Heisenberg Scaling

Heisenberg scaling is the ideal inverse-resource behavior

$$\Delta\theta \propto \frac{1}{R},$$

for the root-mean-square error of estimating a parameter $\theta$ with a declared resource $R$. Equivalently, the mean-square error scales as $R^{-2}$. In the standard noiseless model of $N$ bounded probes acquiring the same phase once, an entangled probe can have quantum Fisher information proportional to $N^2$, whereas independent probes have information proportional to $N$. This gives the familiar contrast

$$\Delta\theta_{\mathrm{HL}} \sim \frac{1}{N}, \qquad \Delta\theta_{\mathrm{SQL}} \sim \frac{1}{\sqrt N}.$$

The symbol $N$ is not self-interpreting. It might mean particles, photons, channel uses, passes through a sample, total interrogation time, energy, or the largest generator eigenvalue made available to the protocol. A claimed scaling law is therefore incomplete until the resource, loss function, prior information, noise model, success rule, and asymptotic limit are fixed.

Required background. Use quantum estimation to specify the parameter and loss, Fisher information for encoded sensitivity, Cramér–Rao bounds for local-unbiasedness assumptions, and the standard quantum limit for the independent-probe comparison.

Scaling exponents and precision bounds

Three claims that are often conflated should be separated:

Claim	Mathematical content	What must be shown
sub-SQL performance	smaller error than a matched independent-probe benchmark at one resource value	comparator, uncertainty, and resource boundary
Heisenberg scaling	error exponent $-1$ in a declared large-$R$ limit	a justified scaling range and control of other changing resources
Heisenberg limit	a lower bound with a model-dependent constant and risk	theorem assumptions, resource definition, and attainability

A protocol may beat the SQL by a constant factor while retaining $R^{-1/2}$ scaling. It may display an $R^{-1}$ slope over a finite interval and later cross back to $R^{-1/2}$ under loss. It may also attain ideal local Fisher information while failing globally because many parameter values produce the same data. None of these statements contradicts the others.

The term is historical rather than a direct application of the position–momentum uncertainty relation. Modern derivations use statistical distinguishability, generator spectra, query complexity, or Bayesian risk.

Noiseless Local Bound from Generator Width

Consider a differentiable unitary family

$$U_\theta = e^{-i\theta G}$$

acting on a pure probe $|\psi\rangle$. Its quantum Fisher information is

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

If $G$ has smallest and largest eigenvalues $g_{\min}$ and $g_{\max}$, Popoviciu’s variance inequality gives

$$(\Delta G)^2_\psi \le \frac{ (g_{\max}-g_{\min})^2 }{4}.$$

Consequently,

$$F_Q \le (g_{\max}-g_{\min})^2.$$

The maximum is attained by an equal superposition of extremal generator eigenstates,

$$|\psi_{\mathrm{ext}}\rangle = \frac{ |g_{\min}\rangle + e^{i\alpha}|g_{\max}\rangle }{\sqrt2}.$$

For $\nu$ independent repetitions, the locally unbiased quantum Cramér–Rao inequality then reads

$$\Delta\theta \ge \frac{1}{ \sqrt\nu\, (g_{\max}-g_{\min}) }.$$

This is a local inequality for the stated estimator class. Treating its right-hand side as an attainable global error requires a measurement, an estimator, and a way to resolve distinct likelihood branches.

Additive generators

Suppose $N$ elementary probes occupy distinct tensor factors, each experiences one use of the encoding, and

$$G_N = \sum_{j=1}^{N}g_j.$$

If every $g_j$ has spectral width

$$w = \lambda_{\max}(g_j) - \lambda_{\min}(g_j),$$

then the width of $G_N$ is $Nw$. The best possible local QFI obeys

$$F_Q^{(N)} \le N^2w^2,$$

and hence

$$\Delta\theta \ge \frac{1}{ Nw\sqrt\nu }.$$

The $N^{-1}$ dependence is Heisenberg scaling with the number of bounded generator uses per round. By contrast, product and parameter-independent separable inputs obey

$$F_Q^{(N)} \le Nw^2,$$

which recovers the independent-probe SQL.

Ancillas that never experience the parameter may enable useful controls or measurements, but they do not enlarge this generator width in the noiseless unitary model. Mixed inputs cannot exceed the pure-state maximum because QFI is convex.

