Noise and Decoherence in Metrology

Noise in quantum metrology is any uncontrolled or incompletely modeled part of a sensing experiment that changes the distribution of recorded data. It may enter during state preparation, parameter encoding, control, storage, transport, or readout. Decoherence is narrower: it is the loss of off-diagonal coherence in a specified reduced description after degrees of freedom or records are ignored. Dephasing, energy relaxation, particle loss, detector errors, calibration drift, and common-mode fluctuations can all limit a sensor, but they are not interchangeable mechanisms.

The metrological question is also more precise than “how much coherence survives?” A useful sensor must keep states corresponding to nearby parameter values distinguishable, deliver that distinguishability through an available measurement, and do so at a favorable rate after every probe, second, calibration, and failed trial is counted. A channel may destroy one kind of coherence while preserving the signal, or preserve a beautiful state while erasing the derivative that carried the signal.

The central examples below compare product and GHZ probes under dephasing, optical probes under loss, and protected modes under correlated noise. Their different outcomes explain why a statement about noise must specify the encoding, controls, and available records.

Required background. Quantum Measurement as Estimation supplies likelihoods and nuisance parameters; Classical and Quantum Fisher Information supplies QFI and its measurement bound; Cramér–Rao Bounds explains local precision and attainability; Quantum Channels and Noise introduces Kraus and master-equation descriptions.

Parameter encoding through a noisy channel

Write one controlled use of a sensor as a parameterized channel

$$\rho_{\theta,\boldsymbol\lambda}^{(u)} = \mathcal E_{\theta,\boldsymbol\lambda}^{(u)}(\rho_0),$$

followed by a POVM $\{M_x^{(u)}\}$. The controls $u$ include preparation, interrogation time, pulse sequence, ancillary systems, and measurement setting. The nuisance vector $\boldsymbol\lambda$ may include dephasing rates, loss, contrast, detuning drift, temperature, and calibration coefficients. The actual likelihood is

$$p(x\mid\theta,\boldsymbol\lambda,u) = \operatorname{Tr} \left[ M_x^{(u)} \mathcal E_{\theta,\boldsymbol\lambda}^{(u)}(\rho_0) \right].$$

This formula prevents several common category errors. Noise is not an additive scalar attached after an ideal sensitivity calculation. It changes the state family, and therefore changes its derivatives, optimal measurement, likelihood support, nuisance correlations, and sometimes the meaning of a resource.

The parameter may enter a Hamiltonian, a decay rate, or a bath state. Ancillas, control, observed environment records, and the fixed resource (time, energy, or channel uses) therefore matter. “Noise restores the SQL” is a theorem only for a specified channel family and allowed operations.

Information in the recorded measurement

For one completed cycle, an implemented measurement has classical Fisher information $F_C$, bounded by the quantum Fisher information $F_Q$ of the output state:

$$F_C(\theta;u) \leq F_Q\!\left[ \mathcal E_{\theta,\boldsymbol\lambda}^{(u)}(\rho_0) \right].$$

If $\nu$ cycles are independent and identically configured, a locally unbiased estimator obeys

$$\operatorname{Var}(\widehat\theta) \geq \frac{1}{\nu F_C} \geq \frac{1}{\nu F_Q}.$$

The output QFI is independent of the chosen measurement, but relies on the specified state and channel model; it is not device-independent certification. It does not promise that the available readout attains the corresponding precision. Noise can rotate the optimal measurement with the unknown parameter, make nuisance parameters nearly degenerate with the signal, or leave information in an inaccessible environment.

Data processing, with an important qualification

If a parameter-independent channel $\Lambda$ acts after encoding, QFI is contractive:

$$F_Q[\Lambda(\rho_\theta)] \leq F_Q[\rho_\theta].$$

Discarding a subsystem, adding parameter-independent detector noise, or coarse-graining a record cannot increase the available information. But the qualification matters: when $\Lambda_\theta$ itself depends on the unknown parameter, it can encode information. Thermometry, loss estimation, noise spectroscopy, and estimation of a decay rate use the “noise” as the signal. Contractivity does not compare different parameter-dependent channel families.

Information rate and dead time

Suppose an interrogation lasts $t$, while preparation, reset, readout, and feedback require dead time $t_{\rm d}$. The cycle time is

$$t_{\rm c}=t+t_{\rm d},$$

so a long run of duration $T_{\rm tot}$ contains approximately $\nu=T_{\rm tot}/t_{\rm c}$ cycles. The relevant local objective is then

$$\dot F_C(t) = \frac{F_C(t)}{t+t_{\rm d}}, \qquad \operatorname{Var}(\widehat\theta) \gtrsim \frac{1}{T_{\rm tot}\dot F_C(t)}.$$

Longer interrogation usually increases signal phase but decreases contrast and the number of repetitions. Maximizing per-shot QFI can therefore select a different operating point from maximizing precision per unit wall-clock time.

