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The Bose--Marletto--Vedral experiment is a family of proposals to test gravitationally induced entanglement between initially independent masses. Sougato Bose and collaborators, and independently Chiara Marletto and Vlatko Vedral, described the idea in 2017. Each mass occupies two distinguishable spatial branches, encoding a qubit. Different branch separations give different Newtonian gravitational potential energies and hence different relative phases.^[ref]

For stationary branches at separations $r_{ab}$ held for duration $\tau$, the ideal model givesThe positive phase follows from negative Newtonian gravitational potential energy in $e^{-iV\tau /\hslash }$. A direct algebraic check separates local rephasing from genuine entanglement: the determinant of the two-by-two coefficient matrix vanishes exactly whenThus a large common phase alone is insufficient. For this ideal pure state, the concurrence is $|\sin (\Delta \varphi /2)|$, reaching one at an odd multiple of $\pi$.

The spin-based proposal uses two Stern-Gerlach interferometers: embedded spins label the spatial branches, the masses interact, and recombination transfers the branch information into measurable spin correlations. Measurements in different spin bases supply an entanglement witness. Unwanted electromagnetic interactions and quantum decoherence must be controlled for the gravitational interpretation.^[ref]

The mediator inference assumes initially unentangled systems, local coupling through a mediator, and no other entangling interaction. Under those assumptions, a mediator described only by commuting observables cannot generate the observed entanglement; at least two incompatible observables are required. This is an indirect test of nonclassical mediation, without selecting a complete high-energy theory of quantum gravity.^[ref]

For a Hermitian perturbation mode $X$ and its time derivative $\dot{X}$, subtract their expectations and use the symmetrized covariance to define variances ${\sigma }_X^2,{\sigma }_{\dot{X}}^2$. A commutator-to-fluctuation diagnostic isFor zero-mean modes this reproduces the usual expression with $⟨XX⟩$ and $⟨\dot{X}\dot{X}⟩$ in the denominator. A finite real Fourier mode or a suitably smeared field must be specified; unregulated coincident field products need not have finite variance.

The Robertson uncertainty relation gives $\mathcal{C}\le 2$ when the denominator is nonzero. Small $\mathcal{C}$ means the commutator is small relative to the marginal fluctuation scales, which can justify classical approximations for some symmetrized correlators. It does not alone show that the quantum state is mixed or that quantum decoherence has occurred. For example, a correlated Gaussian wave packet in a pure state can have a large product of marginal uncertainties while retaining quantum coherence. Which observables and accuracy are being approximated must therefore accompany a claim of effective classicality.

For a pure state $a|0⟩|E_0⟩+b|1⟩|E_1⟩$ with normalized environment states, the upper-right device reduced density matrix entry is $a{b}^{\ast }⟨E_1|E_0⟩$. Distinguishability of the conditional environment states suppresses device coherence. States differing only by a global phase have unit-modulus overlap and do not cause quantum decoherence.

Dynamical decoupling applies a designed sequence of control operations to a quantum system to suppress unwanted evolution, especially coupling to an environment that causes quantum decoherence. The controls repeatedly change how the unwanted interaction acts, so its effects can average away over a cycle. This does not mean that all noise is removed or that arbitrary useful dynamics are automatically preserved.^[ref]

For a simple example, let a qubit have an unwanted constant Quantum Hamiltonian $H=\hslash \Delta Z/2$, where $Z$ is the Pauli Z gate matrix. Free evolution for time $\tau$ gives $U(\tau )=e^{-i\Delta \tau Z/2}$. An ideal Pauli X gate obeys $XZX=-Z$, henceThe two intervals cancel the same constant phase drift in this ideal model. This is the basic refocusing idea behind a spin echo. A changing drift, finite pulse duration or imperfect rotation can leave residual error.