GHZ Phase Amplification

For $N$ qubits, take

$$G_N = J_z = \frac12 \sum_{j=1}^{N}\sigma_z^{(j)}$$

and the GHZ state

$$|\mathrm{GHZ}_N\rangle = \frac{ |0\rangle^{\otimes N} + |1\rangle^{\otimes N} }{\sqrt2}.$$

The state occupies the two extremal eigenspaces of $J_z$. After phase encoding, an irrelevant global phase can be removed to give

$$|\psi_\theta\rangle = \frac{ |0\rangle^{\otimes N} + e^{iN\theta} |1\rangle^{\otimes N} }{\sqrt2}.$$

Thus the relative phase advances $N$ times faster than for one qubit. Since

$$(\Delta J_z)^2 = \frac{N^2}{4},$$

the QFI is

$$F_Q = N^2.$$

An appropriate parity-like readout has probabilities of the form

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

whose classical Fisher information is $N^2$ where both probabilities are positive. Its limiting value is also $N^2$ near zero-probability outcomes, but the score information exactly at those nonregular points is different. A regular quadrature operating point extracts the ideal QFI.

The same formula exposes the global problem: $p_\pm$ is periodic under $\theta\mapsto\theta+2\pi/N$. A fine fringe is sensitive but not uniquely identifying. Without prior localization or additional coarse measurements, the encoded state has $N$ repeated periods in a $2\pi$ interval. This cosine-only readout also has a reflection ambiguity within each period; a second quadrature resolves that reflection but not the $2\pi/N$ periodicity of the state.

The probabilities $(1+\cos\theta)/2$ and $(1+\cos4\theta)/2$ show how phase amplification increases local sensitivity while repeating the same outcome probabilities. Even infinitely many repetitions at this one setting cannot identify a unique phase across the full interval.

NOON States and Optical Phase

For two optical modes, a NOON state is

$$|\mathrm{NOON}\rangle = \frac{ |N,0\rangle + |0,N\rangle }{\sqrt2}.$$

If one arm acquires phase $\phi$, the two components develop relative phase $N\phi$. In the ideal fixed-$N$ model, the state therefore has $F_Q=N^2$ for the relative phase and exhibits $N$-fold fringes.

This example is conceptually clean but experimentally unforgiving. For equal independent transmission $\eta$ in both arms, the probability that all $N$ photons survive is exactly $\eta^N$, and the surviving NOON state retains its phase coherence. Losing a photon reveals which arm was occupied to the environment and removes this coherence. One-arm loss has different survival weights and must not be assigned the same formula. Conditioning only on intact events does not erase that cost: the rejected preparations, sample exposure, and elapsed time remain resources. Optimized lossy states can outperform a coherent-state benchmark by a constant factor, but fixed nonzero independent loss generally removes asymptotic $N^{-1}$ scaling.

Parallel Entanglement and Sequential Passes

Heisenberg scaling is not synonymous with multipartite entanglement. A single coherent two-level probe sent through the same phase element $N$ times undergoes

$$U_\theta^N = e^{-iN\theta g}.$$

Its relative phase and ideal QFI have the same $N$ and $N^2$ dependence as a parallel GHZ protocol. The resource is now $N$ channel uses or $N$ units of generator exposure, not one physical carrier.

Adaptive phase-estimation protocols combine interrogation powers such as $1,2,4,\ldots$ with feedback and repeated readout. They can resolve both coarse and fine phase bits without preparing a large GHZ state. Quantum Phase Estimation develops the algorithmic circuit; here the important metrological point is that controlled powers $U^{2^k}$ cost $2^k$ elementary queries unless the hardware resource model explicitly provides them at another cost.

Counting repeated queries

For a bounded elementary generator, query count gives a simple theoretical comparison. A real sensor may instead be constrained by interrogation and preparation time, energy through the sample, damage, phase-reference power, or ancilla quality. The comparison must keep those constraints explicit rather than counting only simultaneous carriers.

Block size versus total uses

Suppose each round uses an $N$-probe entangled block and there are $\nu$ independent rounds. The total number of elementary channel uses is

$$Q = \nu N.$$

The ideal local bound is

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

At fixed $Q$, increasing the coherent block size improves this local bound, with the formal extreme $N=Q$ giving $1/Q$. But then $\nu=1$, outside the usual repeated-sample regime used to justify efficient local estimators. Global phase-estimation strategies distribute the same budget across different interrogation powers to obtain both range and resolution.

Energy constraints require care

For an unbounded generator such as photon number, a mean-energy constraint does not bound the variance. A state with a tiny amplitude on a very high-energy component can have a large QFI at fixed mean energy. This is not automatically a useful sub-Heisenberg sensor: its likelihood can have rare, fine features, poor global risk, and strong dependence on prior localization.

Maximum energy, total generator action, Bayesian average error, or a finite-difference bound can close different loopholes. The correct choice is part of the physical task, not a cosmetic normalization.

Local Sensitivity Is Not Global Accuracy

The QFI describes infinitesimal distinguishability around a specified $\theta$. A phase is periodic, so a global estimator should use a periodic loss such as

$$C(\widehat\theta-\theta) = 4\sin^2 \left( \frac{ \widehat\theta-\theta }{2} \right),$$

or explicitly declare how phase wraps are unwrapped. Ordinary squared error is suitable only after a branch convention and a sufficiently narrow range have been fixed.