Physically different noise mechanisms

Distinction	First diagnostic	Metrological consequence
dephasing vs dissipation	are basis populations preserved?	phase contrast and energy response degrade differently
erasure vs unflagged loss	is the missing probe identified?	a flagged event can enter the likelihood rather than disappear
independent vs correlated noise	how does covariance scale across probes?	information may remain extensive, saturate, or occupy a protected differential mode
semigroup vs time-inhomogeneous dynamics	does $\mathcal E_{t+s}=\mathcal E_t\mathcal E_s$ hold?	short-time behavior can change asymptotic frequency scaling
unmonitored vs monitored environment	are jump, leakage, or bath records observed?	discarded records can contain recoverable information
known vs unknown noise parameters	are rates and contrasts calibrated?	nuisance correlations reduce effective signal information
signal-parallel vs signal-transverse noise	does control suppress both by the same symmetry?	decoupling or correction may preserve the signal, or erase it

For the dynamical distinctions behind the first row, see Dephasing vs Dissipation. For the precise hierarchy of non-Markovian definitions, see What Non-Markovian Means.

Exact Qubit Dephasing Example

Consider a Ramsey qubit initialized in $|+\rangle$, accumulating an unknown angular frequency $\omega$ for time $t$. Let a real coherence envelope $0\leq\eta(t)\leq1$ describe parameter-independent dephasing. Immediately before the final analysis pulse,

$$\rho_\omega(t) = \frac12 \begin{pmatrix} 1 & \eta(t)e^{-i\omega t} \\ \eta(t)e^{i\omega t} & 1 \end{pmatrix}.$$

The QFI for phase $\phi=\omega t$ is $\eta(t)^2$, and the frequency QFI is

$$F_Q^{(1)}(\omega,t) = t^2\eta(t)^2.$$

A quadrature Ramsey measurement attains this local value when its analysis phase is correctly set. For $N$ independent probes in the same cycle,

$$F_{Q,\mathrm{prod}}^{(N)} = N t^2\eta(t)^2.$$

Thus dephasing reduces the information by the square of the surviving coherence in this model. That compact result is not universal: amplitude damping changes populations as well as coherence, and parameter-dependent noise adds derivative terms of its own. The Dephasing Channel page gives the channel and Kraus representations.

Markovian dephasing and the optimal time

For exponential coherence

$$\eta(t)=e^{-\gamma t},$$

and negligible dead time, the product-state information rate is

$$\dot F_{Q,\mathrm{prod}} = N t e^{-2\gamma t}.$$

Its optimum occurs at

$$t_\star^{\mathrm{prod}} = \frac{1}{2\gamma}, \qquad \dot F_{Q,\mathrm{prod}}^{\max} = \frac{N}{2e\gamma}.$$

Now prepare the $N$-qubit GHZ state. The signal phase is amplified to $N\omega t$, but independent dephasing multiplies its one fragile coherence by $\eta(t)^N=e^{-N\gamma t}$. Its per-cycle QFI and rate are

$$F_{Q,\mathrm{GHZ}}^{(N)} = N^2t^2e^{-2N\gamma t}, \qquad \dot F_{Q,\mathrm{GHZ}} = N^2t e^{-2N\gamma t}.$$

The optimum moves to

$$t_\star^{\mathrm{GHZ}} = \frac{1}{2N\gamma}, \qquad \dot F_{Q,\mathrm{GHZ}}^{\max} = \frac{N}{2e\gamma}.$$

The $N$-fold faster phase accumulation is exactly canceled by an $N$-fold shorter useful interrogation. In this idealized model the optimized GHZ and product strategies have the same information rate. State-preparation time, collective readout error, and nonzero dead time generally make the maximally entangled strategy less favorable, while partially entangled or squeezed states can still improve finite-resource constants in broader models.

With coherence $e^{-\gamma t}$ and zero dead time, product probes have $\gamma\dot F_Q/N=x e^{-2x}$, while an $N=8$ GHZ probe has $\gamma\dot F_Q/N=8x e^{-16x}$. The optimal times differ by eight, but both curves reach $1/(2e)$. The result compares frequency information per total time, not phase information in a single shot.