A2016 solid-state experiment used nitrogen-vacancy centers in diamond to compare decoupling protocols with bounded, finite-duration controls. Long shaped control paths were tested against imposed dephasing and more general noise. The experiment matters because instantaneous perfect pulses are an idealization: a useful sequence must account for the control apparatus as well as the unwanted interaction.^[ref]

Use the Driven two-level Rabi model in the ordered basis $(|e⟩,|g⟩)$, with Pauli matrices satisfying ${\sigma }_z=\mathrm{diag}(1,-1)$. Let the Rabi detuning be $\delta ={\omega }_0-\omega =2\pi ({\nu }_0-\nu )$, in angular frequency, while ${\nu }_0$ and $\nu$ are ordinary frequencies in hertz. Assume two identical square pulses, equal drive oscillation phases, constant positive Rabi angular frequency $\Omega$, constant $\delta$ during pulses and flight, and no quantum decoherence or loss. The rotating-wave approximation and isolation of the chosen two-level system are required, just as for the Driven two-level Rabi model. Free flight means no transverse drive; the reference oscillator continues running.

The pulse and free-flight Hamiltonian operators in the rotating frame areThe Pauli matrix multiplication law gives $(\delta {\sigma }_z+\Omega {\sigma }_x{)}^2=q^{2}I$. Thus the exact special unitary group $\mathrm{SU}(2)$ pulse matrix for this effective Hamiltonian operator, and the flight matrix, areThe sequence is $U_p{U}_f{U}_p$. Starting in $|g⟩$, matrix multiplication and the Born rule giveThis retains the tilted rotation axis and quantum phase accumulated during each finite pulse. At resonance it gives $P_{g\to e}(0)={\sin }^2(\Omega \tau )$; with no flight it reduces to the single square-pulse response for duration $2\tau$. Both checks would fail if arbitrary ideal $\pi /2$ rotations were silently substituted at nonzero Rabi detuning.

For resonant pulse area $\Omega \tau =\pi /2$, $|\delta |/\Omega \ll 1$ and $\tau \ll T$, the central fringe approachesIts half-height points are $|\nu -{\nu }_0|=1/(4T)$, so the central peak's full width at half maximum is $1/(2T)$ in hertz, or $\pi /T$ in angular frequency. This width assumes an effectively flat pulse envelope across that central peak, one flight time and unit contrast; it is neither a universal finite-pulse width nor a frequency uncertainty. Longer flight lets a smaller frequency difference accumulate the same observable quantum phase.

The two pulses here have the same sign and the plotted outcome is transfer to $|e⟩$. The earlier Ramsey interferometry example uses an inverse analysis pulse and reports return to its initial label. Specifying the measured outcome and pulse oscillation phases reconciles the two conventions. In a beam the lengths and speed determine $\tau$ and $T$; in the Quantronium Ramsey interference experiment electronics set the pulse times for a stationary circuit.

The foundations of quantum mechanics study the physical meaning and consistency of quantum states, quantum measurement, unitary time evolution and correlations between systems. The aim is to connect precise mathematical rules with claims about preparation, observation and physical reality.

The measurement problem can be seen in a simple model. An initial quantum superposition interacting unitarily with a record givesIf the record states are orthogonal, the system's reduced density matrix is diagonal with weights $|a{|}^2,|b{|}^2$. This explains suppression of observable interference, while the relation between the joint state and an individual recorded outcome remains a foundational question. Quantum decoherence, the many-worlds interpretation and de Broglie-Bohm theory address different parts of this problem.

John Stewart Bell made a separate question experimentally sharp. In his singlet-pair model, local hidden-variable response functions cannot reproduce all the quantum spin correlations. A Bell test compares measured correlations across analyzer settings with the appropriate Bell inequality. ^[ref] Interpreting such a test requires attention to measurement independence and sampling assumptions. Quantum nonlocality concerns these joint probabilities and is compatible with quantum no-signalling.

The Hadamard gate is the unitary operatorwhere $I$ is the identity operator. It maps the computational basis to the Hadamard basis:

These outputs have equal computational-basis measurement probabilities, but their relative quantum phase differs. Applying another Hadamard gate sends them to different definite outputs. Calling this gate a classical coin toss would discard precisely the information that makes this possible. ^[ref].

Starting from $|0⟩$, compare two quantum circuits, reading matrix products from right to left:Here $Z$ is the Pauli Z gate. After the first Hadamard gate, the probability amplitudes are $(1/\sqrt{2},1/\sqrt{2})$. Inserting $Z$ changes them to $(1/\sqrt{2},-1/\sqrt{2})$ without changing their squared magnitudes. At the final gate, the contributions to output zero cancel and those to output one add. Without $Z$, cancellation occurs at output one instead. This is controlled interference, with the quantum phase determining which answer emerges.

