Standard Quantum Limit

In quantum metrology, the standard quantum limit (SQL) usually names the inverse-square-root precision scaling achieved by independent probes under a fixed single-probe resource constraint. If each experimental round uses $N$ probes and the round is repeated independently $\nu$ times, the attainable local standard-deviation scale in a regular, efficiently estimated model is

$$\Delta\theta_{\mathrm{SQL}} \propto \frac{1}{\sqrt{\nu N}}.$$

The proportionality constant contains the physics of the probe, parameter encoding, interrogation time, contrast, measurement, and units. For a locally unbiased estimator in a regular normalized qubit phase model with product probes,

$$\Delta\theta_{\mathrm{SQL}} \ge \frac{1}{\sqrt{\nu N}}.$$

Required background. Use classical Fisher information and quantum Fisher information to quantify sensitivity, Cramér–Rao bounds for local-unbiasedness conditions, variance and covariance for independent sums, and the Born rule for outcome probabilities.

The name is overloaded. In continuous position measurement, “the SQL” often means an optimum between measurement imprecision and quantum backaction. In laser theory, it can label a model-dependent linewidth. Those are not the same bound as independent-probe scaling. A trustworthy statement names the task and writes the bound rather than relying on the acronym.

Where the 1/√N Law Comes From

Averaging independent outcomes

Suppose one probe produces an outcome $X$ with parameter-dependent mean and variance

$$\mu(\theta) = \mathbb E_\theta[X], \qquad \sigma^2(\theta) = \operatorname{Var}_\theta(X).$$

For $R=\nu N$ independent, identically distributed probe outcomes, the sample mean

$$\overline X_R = \frac{1}{R} \sum_{j=1}^{R}X_j$$

has variance

$$\operatorname{Var}_\theta(\overline X_R) = \frac{\sigma^2(\theta)}{R}.$$

Near an operating point where $\partial_\theta\mu\ne0$, local error propagation gives

$$\Delta\theta \simeq \frac{ \sigma(\theta) }{ \sqrt{\nu N}\, \left| \partial_\theta\mu(\theta) \right| }.$$

The quantum content is in the one-probe distribution: even after technical and detector noise are removed, a probe not prepared in an eigenstate of the readout generally has irreducible outcome variance. Independence then turns that single-probe variance into inverse-square-root scaling.

This argument is local and asymptotic. It assumes a calibrated response, a stable operating point, finite variance, and enough data that the estimator behaves approximately linearly. It does not address phase wrapping, bias, prior information, correlated noise, or an unknown drift shared by every probe.

Fisher-information derivation

Let one measured probe have outcome distribution $p(x\mid\theta)$ and classical Fisher information

$$F_C^{(1)}(\theta) = \sum_x \frac{ \left[ \partial_\theta p(x\mid\theta) \right]^2 }{ p(x\mid\theta) }.$$

Independent likelihoods multiply, log likelihoods add, and Fisher information is additive:

$$F_C^{(R)} = R F_C^{(1)} = \nu N F_C^{(1)}.$$

For a locally unbiased estimator, the classical Cramér–Rao bound therefore gives

$$\operatorname{Var}(\widehat\theta) \ge \frac{1}{ \nu N F_C^{(1)} }.$$

The quantum Fisher information $F_Q$ is the maximum of $F_C$ over all POVMs on that state. Thus

$$F_C^{(1)} \le F_Q^{(1)},$$

and the independent-probe quantum bound is

$$\Delta\theta \ge \frac{1}{ \sqrt{ \nu N F_Q^{(1)} } }.$$

This is the general SQL formula. The familiar unit-prefactor expression follows only when the parameter and generator are normalized so that the best single probe has $F_Q^{(1)}=1$.

Product and Separable Probe Bound

Consider a unitary encoding on $N$ distinguishable probes,

$$U_\theta = \exp(-i\theta H), \qquad H = \sum_{j=1}^{N}h_j.$$

For a pure state, quantum Fisher information is four times the generator variance:

$$F_Q[|\psi\rangle,H] = 4(\Delta H)^2_\psi.$$

For a product input,

$$\rho_{\mathrm{prod}} = \bigotimes_{j=1}^{N}\rho_j,$$

QFI is additive:

$$F_Q[ \rho_{\mathrm{prod}},H ] = \sum_{j=1}^{N} F_Q[\rho_j,h_j].$$

Let the spectral width of each local generator be bounded by

$$\lambda_{\max}(h_j)-\lambda_{\min}(h_j)\le g.$$

The one-probe QFI is therefore at most $g^2$, since a bounded observable has variance at most one quarter of its squared spectral width. Convexity of QFI extends the linear bound to parameter-independent mixtures of product states,

$$\rho_{\mathrm{sep}} = \sum_k p_k \bigotimes_{j=1}^{N} \rho_j^{(k)}:$$ $$F_Q[ \rho_{\mathrm{sep}},H ] \le Ng^2.$$