Dead time changes the optimization

For exponential decay with coefficient $a$ in the exponent, maximizing

$$\frac{t^2e^{-2at}}{t+t_{\rm d}}$$

gives the positive stationary point

$$t_\star = \frac{ 1-2at_{\rm d} + \sqrt{1+12at_{\rm d}+4a^2t_{\rm d}^2} }{4a}.$$

Use $a=\gamma$ for product probes and $a=N\gamma$ for GHZ probes, together with their different prefactors. A short GHZ interrogation pays the fixed overhead many more times. In clock language, dead time can also alias local oscillator noise into the measurement band; that system effect is not captured by the simple independent-cycle formula.

Readout contrast is another channel

If a symmetric binary readout has net contrast $A$ and is operated at quadrature, its local Fisher information becomes

$$F_C(\omega,t) = A^2 t^2\eta(t)^2.$$

The factors $A$ and $\eta$ may look identical in one fringe, but they arise at different stages and respond differently to controls. A calibration that treats $A$ as known can overstate precision if contrast drift is correlated with frequency. The multiparameter Fisher matrix, not a scalar contrast correction, is then required.

Photon Loss and Fragile Optical States

Loss couples the sensed optical mode to inaccessible modes. It reduces detected energy, leaks which-path information, and turns pure inputs into mixed states. It must be placed at a declared reference plane: source loss, propagation loss, sample absorption, collection loss, and detector inefficiency are physically different even when they share a transmissivity.

Consider a two-arm interferometer with differential-phase encoding $U_\phi=e^{-i\phi(n_a-n_b)/2}$, independent loss of the same fixed transmissivity $0<\tau<1$ in both arms, and inaccessible loss environments. Let $\overline N$ count the total incident photons in both arms. The channel bound (Demkowicz-Dobrzański, Jarzyna, and Kołodyński) gives

$$F_Q \leq \frac{\tau}{1-\tau}\,\overline N, \qquad \Delta\phi \geq \sqrt{ \frac{1-\tau}{\tau\overline N} }.$$

Here $\Delta\phi$ is a local Cramér–Rao precision benchmark; for independent repetitions, the total incident energy is counted. A single lossy mode with generator $n_a$, a lossless external phase reference, and signal photons $\overline N_a$ instead has the bound $F_Q\leq4\tau\overline N_a/(1-\tau)$. The factor cannot be selected without specifying the generator and reference. In both cases, fixed independent loss limits the asymptotic QFI to linear growth with the stated energy. Entanglement and squeezing may still improve the constant, sometimes substantially, but do not restore asymptotic $\overline N^2$ QFI.

The bound becomes loose as $\tau\to1$ and should not be used as a finite-$N$ interpolation formula. It also does not apply unchanged to heralded erasure, correlated loss, monitored environments, nonlinear sample interactions, or a different resource definition.

Why NOON coherence is fragile

For the equal-arm loss model above, an $N$-photon NOON state retains its phase-sensitive coherence only when every photon survives. That block has weight $\tau^N$, and all lost-photon blocks carry no differential phase. The unconditional output QFI is exactly

$$F_{Q,\mathrm{NOON}} = N^2\tau^N.$$

The $N^2$ ideal sensitivity is eventually overwhelmed by exponential loss. Optimal lossy probes distribute number more gently than a NOON state, and coherent light mixed with squeezed vacuum can approach the relevant large-energy bound in important interferometric regimes. The lesson is not that nonclassical light is useless; it is that the most fragile noiseless optimum is rarely the noisy optimum. See Mach–Zehnder Interferometry for optical probe and measurement details.

Incident, interacting, and detected resources

Three photon numbers should not be silently identified:

$$\overline N_{\rm in}, \qquad \overline N_{\rm sample}, \qquad \overline N_{\rm det}.$$

A photosensitive sample may make $\overline N_{\rm sample}$ the scarce resource, while a remote-sensing link may be limited by transmitted energy and aperture. Detector loss can sometimes be improved without increasing sample exposure. Every quantum and classical comparator must use the same resource boundary.

General Bounds from Channel Geometry

For independent uses of a parameterized channel, bounds can be formulated without optimizing over an exponentially large input state. Let $\{K_j(\theta)\}$ be Kraus operators and dots denote $\theta$ derivatives. Define, for a chosen Kraus representation,

$$\alpha = \sum_j \dot K_j^\dagger\dot K_j, \qquad \beta = i\sum_j \dot K_j^\dagger K_j.$$

Kraus representations are not unique. A parameter-dependent unitary rotation on the environment changes $\alpha$ and $\beta$ without changing the channel. Optimizing over this freedom gives a channel-extension bound of the form

$$F_Q^{(N)} \leq 4\min_h \left[ N\lVert\alpha_h\rVert + N(N-1)\lVert\beta_h\rVert^2 \right].$$

Here $h$ generates the Kraus-space rotation and $\lVert\cdot\rVert$ is the operator norm. If a representation can be chosen with $\beta_h=0$ and bounded $\alpha_h$, the quadratic term vanishes:

$$F_Q^{(N)} \leq 4N\min_{h:\,\beta_h=0} \lVert\alpha_h\rVert.$$

That establishes at-most-linear asymptotic QFI for the specified independent channel, even allowing entangled probes and passive ancillas. It does not say that every measurement attains the bound, and it does not automatically cover correlated channels, adaptive memory effects, or a control model outside the derivation.