The same calculation works for any phase gate $P_{\varphi }=\mathrm{diag}(1,e^{i\varphi })$:For example, $\varphi =\pi /3$ gives $P(1)=1/4$, while $\varphi =\pi /2$ gives $P(1)=1/2$. The output-one amplitude at $\varphi =\pi /2$ is $(1-i)/2$: its two complex contributions add before the Born rule is applied. A phase gate alone leaves the populations unchanged; recombination converts its relative quantum phase into observable probabilities.

If quantum decoherence completely removes the computational basis coherence between the gates, the intermediate density matrix becomes $I/2$. Both the phase gate and final Hadamard gate leave $I/2$ unchanged, so $P(1)=1/2$ for every $\varphi$. The disappearing fringe explains why preserving quantum coherence matters: it enables the gate sequence to carry phase information into the final measurement.

A physicist who coauthored an optical simulation of controlled quantum decoherence. The experiment used a photon's polarization as a qubit and manipulated its coupling to frequency degrees of freedom to engineer dephasing dynamics. The platform lets different environments be compared through the evolution of the same observable quantum system.^[ref]

A physicist who coauthored a study of experimentally controlled dephasing in a qubit. Its optical platform simulates how coupling to an environment changes coherence, including dynamics inspired by a Transverse-field Ising model. This provides an experimental way to compare environment models and their consequences for quantum decoherence.^[ref]

An interpretation retaining universal unitary evolution of the quantum state, with definite measurement records described relative to outcome branches. Quantum decoherence explains the practical isolation and stable record basis of emergent branches. It does not insert a physical collapse selecting just one component. The emergence of branch structure and justification of Born rule weights involve additional interpretive arguments beyond writing the Schrödinger equation.

The quantum measurement problem asks how a unitary superposition of system and apparatus outcomes gives rise to one definite observed outcome. Quantum decoherence explains the suppression of interference between outcome branches, but by itself does not select one branch.

A quantum computing apparatus needs a scalable quantum register of controllable qubits, reproducible initialization, a universal quantum gate set, sufficiently long quantum coherence, and selective readout. A qubit is a two-dimensional state space; several qubits occupy a tensor product space and can develop entanglement. Quantum state preparation must establish a known low-entropy input and supply fresh ancillas during quantum error correction. A universal quantum gate set may use arbitrary single-qubit rotations with an entangling two-qubit gate such as controlled-NOT gate. Quantum coherence and control fidelity must suffice for the intended circuit, and measurement must recover chosen output probabilities without uncontrolled disturbance of other qubits. These are the five DiVincenzo criteria.^[ref]

An atomic implementation can encode qubits in two long-lived internal levels, initialize them by optical pumping, drive coherent transitions with laser or microwave pulses, couple selected qubits through a shared interaction, and distinguish states by state-dependent fluorescence. For trapped ions, a shared vibrational mode can mediate entangling gates, so motional control and cooling are part of the computational apparatus. The motion is a resource only while uncontrolled heating does not erase the required quantum coherence.

The computational resource is coherent control of amplitudes and entanglement, with interference making useful outcomes likely. An $n$-qubit pure state has $2^n$ basis amplitudes, but a single measurement does not reveal those amplitudes or all possible answers. Algorithms exploit structure rather than simply reading out parallel classical computations. The Shor algorithm gives factoring with polynomial resource scaling in the input bit length, compared with the best known classical algorithms' superpolynomial scaling; an exponential classical lower bound has not been proved.^[ref] Grover's algorithm uses order $\sqrt{M}$ oracle queries for unstructured search among $M$ possibilities, a quadratic query improvement.^[ref] Quantum simulation is a further motivation because interacting quantum systems can require exponentially many classical amplitudes for a direct generic representation.

The main challenges are quantum decoherence, fluctuating control fields, imperfect gates, leakage out of the qubit subspace, readout errors, crosstalk, and the difficulty of scaling connectivity and calibration. Faster control can introduce extra errors, while stronger isolation can make preparation and readout harder. Quantum error correction encodes logical information redundantly in several physical qubits and extracts error syndromes without measuring the unknown logical state itself. Fault-tolerant quantum computation prevents the correction apparatus and gates from spreading a few faults into uncorrectable logical errors. It requires error rates below an architecture- and noise-dependent threshold and substantial resources; there is no universal numerical threshold for all implementations. Advantages are task-dependent, while reliable scaling requires both coherent hardware and error control.