Consequently,

$$\Delta\theta_{\mathrm{sep}} \ge \frac{1}{ g\sqrt{\nu N} }.$$

For qubits with $h_j=\sigma_z^{(j)}/2$, the spectral width is $g=1$, giving the normalized SQL.

The local resource bound is essential. A bosonic probe with unbounded energy can have unbounded generator variance, so “one probe” is not a complete resource statement. For light, specify photon-number sectors, energy through the sample, or channel uses and the allowed state class. A fixed mean photon number alone does not bound local QFI over arbitrary states: a tiny coherent amplitude in a very high-number sector can make the variance arbitrarily large. Such local-QFI growth does not by itself give useful global phase precision. For spectroscopy, interrogation time enters the generator.

Why generator variance can mislead

For pure states under unitary encoding, $F_Q=4(\Delta H)^2$. For mixed states, only

$$F_Q[\rho,H] \le 4(\Delta H)^2_\rho$$

is guaranteed. The variance can contain classical uncertainty that carries no phase information.

For example, consider the incoherent mixture

$$\rho_{\mathrm{mix}} = \frac12 |0\rangle\langle0|^{\otimes N} + \frac12 |1\rangle\langle1|^{\otimes N}$$

with $H=J_z$. Its variance scales as $N^2$, but

$$[\rho_{\mathrm{mix}},J_z]=0,$$

so the state does not change under $\exp(-i\theta J_z)$ and

$$F_Q[\rho_{\mathrm{mix}},J_z] = 0.$$

A large collective variance is therefore not by itself evidence of sub-SQL sensitivity.

Binomial Phase Example

Take one two-outcome probe with

$$\begin{aligned} p_+(\theta) &= \frac{ 1+\sin\theta }{2}, \\ p_-(\theta) &= \frac{ 1-\sin\theta }{2}. \end{aligned}$$

The one-probe Fisher information is

$$\begin{aligned} F_C^{(1)} &= \sum_{s=\pm} \frac{ \left( \partial_\theta p_s \right)^2 }{ p_s } \\ &= 1, \end{aligned}$$

at regular points where both outcomes have positive probability. The limit at a zero-probability outcome is also one, but the score information exactly at that nonregular point differs; the regular bound cannot be asserted there by continuity alone.

For $R$ independent probes, the number $K$ of plus outcomes is binomial:

$$K \sim \operatorname{Binomial} \left( R, \frac{1+\sin\theta}{2} \right).$$

At the mid-fringe point $\theta=0$,

$$\mathbb E[K] = \frac{R}{2}, \qquad \operatorname{Var}(K) = \frac{R}{4}.$$

The estimator

$$\widehat\theta = \frac{2K}{R}-1$$

has mean $\sin\theta$, so it is locally unbiased at zero and has exactly, at that point,

$$\Delta\widehat\theta = \frac{1}{\sqrt R} = \frac{1}{\sqrt{\nu N}}.$$

The same likelihood is realized by ideal Ramsey or two-path interference experiments after a convention-dependent choice of phase origin. The SQL is not tied to one apparatus; it is the information scaling of the independent binary trials.

Projection Noise of Independent Spins

For $N$ two-level probes, define collective spin

$$\mathbf J = \frac12 \sum_{j=1}^{N} \boldsymbol{\sigma}^{(j)}.$$

Prepare each spin along $+x$. The product state has

$$\langle J_x\rangle = \frac N2, \qquad (\Delta J_y)^2 = (\Delta J_z)^2 = \frac N4.$$

Let the parameter generate a rotation about $z$:

$$U_\theta = \exp(-i\theta J_z).$$

Near $\theta=0$,

$$\left. \frac{\partial}{\partial\theta} \langle J_y\rangle_\theta \right|_{\theta=0} = \langle J_x\rangle = \frac N2.$$

Error propagation for one round gives

$$\Delta\theta = \frac{ \Delta J_y }{ \left| \partial_\theta\langle J_y\rangle \right| } = \frac{ \sqrt N/2 }{ N/2 } = \frac{1}{\sqrt N}.$$

After $\nu$ independent rounds,

$$\Delta\theta = \frac{1}{\sqrt{\nu N}}.$$

This is quantum projection noise: each spin measurement gives a discrete outcome even when state preparation and readout are ideal. Detector noise is additional and must be calibrated separately. Ramsey Interferometry develops the pulse sequence and physical spectroscopy.