These bounds explain why weak local dephasing, depolarization, relaxation, or loss often changes the large-$N$ result discontinuously: ideal quadratic QFI can persist over a finite preasymptotic region, yet the true asymptote is linear. A handful of small-$N$ data points cannot distinguish a long crossover from a new scaling law.

Short-Time Dynamics and the Zeno Scaling

Exponential decay at arbitrarily short times is an idealization. Many microscopic models begin quadratically. Suppose independent dephasing has

$$\eta(t) = 1-c t^\beta+o(t^\beta), \qquad \beta>1.$$

Assume $c>0$, independent noise across probes, and independent repetitions (for example, with an appropriate bath reset). For an $N$-qubit GHZ state at short times,

$$\eta(t)^{2N} \simeq e^{-2Nc t^\beta},$$

and, neglecting dead time, its information rate scales as

$$\dot F_{Q,\mathrm{GHZ}} \simeq N^2t e^{-2Nc t^\beta}.$$

Optimization gives

$$t_\star \propto N^{-1/\beta}, \qquad \dot F_Q^{\max} \propto N^{2-1/\beta}.$$

At fixed total time the variance can therefore scale as $N^{-(2-1/\beta)}$. For quadratic short-time decay, $\beta=2$, this is the Zeno scaling

$$\operatorname{Var}(\widehat\omega) \propto N^{-3/2}, \qquad \Delta\omega \propto N^{-3/4}.$$

This improvement is neither the noiseless Heisenberg law nor a generic reward for “non-Markovianity.” The decisive assumption is the short-time departure from a semigroup, together with access to interrogation times that shrink into that regime and repeated data with the assumed independence. Resetting a probe alone need not reset a slowly fluctuating environment. Finite control bandwidth, dead time, spatial correlations, and a later exponential regime can remove the asymptotic window. Information backflow is one possible feature of memory, but it is not the criterion behind the bound above (Smirne and colleagues).

For example, if a fixed nonzero dead time dominates the shrinking interrogation, the same GHZ calculation instead maximizes $N^2t^2e^{-2Nct^\beta}/t_{\rm d}$. It gives $t_\star^\beta\simeq1/(Nc\beta)$ and rate $\dot F_Q^{\max}\propto N^{2-2/\beta}$. For quadratic decay this is linear in $N$, so the zero-dead-time $N^{3/2}$ rate does not survive this limit.

Correlated Noise Can Saturate or Protect

Correlations change which collective mode contains signal and which contains noise. A classical Gaussian example already shows the geometry. Let one simultaneous sensor record be

$$\mathbf y = \theta\mathbf s+\boldsymbol\epsilon, \qquad \boldsymbol\epsilon \sim \mathcal N(\mathbf0,\Sigma).$$

For known, positive-definite, parameter-independent $\Sigma$, its Fisher information is

$$F_C = \mathbf s^{\mathsf T} \Sigma^{-1} \mathbf s.$$

For a common signal $\mathbf s=\mathbf1$ and covariance

$$\Sigma = \sigma_{\rm i}^2 I + \sigma_{\rm c}^2\mathbf1\mathbf1^{\mathsf T},$$

the result is

$$F_C = \frac{N}{\sigma_{\rm i}^2+N\sigma_{\rm c}^2}.$$

Independent noise alone gives information proportional to $N$. Any fixed common-mode component makes the information saturate at $1/\sigma_{\rm c}^2$ as $N\to\infty$. Treating the records as independent would report fictitious precision.

Conversely, a differential signal vector satisfying $\mathbf1^{\mathsf T}\mathbf s=0$ lies outside the common-noise mode. A differential estimator or decoherence-free subspace can reject that noise. But it also rejects any signal with the same common-mode symmetry. Protection is possible when signal and noise occupy distinguishable operator or spatial subspaces, not merely because noise is correlated. Distributed Quantum Sensing develops this geometry for sensor networks and entangled probes.