Write $c=⟨e_0|e_1⟩=v{e}^{i\alpha }$. The Cauchy-Schwarz inequality gives $0\le v\le 1$ because the two environment states are normalized. Immediately after the interaction, the joint quantum state isApplying the last Hadamard gate givesSince the environment is unobserved, the Born rule uses the squared norm of the environment-valued coefficient of $|0⟩$:Thus the Hadamard interference with environment overlap formula isThe plus sign follows from the explicitly chosen phase gate and the order $⟨e_0|e_1⟩$ of the inner product. Equivalently, before the last Hadamard gate the partial trace over the environment givesThe complex environment overlap in qubit dephasing both reduces the quantum coherence by $v$ and adds a relative quantum phase $\alpha$. Orthogonal environment states have $v=0$ and give $P_0=1/2$ for every $\phi$. If $v=1$, the quantum interference fringe has full contrast and is only shifted; the qubit has not become mixed. For $v=0$, $\alpha$ is immaterial.

Moving the interaction to between the first Hadamard gate and the phase gate givesthe identical joint quantum state. Therefore the expression is unchanged. This is the commutation of computational-basis dephasing and phase gates; it uses the specified population-preserving interaction with the same environment states, rather than an arbitrary model of quantum decoherence.

Normalize the second qubit as $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$. Since a Pauli X gate changes $|-⟩$ to $-|-⟩$, the Boolean quantum oracle acts asThis quantum phase kickback leaves the second qubit uncorrelated with the first. Put $b=f(0)\oplus f(1)$: $b=0$ for a constant function and $b=1$ for a balanced Boolean function. Apart from the irrelevant global phase $(-1{)}^{f(0)}$, the first qubit and environment reachThe population-preserving interaction commutes with the quantum oracle on this answer-register input, so its placement before or after the oracle, between the two Hadamard gates, gives the same result. Substitute $\phi =\pi b$ into b:The printed Deutsch algorithm uses the rule zero means constant function; one means balanced Boolean function. With that rule, the Deutsch algorithm with pure dephasing succeeds for either class withFor real nonnegative environment overlap this is $(1+v)/2$, ranging from certainty at $v=1$ to random guessing at $v=0$. For general complex overlap, a coherent shift can invert the printed answer: for example $v=1,\alpha =\pi$ makes the printed rule always wrong. If the overlap is known but the printed quantum measurement is retained, choose the better labeling of its outcomes; this givesThere is a further improvement if a known $\alpha$ can be compensated by the phase gate $P_{-\alpha }$ before the final Hadamard gate. It removes the coherent shift without reversing any actual quantum decoherence, and yieldsThis is optimal when only the first qubit is accessible, with equal prior probabilities for the two classes. Before the final Hadamard gate, their reduced density matrices areThe difference ${\rho }_0-{\rho }_1$ has eigenvalues $\pm v$, giving trace distance $v$. The Holevo–Helstrom theorem therefore bounds the success probability by $(1+v)/2$, exactly attained by the compensated procedure. A known phase shift and loss of coherence have different effects: at $v=0$ the two density matrices coincide and no measurement on the available qubit can reveal the answer.

The five DiVincenzo criteria are: a scalable collection of well-characterized qubits; preparation in a known fiducial quantum state; relevant coherence times long compared with quantum gate times; a universal quantum gate set; and qubit-specific quantum measurement. The criterion is useful control of the whole computation, including errors, rather than merely observing two resolved levels.^[ref]

For trapped ions, the implementation is:

For trapped neutral atoms, the corresponding implementation is:

Thus neither platform is forbidden from meeting a criterion in principle. Weak neutral-ground-state interactions make entangling control harder, but claiming that trapped neutral atoms cannot implement a universal quantum gate set would be false. Gate-model quantum algorithms using Rydberg blockade had already been demonstrated in a programmable neutral-atom processor by 2022.^[ref]

Take the Hadamard gate to be $H=2^{-1/2}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$ and the phase gate to be $P(\varphi )=\mathrm{diag}(1,e^{i\varphi })$. This fixes the sign convention for the interference phase.