Optical Shot Noise

An ideal coherent optical mode has Poisson photon-number statistics,

$$\operatorname{Var}(n) = \langle n\rangle = \overline n.$$

For a balanced lossless interferometer with a coherent input and vacuum in the other input, photon counts at the two outputs are independent Poisson variables with means $\mu_\pm=\overline n(1\pm\cos\phi)/2$. Away from zero-mean outputs, their information adds to

$$F_C=\frac{(\partial_\phi\mu_+)^2}{\mu_+}+\frac{(\partial_\phi\mu_-)^2}{\mu_-}=\overline n.$$

The phase uncertainty therefore scales as

$$\Delta\phi_{\mathrm{shot}} \ge \frac{1}{\sqrt{\overline n}}.$$

For $\nu$ identical independent intervals,

$$\Delta\phi_{\mathrm{shot}} \ge \frac{1}{ \sqrt{ \nu\overline n } }.$$

This is the optical shot-noise limit for the declared coherent-state model. It depends on what optical modes and phase references are counted, whether $\overline n$ is incident or detected, and whether loss is assigned to the state, channel, or detector. The full interferometer likelihood and dark-fringe subtleties belong to Interferometers.

Shot Noise, Projection Noise, and the SQL

The terms overlap but are not synonyms in every context.

Term	Typical random variable	Independent-probe behavior	Essential declaration
projection noise	outcomes of uncorrelated atoms, spins, or qubits	variance of a collective count is proportional to $N$	state, basis, contrast, and readout model
optical shot noise	coherent-state photoelectrons or photon counts	Poisson variance equals the mean count	incident or detected flux, modes, bandwidth, and efficiency
standard quantum limit	estimator precision under a named independent or separable-probe class	$\Delta\theta\propto(\nu N)^{-1/2}$	task, generator, resources, estimator, and comparator
technical noise	drift, electronics, classical intensity noise, vibration, calibration	need not average as $1/\sqrt{\nu N}$	spectrum, correlation time, calibration, and subtraction

“Shot-noise limited” means measured noise agrees with a declared shot-noise model over a specified operating range. It does not mean every uncertainty is quantum, nor does it establish accuracy.

Loss, interrogation time, and repeated passes

A comparison must fix the counted probes, total time, interaction strength, allowed inputs, prior range, and estimator conditions. The following changes illustrate why a numerical SQL can move even though the inverse-square-root argument is unchanged.

Detection loss

If each of $R_{\mathrm{in}}$ independent probes is detected with parameter-independent probability $\eta$ and the loss outcome carries no signal information, the expected detected count is

$$R_{\mathrm{det}} = \eta R_{\mathrm{in}}.$$

For the normalized model,

$$\Delta\theta \gtrsim \frac{1}{ \sqrt{ \eta R_{\mathrm{in}} } }.$$

Reporting $1/\sqrt{R_{\mathrm{det}}}$ while comparing against a benchmark charged for all incident probes can manufacture an apparent gain. The numerator and denominator must use the same boundary.

Interrogation time

If a frequency $\omega$ generates phase $\theta=\omega T$, then

$$\Delta\omega_{\mathrm{SQL}} \ge \frac{1}{ T\sqrt{\nu N} }$$

in the ideal normalized model. But increasing $T$ may reduce contrast, increase decoherence, narrow dynamic range, and reduce the number of rounds available in a fixed total time. A frequency benchmark must count

$$T_{\mathrm{tot}} \simeq \nu \left( T+T_{\mathrm{dead}} \right),$$

not only the coherent interrogation interval.

Multiple passes

A probe that traverses a phase element $m$ times accumulates $m\theta$. Treating it as one free probe while comparing against a single-pass strategy undercounts the interaction. A matched resource may be the total number of channel uses,

$$R_{\mathrm{use}} = mR_{\mathrm{probe}},$$

or the total energy deposited in a sample.