Noise Parameters as Nuisance Parameters

Suppose the signal $\theta$ is estimated jointly with unknown contrast, dephasing, or drift parameters $\boldsymbol\lambda$. Partition the Fisher matrix as

$$F = \begin{pmatrix} F_{\theta\theta} & F_{\theta\lambda} \\ F_{\lambda\theta} & F_{\lambda\lambda} \end{pmatrix}.$$

For a regular, jointly identifiable model with invertible $F_{\lambda\lambda}$, the effective local information for $\theta$ is the Schur complement

$$F_{\rm eff} = F_{\theta\theta} - F_{\theta\lambda} F_{\lambda\lambda}^{-1} F_{\lambda\theta}.$$

A singular nuisance block requires restriction to identifiable directions and an explicit check that the desired signal can be estimated; an arbitrary matrix inverse or prior cannot supply missing data. If the signal derivative resembles a contrast or drift derivative, $F_{\rm eff}$ can be much smaller than $F_{\theta\theta}$. Separate calibration data add information to $F_{\lambda\lambda}$, but the associated probes and time belong in an end-to-end resource comparison. A prior can regularize weakly identified nuisance directions, but then the reported bound is Bayesian and prior-dependent.

When decoherence is the signal

For the dephasing qubit above with $\eta=e^{-\gamma t}$ and known phase, the QFI for the decay rate is

$$F_Q(\gamma,t) = \frac{t^2e^{-2\gamma t}} {1-e^{-2\gamma t}}.$$

The channel now encodes $\gamma$ through the Bloch-vector length. Noise spectroscopy and relaxometry exploit this dependence. Calling decoherence purely detrimental would miss the parameter being measured.

Control Is a Signal-and-Noise Filter

A control sequence changes both the signal and noise couplings. For a qubit with toggling-frame Hamiltonian $H/\hbar=y(t)[s(t)+\xi(t)]Z/2$, take $s,\xi$ in angular-frequency units. The target phase is

$$\phi_s(T)=\int_0^T y(t)s(t)\,dt.$$

For zero-mean stationary classical Gaussian noise, define the two-sided angular-frequency spectrum and filter by

$$\begin{aligned} S_{\xi\xi}(\omega)&=\int_{-\infty}^{\infty} e^{i\omega\tau}\langle\xi(\tau)\xi(0)\rangle\,d\tau,\\ Y_T(\omega)&=\int_0^T y(t)e^{i\omega t}\,dt. \end{aligned}$$

The random phase is $\phi_\xi=\int y\xi\,dt$. Its Gaussian characteristic function gives coherence $\eta(T)=e^{-\chi(T)}$, where

$$\chi(T)=\frac12\operatorname{Var}(\phi_\xi) =\frac{1}{4\pi}\int_{-\infty}^{\infty} S_{\xi\xi}(\omega)|Y_T(\omega)|^2\,d\omega.$$

The factor $1/2$ is essential: phase variance and the coherence-decay exponent are different quantities. Other spectral conventions redistribute factors of $2$ or $2\pi$, so the spectrum and filter must be defined together.

The same modulation $y(t)$ determines both signal response and noise filter. Spin echo suppresses quasistatic detuning, but it also cancels a static signal coupled through the same operator. For AC sensing, pulse timing can make the target add coherently while low-frequency drift cancels. Dynamical Decoupling derives filter functions and explains sequence limitations.

Useful control strategies include:

• choosing a clock transition or sweet spot with small nuisance derivative;
• common-mode rejection and differential encoding;
• dynamical decoupling matched to a target waveform;
• adaptive interrogation times that balance range and coherence;
• optimal control that shapes both signal generator and dissipation;
• reservoir engineering or autonomous stabilization;
• monitoring emitted quanta or leakage flags when their records are informative;
• quantum error detection or correction when signal and error actions can be separated.

Every strategy has overhead and bandwidth. Pulse errors, finite duration, ancilla dephasing, recovery latency, and calibration drift belong in the implemented channel, not in a footnote after the ideal result.

Environment Monitoring and Conditional Records

An unconditional master equation averages over possible environment records. If emitted photons, quantum jumps, erasures, or ancilla syndromes are observed, the full record can retain more information than the averaged sensor state. Schematically,

$$\rho_\theta = \sum_r p(r\mid\theta)\rho_{\theta\mid r}.$$

Keeping the classical label $r$ produces a classical-quantum state whose QFI is

$$F_Q^{\rm record} = F_C[p(r\mid\theta)] + \sum_r p(r\mid\theta) F_Q[\rho_{\theta\mid r}].$$

Discarding $r$ cannot increase information. This identity explains why trajectory monitoring, heralded erasure, and syndrome records can outperform an unmonitored reduced-state protocol. It also gives the correct accounting for postselection: a high-information successful branch is weighted by its success probability, and parameter-dependent success itself carries classical information. Quantum Trajectories develops conditional states and monitored records.