For the coherent quantum circuit, the successive quantum states areThe Born rule therefore gives the concise answer

With the stated quantum decoherence after the phase gate, the joint pure state before the last Hadamard gate isAfter that Hadamard gate, its environment coefficient at output zero is $(|e_0⟩+e^{i\varphi }|e_1⟩)/2$. Squaring its norm sums over every unobserved environment outcome, soThe Cauchy-Schwarz inequality gives $0\le v\le 1$. In this single-qubit interference with conditional environment states, $v$ is the interference visibility: orthogonal environment states erase the fringe, while a unit-modulus overlap shifts its phase without reducing visibility. The plus sign in $\varphi +\alpha$ follows from the specified overlap $⟨e_0|e_1⟩$, rather than its complex conjugate.

If the same quantum decoherence occurs before the phase gate, the joint pure state is first $(|0⟩|e_0⟩+|1⟩|e_1⟩)/\sqrt{2}$. The phase gate then produces exactly the same state used above. Equivalently, the controlled environment interaction commutes with the diagonal phase gate. The probability is unchanged.

For the Deutsch algorithm, use the reversible Boolean quantum oracleThe algebraic oracle map printed in the source has its arguments misplaced: taken literally, it sends two distinct answer inputs to the same output for constant $f$, so it cannot be a unitary operator. The corrected map above agrees with the diagram's input-control, answer-target arrangement. Put the answer qubit in $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$. Since the Pauli X gate satisfies $X|-⟩=-|-⟩$, quantum phase kickback givesWrite $b=f(0)\oplus f(1)$. Up to the irrelevant global phase $(-1{)}^{f(0)}$, the input qubit acquires relative phase $(-1{)}^b$, so its ideal final computational-basis measurement returns $b$: zero for a constant function and one for a balanced function.

Let the specified quantum decoherence occur in the interference portion between the two Hadamard gates. It commutes with the oracle's effective diagonal phase on the input, so the same analysis applies whether it occurs just before or just after the query. The two likelihoods areThus the diagram's usual zero-for-constant decision rule hasThis is the reliability for either promised class separately, with no prior needed. For equally likely classes, if $v\cos \alpha$ is known to be negative, reversing the interpretation gives $(1+|v\cos \alpha |)/2$. A known environment phase can instead be compensated by the phase gate $P(-\alpha )$ before the final Hadamard gate, giving success probability $(1+v)/2$. In particular, $v=0$ gives no information and $v=1$ permits certainty after phase compensation. These refinements distinguish a shifted fringe from lost quantum coherence; with the unmodified diagram a phase shift alone can reverse the apparent answer. Quantum decoherence after the final Hadamard gate leaves its computational populations unchanged, so that different placement would not spoil the measured answer.

Write both the input and output integer labels in most-significant-bit-first order, $x={\sum }_{j=1}^n{x}_j{2}^{n-j}$ and $y={\sum }_{j=1}^n{y}_j{2}^{n-j}$. Factoring the Fourier phase over the bits of $y$ gives the product decomposition of a Fourier phase state:Here $0.x_l\cdots x_n$ denotes the binary fraction ${\sum }_{r=l}^n{x}_r/2^{r-l+1}$; the integer part of $x/2^j$ does not affect the phase.

Build a dyadic quantum Fourier transform circuit as follows. Process wires $j=1,\dots ,n$ in that order. First apply a Hadamard gate to wire $j$. Then, for each $l=j+1,\dots ,n$, apply the controlled phase gate $R_{l-j+1}$ with control wire $l$ and target wire $j$. On a computational-basis state, every unprocessed control is still the bit $x_l$, so wire $j$ becomesOnce a wire has been processed it is not touched by later stages. Reversing the order of the output wires gives exactly the Fourier product above. Agreement on every computational-basis state proves agreement as linear operators on all superpositions, even though the intermediate operation on a general input can create entanglement.

There are $n$ Hadamard gates and $n(n-1)/2$ controlled phase gates. Output reversal may be treated as wire relabelling. If it must instead be implemented physically while using only the requested gate family, use $\lfloor n/2\rfloor$ swap operators. Each swap is three controlled-NOT gates, andThus the swaps use only Hadamard gates and Controlled-Z gates, which are controlled $R_1$ gates. An exact physical network haselementary gates from the stated family. This implements the positive-exponent Fourier transform with the correct output order in quadratic size. Omitting the output reversal without relabelling the wires would produce the reversed-order transform instead.