Scaling, Constants, and Floors

The four curves are explicit mathematical comparisons, not experimental data. The $R^{-1/2}$ benchmark and a constant-factor improvement $0.7R^{-1/2}$ have the same log–log slope. Adding a resource-independent variance $0.03^2$ bends the curve upward toward a standard-deviation floor of $0.03$. The ideal $R^{-1}$ curve has a different exponent; its attainability requires a different physical strategy and assumptions.

Suppose an uncertainty model is

$$(\Delta\theta)^2 = \frac{A}{R} + B.$$

The first term averages as independent noise. The second is a floor from a correlated fluctuation, calibration uncertainty, or unresolved systematic. The crossover occurs near

$$R_\times = \frac{A}{B}.$$

Below $R_\times$, a log–log fit may look SQL-like. Above it, collecting more probes produces little improvement.

Three claim types must be separated:

• SQL scaling: an exponent compatible with $R^{-1/2}$;
• sub-SQL constant: lower variance than a matched independent-probe benchmark at a specified $R$;
• super-classical scaling: an exponent better than $-1/2$ over a justified range.

A state can provide a useful constant-factor gain while retaining $R^{-1/2}$ asymptotic scaling. Conversely, a fitted steep slope over a narrow pre-asymptotic interval does not establish a new asymptotic law.

What Beating the SQL Means

For the normalized phase problem, define a metrological noise parameter

$$\xi^2 = \nu N (\Delta\theta)^2.$$

The independent-probe benchmark has $\xi^2\ge1$. A measured value

$$\xi^2<1$$

supports sub-SQL performance only if the same losses, duration, prior, estimator, and accepted-run rule are used for the quantum strategy and comparator.

For collective spins, the Wineland squeezing parameter is

$$\xi_R^2 = \frac{ N(\Delta J_\perp)^2 }{ \left| \langle\mathbf J\rangle \right|^2 },$$

with the transverse direction and phase response chosen consistently. $\xi_R^2<1$ certifies a phase sensitivity below the coherent-spin-state benchmark under those conventions.

For $N$ qubits and a collective linear generator, the condition

$$F_Q[\rho,J_{\mathbf n}] > N$$

implies that $\rho$ is entangled and has sub-SQL potential for that local estimation problem. This does not guarantee that an available measurement attains the QFI, that the advantage survives loss, or that the full sensor outperforms a classical instrument.

The SQL Is Not a Universal Quantum Bound

It is not the Heisenberg uncertainty relation

The SQL is an estimation benchmark for a strategy class. The Robertson uncertainty relation constrains variances of two observables in one state. One cannot derive a complete sensor limit merely by inserting a position–momentum or number–phase slogan.

Entanglement can change the ideal scaling

Entangled probes can have QFI proportional to $N^2$ in noiseless local models, suggesting

$$\Delta\theta \propto \frac{1}{N\sqrt\nu}.$$

That ideal Heisenberg scaling requires careful counting of generator range, channel uses, interactions, time, and prior information. It also does not imply an end-to-end advantage.

Noise can restore inverse-square-root scaling

For many common independent loss and decoherence models, asymptotically optimal quantum strategies retain at most a constant-factor advantage over $N^{-1/2}$ scaling. This is a broad and important result, not a universal theorem for every correlated, non-Markovian, error-corrected, or controlled sensing model. The noise channel and available controls decide the bound.

Classical improvements can beat a weak baseline

Feedback, a better estimator, more complete likelihood, higher collection efficiency, longer interrogation, or a better operating point can outperform an earlier “shot-noise” number without using a nonclassical probe. The correct comparator is the best allowed independent-probe strategy under matched resources, not a convenient historical implementation.

Continuous-Measurement SQL Is Different

In a weak continuous position measurement, increasing measurement strength can reduce imprecision while increasing force backaction. A generic added displacement-noise spectrum has the structure

$$S_{xx}^{\mathrm{add}}(\omega) ={} S_{xx}^{\mathrm{imp}}(\omega) + \left| \chi(\omega) \right|^2 S_{FF}^{\mathrm{BA}}(\omega) + 2\operatorname{Re} \left[ \chi(\omega)^* S_{xF}(\omega) \right],$$

Here $x_{\mathrm{add}}(\omega)=x_{\mathrm{imp}}(\omega)+\chi(\omega)F(\omega)$. Use two-sided symmetrized spectra with $S_{xF}(\omega)=\int dt\,e^{i\omega t}\langle\{x_{\mathrm{imp}}(t),F(0)\}\rangle/2$ for stationary zero-mean noise. Then $\chi$ is the mechanical susceptibility, $S_{xx}^{\mathrm{imp}}$ is imprecision, and $S_{FF}^{\mathrm{BA}}$ is backaction-force noise. This cross-spectrum convention fixes the complex conjugate in the interference term.