Quantum Error-Corrected Sensing

Ordinary quantum error correction protects an unknown logical state. In metrology, recovery must do something subtler: suppress the dominant errors while preserving a nontrivial logical action of the signal Hamiltonian. Correcting the operator that generates the signal can make a perfectly stable but perfectly insensitive sensor.

Consider a finite-dimensional Markovian model

$$\frac{d\rho}{dt} = -i\omega[G,\rho] + \sum_j \left( L_j\rho L_j^\dagger - \frac12\{L_j^\dagger L_j,\rho\} \right).$$

Define the real Lindblad span

$$\mathcal S = \operatorname{span}_{\mathbb R} \left\{ I, L_j^{\rm H}, iL_j^{\rm AH}, (L_j^\dagger L_k)^{\rm H}, i(L_j^\dagger L_k)^{\rm AH} \right\},$$

where $A^{\rm H}=(A+A^\dagger)/2$ and $A^{\rm AH}=(A-A^\dagger)/2$. Under the strong assumptions of noiseless ancillas and arbitrarily fast, accurate control and recovery, Heisenberg scaling in total sensing time is attainable if and only if

$$G\notin\mathcal S.$$

This is the Hamiltonian-not-in-Lindblad-span, or HNLS, condition.

Two one-qubit geometries

If both signal and dephasing act through $Z$,

$$G=Z/2, \qquad L=\sqrt{\gamma/2}\,Z,$$

then $G\in\mathcal S$. Any code that makes $Z$ errors logically invisible also removes the DC $Z$ signal at leading order. Fast ideal QEC cannot restore Heisenberg scaling for that model.

This choice of $L$ gives coherence decay $e^{-\gamma t}$, matching the Ramsey convention above. If instead $G=X/2$ with the same $L$, then $G\notin\mathcal S$. With a noiseless ancilla, the codewords

$$|0_L\rangle = |+\rangle_S|0\rangle_A, \qquad |1_L\rangle = |-\rangle_S|1\rangle_A$$

send a sensor $Z$ error outside the code space, while $X_S/2$ acts as a logical $Z_L/2$ signal. Frequent syndrome extraction can in principle suppress the noise without erasing the signal.

The theorem is not an off-the-shelf hardware guarantee. Real ancillas are noisy; recoveries consume time; controls have finite strength; higher-order errors accumulate; and syndrome measurements can introduce bias. A useful experiment must compare the corrected sensor with the best uncorrected and classical strategies at the same elapsed time, sensing volume, signal exposure, bandwidth, and hardware inventory. Why Quantum Error Correction Is Possible explains the Knill–Laflamme condition and general recovery theory.

Worked precision comparisons

Markovian Ramsey sensor

Take $N=100$ independent qubits, coherence rate $\gamma=10\ \mathrm{s}^{-1}$, total averaging time $T_{\rm tot}=10\ \mathrm s$, and negligible dead time. The optimal product interrogation is

$$t_\star = \frac{1}{2\gamma} = 0.050\ \mathrm s.$$

The total QFI ceiling is

$$F_Q^{\rm tot} = T_{\rm tot} \frac{N}{2e\gamma} \simeq 18.4\ \mathrm{s}^2,$$

so

$$\Delta\omega \geq \frac{1}{\sqrt{F_Q^{\rm tot}}} \simeq 0.233\ \mathrm{rad\,s^{-1}}.$$

An ideal 100-qubit GHZ probe uses $t_\star=1/(2N\gamma)=0.50\ \mathrm{ms}$ and has the same total-QFI ceiling. It requires one hundred times as many preparation and readout cycles. Even a small fixed overhead therefore breaks the formal tie against it.

Lossy optical phase

For equal loss in both arms, let $\tau=0.90$ and count $\overline N=10^6$ total incident photons. The channel bound gives

$$\Delta\phi \geq \sqrt{ \frac{0.10}{0.90\times10^6} } \simeq 3.33\times10^{-4}\ \mathrm{rad}.$$

A coherent-state shot-noise scale based on the detected mean $\tau\overline N$ is approximately $1/\sqrt{\tau\overline N}=1.05\times10^{-3}\ \mathrm{rad}$. Thus this particular ultimate bound allows at most a factor $1/\sqrt{1-\tau}\simeq3.16$ improvement in standard deviation over that benchmark. It does not prove that a realizable source and receiver attain the factor.

