

Include Alice's detector and environment before taking a partial trace. Ideal measurement produces the global stateOrthogonality of the two record states removes their cross terms from the reduced state of the particles:This nonselective quantum measurement has changed the initially entangled singlet into a separable, classically anticorrelated reduced state. Its entropy is one bit, although the full particles-plus-records state is still pure under the assumed unitary dynamics. The reduced density matrix is an improper mixture; tracing the environment does not by itself assert a literal collapse of that global state.

The Born rule gives Alice outcomes zero and one with equal probability. Conditional on a recorded result, the selective quantum measurement assigns $|01⟩$ or $|10⟩$ to the particles. The unitary measurement coupling and orthogonal records explain decoherence in this basis; they do not supply an additional rule selecting a unique outcome. That is part of the measurement problem. In a collapse postulate the individual outcome is stochastic; other interpretations account for outcome experience differently while retaining these operational probabilities.

When Bob makes the same kind of measurement, the global state becomesTracing both laboratories gives the same reduced matrix, ${\rho }_{AB}^{(2)}={\rho }_{AB}^{(1)}$. Bob has added a correlated record, not restored the lost reduced coherence. If Alice's outcome is known, Bob's opposite outcome is certain in this ideal common basis. Without that record, his marginal is $I/2$, so his outcome remains equiprobable. Alice cannot choose her random result or use this correlation for faster-than-light signalling, consistently with quantum no-signalling.

These common-basis measurements alone do not distinguish quantum predictions from a local source sending opposite preassigned bits. To test that explanation, prepare fresh singlets and let the observers independently choose among incompatible measurement axes before recording each result. For singlet spin measurements along unit vectors $\mathbf{a},\mathbf{b}$, quantum mechanics predicts $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot \mathbf{b}$. The CHSH inequality bounds $|S|\le 2$ for Bell-local models with measurement-independent settings. Choosinggives $S=E(\mathbf{a},\mathbf{b})+E(\mathbf{a},{\mathbf{b}}^{\prime })+E({\mathbf{a}}^{\prime },\mathbf{b})-E({\mathbf{a}}^{\prime },{\mathbf{b}}^{\prime })=-2\sqrt{2}$. A suitable Bell test constrains that class of explanations. Independently chosen settings, control of detection bias and separation of the measurement events matter when interpreting an actual experiment.

If Alice first performs the fixed-axis irreversible measurement described above, the resulting mixed state has $E(\mathbf{a},\mathbf{b})=-a_z{b}_z$ and cannot supply that singlet violation. The incompatible settings must belong to the Bell measurement itself. Moreover, Bell violations do not choose among all interpretations of quantum mechanics: interpretations sharing the Born predictions remain observationally equivalent for this experiment. Objective-collapse modifications could instead require additional interference tests with specified departures from unitary dynamics.

Quantum cryptography includes quantum key distribution, which establishes a shared secret key rather than transmitting an arbitrary plaintext directly. In BB84, Alice randomly prepares one of the two computational-basis or two complementary-basis qubit states; Bob randomly measures in either basis. They announce bases over an authenticated classical channel and retain the positions where their choices agree, about half in the ideal unbiased case. They test a subset for errors, reconcile the remaining strings, and apply privacy amplification to remove a bounded eavesdropper's information. A protocol can abort when the observed data do not support a secret key.^[ref]

Nonorthogonal states cannot be perfectly copied by a state-independent device, by the no-cloning theorem. An intercept-resend attacker who measures each signal in a random BB84 basis chooses the wrong basis half the time. On those sifted positions Bob has a one-half error probability, giving the ideal BB84 intercept-resend error rate $1/4$. This simple calculation demonstrates detectable disturbance for that attack; it is not a complete security proof for arbitrary attacks.

Authentication is essential: otherwise an adversary can impersonate both participants. Error correction reveals information that must be included in the privacy budget, and losses, finite sample sizes, multiphoton emission and detector behavior belong in the security model. Entanglement-based variants can use Bell correlations, but observing some entanglement or a Bell violation alone does not replace a quantitatively specified key-distillation protocol. The final key can subsequently be used in a suitable classical encryption scheme.

For quantum teleportation, Alice holds an unknown input $|\psi {⟩}_1$ and one member of a shared $|{\Phi }^{+}{⟩}_{23}$ pair; Bob holds qubit 3. Alice applies a CNOT gate from 1 to 2 and a Hadamard gate on 1, then measures both in the computational basis. Expansion gives the Bell-basis teleportation identityEvery outcome pair $(m,n)$ has probability $1/4$, independently of the input. Alice sends these two classical bits, and Bob appliesThe order of the two corrections matters only by a global sign for the doubly nonzero outcome. The transformation also transfers entanglement with an untouched reference, because the identity is linear in the input amplitudes.

Before receiving the bits, Bob must average over all four Pauli-related states, obtaining $I/2$ for any input. Thus teleportation requires classical communication and does not send information faster than light. The original input is consumed by Alice's measurement, so no second copy remains and the no-cloning theorem is respected. Ideal qubit teleportation consumes one shared Bell pair and two classical bits, as well as local gates and measurements. Imperfect entanglement, gate error, detection error or lost classical records reduce practical fidelity; teleportation transfers the quantum state, not the underlying material particle.

With both final splitters removed, the path labels map directly into detector labels:In particular $P_{RR}(C^{+},C^{-})=0$, because joint occupation of the two intersecting arms has already annihilated. Each of the other three charged-particle coincidences has unconditional probability $1/4$.

Apply the final splitter only to the first ket. The $|dd⟩$ amplitudes from $i|uv⟩$ and $|vv⟩$ cancel, givingThus detection at $D^{+}$ occurs only in the coincidence $D^{+},C^{-}$, with probability $1/8$. The other two nonzero coincidence probabilities are $P(C^{+},C^{-})=1/8$ and $P(C^{+},D^{-})=1/2$. The cancellation is quantum interference between indistinguishable surviving alternatives.

Interchanging the two wings givesNow detection at $D^{-}$ occurs only together with $C^{+}$, with probability $1/8$. The remaining nonzero coincidence probabilities are $P(C^{+},C^{-})=1/8$ and $P(D^{+},C^{-})=1/2$.