Under a particular uncorrelated quantum-limited detector model, minimizing the first two terms produces a frequency-dependent SQL. Correlations, variational readout, backaction evasion, or a quantum nondemolition observable can alter that optimum. This spectral tradeoff is conceptually related to quantum estimation but is not the $\nu N$ independent-probe SQL derived above.

Expressing a metrological gain

If variances are compared, a gain factor can be written

$$G = \frac{ (\Delta\theta_{\mathrm{SQL}})^2 }{ (\Delta\theta_{\mathrm{test}})^2 }.$$

The gain in decibels is

$$G_{\mathrm{dB}} = 10\log_{10}G.$$

For example, $3$ dB corresponds to approximately half the variance, not half the standard deviation. State the matched physical benchmark and estimator assumptions alongside this number.

Exercises

1. Fisher information of a binary fringe

For

$$p_\pm(\theta) = \frac{1\pm\sin\theta}{2},$$

show that one probe has $F_C^{(1)}=1$. What precision bound follows for $R$ independent probes?

Solution

The derivatives are

$$\partial_\theta p_\pm = \pm\frac{\cos\theta}{2}.$$

Hence

$$\begin{aligned} F_C^{(1)} &= \frac{\cos^2\theta}{4} \left( \frac{1}{p_+} + \frac{1}{p_-} \right) \\ &= \frac{\cos^2\theta}{4} \frac{p_++p_-}{p_+p_-} \\ &= 1, \end{aligned}$$

because $p_+p_-=\cos^2\theta/4$ at regular points. The limiting information is one near a zero-probability outcome, but the ordinary regularity assumptions fail at that point. For regular operating points, additivity gives $F_C^{(R)}=R$, so a locally unbiased estimator satisfies

$$\Delta\theta \ge \frac{1}{\sqrt R}.$$

2. Additivity for independent likelihoods

Let $X_1,\ldots,X_R$ be independent with one-probe score

$$s_j = \partial_\theta \log p(X_j\mid\theta).$$

Show that the total Fisher information is $R F_C^{(1)}$ under the usual regularity condition $\mathbb E[s_j]=0$.

Solution

The joint score is

$$S = \sum_{j=1}^{R}s_j.$$

Therefore

$$\mathbb E[S^2] = \sum_j \mathbb E[s_j^2] + \sum_{j\ne k} \mathbb E[s_js_k].$$

Independence and zero mean give

$$\mathbb E[s_js_k] = \mathbb E[s_j]\mathbb E[s_k] = 0$$

for $j\ne k$. Each diagonal term is $F_C^{(1)}$, so

$$F_C^{(R)} = \mathbb E[S^2] = R F_C^{(1)}.$$

3. Coherent-spin SQL

A coherent spin state has $\langle J_x\rangle=N/2$ and $(\Delta J_y)^2=N/4$. A small $z$ rotation gives $\partial_\theta\langle J_y\rangle=N/2$. Derive the one-round phase uncertainty and evaluate it for $N=10^6$.

Solution

Error propagation gives

$$\Delta\theta = \frac{ \sqrt{N}/2 }{ N/2 } = \frac{1}{\sqrt N}.$$

For $N=10^6$,

$$\Delta\theta = 10^{-3}\ \mathrm{rad}$$

in the ideal local model.

4. Detection efficiency

An experiment sends $R_{\mathrm{in}}=10^8$ independent normalized probes and has efficiency $\eta=0.64$. Find the detection-limited SQL if no information is obtained from lost probes. Compare it with the incorrectly loss-free value.

Solution

The expected detected resource is

$$R_{\mathrm{det}} = 0.64\times10^8 = 6.4\times10^7.$$

Thus

$$\Delta\theta_{\mathrm{SQL}} \gtrsim \frac{1}{\sqrt{6.4\times10^7}} = 1.25\times10^{-4}.$$

Ignoring loss would give $10^{-4}$. The loss increases the uncertainty by $1/\sqrt{0.64}=1.25$.

5. Statistical-to-floor crossover

Suppose

$$(\Delta\theta)^2 = \frac{10^{-2}}{R} + 10^{-10}.$$

Find the resource $R_\times$ where the two contributions are equal and the asymptotic uncertainty floor.