Common-mode floor

Set $\sigma_{\rm i}^2=1$ and $\sigma_{\rm c}^2=0.01$ in the correlated Gaussian model. For $N=100$ sensors,

$$F_C = \frac{100}{1+100(0.01)} = 50,$$

rather than the independently modeled value $100$. For $N=1000$, $F_C\simeq90.9$, already near the asymptotic ceiling $100$. Adding sensors cannot average away the shared fluctuation.

A longer coherence time or a larger modeled QFI does not by itself establish better measured precision. The useful checks are the implemented likelihood, its signal response, the complete record including failures, and elapsed time. Keep the chronological data when testing drift; pooling correlated cycles as independent observations can create an apparent gain.

Exercises

Exercise 1: QFI of a dephased Ramsey qubit

For

$$\rho_\phi = \frac12 \left[ I + \eta(\cos\phi\,X+\sin\phi\,Y) \right],$$

with parameter-independent $0\leq\eta<1$, use the qubit Bloch-vector formula

$$F_Q = |\partial_\phi\mathbf r|^2 + \frac{(\mathbf r\cdot\partial_\phi\mathbf r)^2} {1-|\mathbf r|^2}$$

to derive the phase and frequency QFIs.

Solution

The Bloch vector is

$$\mathbf r = \eta(\cos\phi,\sin\phi,0),$$

so

$$\partial_\phi\mathbf r = \eta(-\sin\phi,\cos\phi,0).$$

Therefore $|\partial_\phi\mathbf r|^2=\eta^2$ and $\mathbf r\cdot\partial_\phi\mathbf r=0$. Hence

$$F_Q(\phi)=\eta^2.$$

For $\phi=\omega t$, the chain rule gives

$$F_Q(\omega,t) = \left(\frac{\partial\phi}{\partial\omega}\right)^2 F_Q(\phi) = t^2\eta(t)^2.$$

Exercise 2: A Kraus bound for phase dephasing

Take $0<\eta<1$, $a=\sqrt{(1+\eta)/2}$, $b=\sqrt{(1-\eta)/2}$, and phase-channel Kraus operators $K_0=a e^{-i\phi Z/2}$ and $K_1=bZ e^{-i\phi Z/2}$. At $\phi=0$, rotate them by $u(\phi)=e^{-i\phi h}$ with $h_{00}=h_{11}=0$ and $h_{01}=h_{10}=-1/(4ab)$. Show that the resulting channel-extension bound is linear in the number $N$ of parallel uses.

Solution

The rotated derivatives are

$$\dot{\widetilde K}_0=-i(a/2+hb)Z, \qquad \dot{\widetilde K}_1=-i(b/2+ha)I,$$

where $h=-1/(4ab)$ denotes the off-diagonal matrix element. Thus $\beta=-[1/2+2hab]Z=0$, while

$$\alpha=[(a/2+hb)^2+(b/2+ha)^2]I =\frac{\eta^2}{4(1-\eta^2)}I.$$

The bound is therefore $F_Q^{(N)}\leq N\eta^2/(1-\eta^2)$. For fixed $\eta<1$ it excludes asymptotic $N^2$ growth for this parallel phase-channel model, including passive ancillas. It need not be tight at finite $N$ and diverges as $\eta\to1$. In frequency sensing $\phi=\omega t$ and $\eta=\eta(t)$, choosing an $N$-dependent interrogation time requires optimizing the resulting time-dependent bound anew.

Exercise 3: Optimal interrogation with dead time

Maximize

$$f(t)=\frac{t^2e^{-2at}}{t+d}$$

for $a>0$ and $d\geq0$. Check the $d=0$ limit.

Solution

For $t>0$, logarithmic differentiation gives

$$\frac{f'(t)}{f(t)} = \frac{2}{t} -2a -\frac{1}{t+d}.$$

Setting this to zero and clearing denominators gives

$$2at^2+(2ad-1)t-2d=0.$$

The positive root is

$$t_\star = \frac{ 1-2ad+\sqrt{1+12ad+4a^2d^2} }{4a}.$$

At $d=0$, this reduces to $t_\star=1/(2a)$. The other quadratic root is nonpositive and is not an interrogation time.

Exercise 4: Loss bound and maximum allowed gain

For equal transmissivity $\tau=0.75$ in both arms and total incident mean photon number $\overline N=4\times10^5$, evaluate the channel loss bound. Compare it with the detected-photon coherent scale $1/\sqrt{\tau\overline N}$.