For the perturbation bound, the operator norm in the source is equivalently $\Vert A\Vert ={\sup }_{\Vert \psi \Vert =1}\Vert A\psi \Vert$. It satisfies the triangle inequality, is unchanged by left or right multiplication by a unitary operator, and a unitary operator has norm one. The exact telescoping identity isIt follows by replacing one factor at a time; alternatively all intermediate products cancel in the sum. Taking the operator norm and using unitary invariance gives the quantum circuit gate-error telescoping boundThe original PDF assumes strict $\Vert U_j-V_j\Vert <ϵ$, so its strict conclusion is justified. If the hypothesis were weakened to a non-strict inequality, the conclusion would likewise be $\le mϵ$. The TeX aid misreads this hypothesis; the PDF resolves it.

Apply the non-strict version to each approximate unitary gate in the Fourier network. Extending a gate by the identity on other wires leaves its operator norm error unchanged. Including the physical swap decompositions, all gates are of the required type and can obey the stated accuracy. HenceWith exact output relabelling rather than physical swaps the numerator is just $n(n+1)/2$. The proof retains gate phases and makes no assumption of cancellation between coherent errors.

For every normalized input the output-state norm error is at most this same bound. If $M$ is any positive operator satisfying $0\le M\le I$, the change in its measurement in quantum mechanics outcome probability is at most $2\delta$, where $\delta =\Vert U_n-{\mathrm{QFT}}_{2^n}\Vert$: expand the difference of the two expectations into two terms, each bounded by $\delta$ using the Cauchy-Schwarz inequality. Thus inverse-polynomial gate accuracy gives an inverse-polynomial total error, rather than requiring exponential precision in the number of qubits. For the phase gates this corresponds to specifying angles with $O(\log n)$ bits of precision, since $|e^{i\theta }-e^{i{\theta }^{\prime }}|\le |\theta -{\theta }^{\prime }|$. This is a robustness statement for unitary gate approximations; it does not by itself correct quantum decoherence, correlated stochastic faults, or limitations of hardware connectivity.

Let $B={\sum }_k{g}_k{Z}_k$. In the Zurek spin-bath model, $H=-Z_D\otimes B$, so unitary time evolution with $\hslash =1$ is $U(t)=e^{it{Z}_D\otimes B}$. The bath Hamiltonian terms commute, and device states $|0⟩,|1⟩$ have $Z_D$ eigenvalues $+1,-1$. Hencewhere the two normalized conditional bath states areTake each bath factor normalized, $|{\alpha }_k{|}^2+|{\beta }_k{|}^2=1$, and $|a{|}^2+|b{|}^2=1$. This entails no restriction: if only the product is initially normalized, divide each nonzero factor by its norm; the product of these norms is one.

Taking the partial trace over the bath gives the reduced density matrixThe orientation of this conditional environment overlap fixes the sign of the phase in the upper-right entry. Factorizing the overlap yields the decoherence factorThe populations are conserved because $[H,Z_D]=0$. Only phase coherence can be reduced. In particular,Quantum decoherence here results from distinguishable conditional bath states, although the complete system remains in a pure state under unitary time evolution.

For these three couplings, the decoherence factor becomesIt is periodic with period $2$. To locate its extrema, put $c=\cos \pi t$; then $z=c(1+c)/2$. Its maximum is $1$ at $t=0,2,4$, and its minimum is $-1/8$ where $c=-1/2$, namely $t=2/3,4/3,8/3,10/3$. The zeros on the displayed interval are $t=1/2,1,3/2,5/2,3,7/2$. The half-integer zeros are crossings; the zeros at $1$ and $3$ are double zeros, where the curve touches zero from below.

The finite spin-bath coherence recurrence returns the device to full coherence every two time units. Negative $z$ indicates a relative phase change and is not a negative probability. When $ab\ne 0$, the device reduced density matrix becomes diagonal at each zero, but quantum decoherence is not irreversible in this finite bath. The sketch explicitly displays both loss of coherence and its revival.

The exact mode functions satisfy the Wronskian normalizationUsing the Dirac delta function generated by the ladder-operator commutator therefore gives the exact canonical commutation relationThis remains true on superhorizon scales, so the exact operators do not literally commute. A claim of classical behavior must instead compare this fixed commutator with the growing statistical fluctuations, or appeal to quantum decoherence.