Solution

Set the two variance terms equal:

$$\frac{10^{-2}}{R_\times} = 10^{-10}.$$

Therefore

$$R_\times = 10^8.$$

For $R\gg10^8$, the variance approaches $10^{-10}$, so the standard deviation approaches

$$\Delta\theta_{\mathrm{floor}} = 10^{-5}.$$

6. A variance trap

For

$$\rho = \frac12 |0\rangle\langle0|^{\otimes N} + \frac12 |1\rangle\langle1|^{\otimes N},$$

compute $(\Delta J_z)^2$ and explain why the state has zero QFI for a $J_z$-generated phase.

Solution

The two components have $J_z$ eigenvalues $+N/2$ and $-N/2$. The mean is zero and

$$(\Delta J_z)^2 = \frac12 \left( \frac N2 \right)^2 + \frac12 \left( -\frac N2 \right)^2 = \frac{N^2}{4}.$$

Nevertheless, $\rho$ is diagonal in the $J_z$ basis, so

$$e^{-i\theta J_z} \rho e^{i\theta J_z} = \rho.$$

No measurement can infer $\theta$ from an unchanged state, and $F_Q[\rho,J_z]=0$. For mixed states, generator variance can be classical rather than phase-sensitive.

7. Constant gain versus scaling gain

A protocol achieves

$$\Delta\theta = \frac{0.8}{\sqrt R}.$$

Does it beat the unit-prefactor SQL? Does it exhibit Heisenberg scaling? Compute its variance gain in decibels.

Solution

At matched $R$, its variance is $0.64/R$ rather than $1/R$, so it beats the unit-prefactor SQL by

$$G = \frac{1}{0.64} = 1.5625.$$

The decibel gain is

$$G_{\mathrm{dB}} = 10\log_{10}(1.5625) \approx 1.94\ \mathrm{dB}.$$

Its exponent remains $-1/2$, so this is a constant-factor sub-SQL gain, not Heisenberg scaling.

8. Count multiple passes

$R$ probes each pass through a phase sample $m$ times, producing a local phase response $m$ times larger than a single pass. The raw error propagation gives $\Delta\theta=1/(m\sqrt R)$. Express this in terms of the total channel uses $R_{\mathrm{use}}=mR$.

Solution

Since $R=R_{\mathrm{use}}/m$,

$$\Delta\theta = \frac{1}{ m\sqrt{R_{\mathrm{use}}/m} } = \frac{1}{ \sqrt{ mR_{\mathrm{use}} } }.$$

At fixed pass number $m$, the scaling with total channel uses remains $R_{\mathrm{use}}^{-1/2}$, though the prefactor improves. If $m$ itself grows with the resource, interrogation time, loss, and sample exposure must also be included before assigning a scaling class.

9. Why unbiasedness matters

The estimator $\widehat\theta=0$ has zero variance for every data set. Why does it not violate the Cramér–Rao bound for an unknown $\theta$?

Solution

Its expectation is always zero, so its bias is

$$b(\theta) = \mathbb E_\theta[\widehat\theta]-\theta = -\theta.$$

It is unbiased in value at $\theta=0$ but is not locally unbiased anywhere: the derivative of its mean is zero instead of one. Its mean-square error is

$$\mathbb E_\theta[ (\widehat\theta-\theta)^2 ] = \theta^2.$$

Variance alone is not a valid risk for comparing arbitrarily biased estimators. One must use an unbiased or bias-aware bound, a local minimax risk, or a Bayesian risk with a declared prior.

10. Mean photon number is not a variance bound

For an integer $M>\bar n$, consider

$$|\psi_M\rangle=\sqrt{1-\bar n/M}|0\rangle+\sqrt{\bar n/M}|M\rangle.$$

At fixed mean photon number $\bar n$, compute the local QFI for a phase generated by the photon-number operator. Does its growth prove arbitrarily accurate global estimation?

Solution

The moments are $\langle n\rangle=\bar n$ and $\langle n^2\rangle=\bar n M$, hence

$$F_Q=4\operatorname{Var}(n)=4(\bar n M-\bar n^2).$$

It grows without bound as $M$ grows, despite fixed mean energy. But the encoded state is periodic under $\theta\mapsto\theta+2\pi/M$, with many aliases in a broad phase interval. Local QFI alone gives no global-risk guarantee. Preparation, a usable phase reference, the prior, and an actual inference protocol must be specified.

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