Solution

The loss bound is

$$\Delta\phi \geq \sqrt{ \frac{0.25}{0.75(4\times10^5)} } = 9.13\times10^{-4}\ \mathrm{rad}.$$

The coherent scale is

$$\Delta\phi_{\rm coh} \simeq \frac{1}{\sqrt{0.75(4\times10^5)}} = 1.83\times10^{-3}\ \mathrm{rad}.$$

The bound therefore allows at most a factor of two reduction in standard deviation relative to this benchmark, equal to $1/\sqrt{1-\tau}$. It is an allowed ceiling, not an achieved protocol.

Exercise 5: Common-mode Fisher information

Use the Sherman–Morrison identity to invert

$$\Sigma = \sigma_{\rm i}^2I + \sigma_{\rm c}^2\mathbf1\mathbf1^{\mathsf T}$$

and derive $F_C=\mathbf1^{\mathsf T}\Sigma^{-1}\mathbf1$.

Solution

The rank-one inverse is

$$\Sigma^{-1} = \frac{1}{\sigma_{\rm i}^2}I - \frac{\sigma_{\rm c}^2} {\sigma_{\rm i}^2(\sigma_{\rm i}^2+N\sigma_{\rm c}^2)} \mathbf1\mathbf1^{\mathsf T}.$$

Using $\mathbf1^{\mathsf T}\mathbf1=N$ gives

$$\begin{aligned} F_C &= \frac{N}{\sigma_{\rm i}^2} - \frac{\sigma_{\rm c}^2N^2} {\sigma_{\rm i}^2(\sigma_{\rm i}^2+N\sigma_{\rm c}^2)} \\ &= \frac{N}{\sigma_{\rm i}^2+N\sigma_{\rm c}^2}. \end{aligned}$$

Exercise 6: Does the code preserve the signal?

For codewords

$$|0_L\rangle=|+\rangle_S|0\rangle_A, \qquad |1_L\rangle=|-\rangle_S|1\rangle_A,$$

show that a sensor error $Z_S$ is detectable and that the signal $X_S$ acts nontrivially within the code.

Solution

Since $Z|+\rangle=|-\rangle$ and $Z|-\rangle=|+\rangle$,

$$Z_S|0_L\rangle = |-\rangle_S|0\rangle_A, \qquad Z_S|1_L\rangle = |+\rangle_S|1\rangle_A.$$

Both states are orthogonal to both codewords, so $PZ_SP=0$ and the error can be detected without learning the logical amplitude. Meanwhile,

$$X_S|0_L\rangle=|0_L\rangle, \qquad X_S|1_L\rangle=-|1_L\rangle.$$

Thus $PX_SP=Z_L$ is a nontrivial logical generator. The code separates the transverse signal from the dephasing error in the ideal model.

Exercise 7: Postselection accounting

A protocol succeeds with parameter-independent probability $q=0.1$. Its successful records have Fisher information $F_s=50$; failures are discarded and carry no parameter information. What is the information per attempted trial? How does the answer change conceptually if $q=q(\theta)$?

Solution

For parameter-independent success, the complete-record Fisher information is

$$F_{\rm attempt} = qF_s = 0.1(50) = 5.$$

Reporting $50$ per attempt would overstate the information by a factor of ten. If $q$ depends on $\theta$, the success/failure label contributes binary Fisher information

$$F_{\rm flag} = \frac{[\partial_\theta q(\theta)]^2} {q(\theta)[1-q(\theta)]},$$

and the conditional successful and failed branches must also be weighted by their probabilities. Discarding the failure label can introduce both lost information and selection bias.

Exercise 8: Derive the decay-rate QFI

For a qubit with Bloch vector $\mathbf r=(e^{-\gamma t},0,0)$ and $\gamma>0$, derive the QFI for $\gamma$ and inspect its short-time limit.

Solution

The derivative is

$$\partial_\gamma\mathbf r = (-t e^{-\gamma t},0,0).$$

The derivative is parallel to $\mathbf r$, so the qubit Bloch formula gives

$$\begin{aligned} F_Q(\gamma,t) &= t^2e^{-2\gamma t} + \frac{t^2e^{-4\gamma t}} {1-e^{-2\gamma t}} \\ &= \frac{t^2e^{-2\gamma t}} {1-e^{-2\gamma t}}. \end{aligned}$$

For $t\to0$, $1-e^{-2\gamma t}\simeq2\gamma t$, so

$$F_Q(\gamma,t) \simeq \frac{t}{2\gamma}.$$

The information per shot tends to zero even though the state approaches a pure state and the Bloch-radius derivative is measured relative to a small mixedness. Decoherence is the parameter-encoding mechanism in this task.

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