A fixed-budget covariant solution

Let $Q\ge1$, let the generator have integer eigenvalues $n=0,1,\ldots,Q$, and take a uniform prior over one phase period. The sine state

$$|\psi_Q\rangle = \sqrt{ \frac{2}{Q+2} } \sum_{n=0}^{Q} \sin \left[ \frac{(n+1)\pi}{Q+2} \right] |n\rangle$$

with the canonical covariant phase measurement minimizes the average periodic cost in this model (Berry and Wiseman, 2000). To see the origin of the sine coefficients, write a normalized probe as $\sum_{n=0}^Q c_n|n\rangle$, with real nonnegative coefficients after aligning their phases. The mean cost is

$$\overline C=2-2\sum_{n=0}^{Q-1}c_nc_{n+1}.$$

Maximizing the adjacent-coefficient sum is the largest-eigenvalue problem for a tridiagonal matrix with off-diagonal entries $1/2$ and boundary values $c_{-1}=c_{Q+1}=0$. Its largest eigenvector is the sine state and its eigenvalue is $\cos[\pi/(Q+2)]$. Therefore $\overline C_{\min}=2\{1-\cos[\pi/(Q+2)]\}$, with first circular moment

$$\left| \left\langle e^{i(\widehat\theta-\theta)} \right\rangle \right| = \cos \left( \frac{\pi}{Q+2} \right).$$

Writing this moment as $S$, the Holevo phase variance is $V_H=S^{-2}-1$, hence

$$V_H = \tan^2 \left( \frac{\pi}{Q+2} \right) \sim \frac{\pi^2}{Q^2}.$$

This is a global Heisenberg-scaling result, but its constant differs from the unit-prefactor local QFI expression. The difference is not a contradiction: the tasks and risks differ.

Prior information is a resource

If the prior width already shrinks as $1/N$, a GHZ fringe can be confined to one branch and the local calculation becomes operational. If that prior was obtained from additional probes, those probes count toward the total cost. If it comes from stable external knowledge, the comparison must grant the same knowledge to the benchmark.

A robust protocol often uses a small fraction of its budget for coarse localization and the rest for fine estimation. Multiscale adaptive schemes make this explicit. Reporting only the finest interrogation power hides the cost of disambiguation.

Entanglement Is One Route, Not the Definition

For a linear collective generator and bounded local probes, multipartite entanglement is necessary to exceed the separable-state QFI bound $F_Q\le Nw^2$. It is therefore a resource for sub-SQL local sensitivity in that specific model.

Yet entanglement is neither a sufficient guarantee of useful metrology nor a universal prerequisite for $1/Q$ scaling. A GHZ state measured in the wrong basis carries no extracted Fisher information. A sequential protocol can achieve the same query scaling with one probe. Adaptive feedback, quantum memories, or coherent control can trade spatial entanglement for temporal coherence.

The operational question is not simply whether a state is entangled. It is whether the complete preparation–encoding–measurement–estimation protocol beats the best allowed comparator under the same resource and noise model.

Nonlinear Encodings and Apparent Super-Heisenberg Laws

Suppose the unknown coupling multiplies a $k$-body generator whose spectral width grows as $N^k$. The same generator-width argument permits

$$F_Q = O(N^{2k}), \qquad \Delta\theta = O(N^{-k})$$

for optimized entangled inputs. Product inputs can in some models achieve $O(N^{-(k-1/2)})$. Relative to particle number alone, these exponents are steeper than $N^{-1}$.

They do not violate the linear-query Heisenberg bound. The interaction graph contains a growing number of parameter-coupled terms, and the generator strength itself scales with $N$. Calling the result “super-Heisenberg” can be useful shorthand only when the nonlinear Hamiltonian and the charged resources are stated explicitly.

Preparation time is especially important near a quantum critical point. Susceptibility may grow rapidly with system size while the gap closes, so adiabatic preparation takes longer. A particle-only fit can then assign the benefit to criticality while omitting the time that generated the sensitive state. Similar care is required when control power or an externally supplied nonlinearity grows with $N$.

Independent Noise Usually Changes the Asymptote

Ideal $N^2$ QFI relies on coherence across generator eigenstates separated by $O(N)$. That same macroscopic separation often makes the probe fragile. Independent loss, dephasing, spontaneous emission, or imperfect visibility can suppress the coherence faster as the block grows.

For broad classes of independent, nonzero noise channels, channel-extension and purification bounds imply

$$F_Q^{(N)} \le c(\eta)N + o(N),$$

where $c(\eta)$ depends on the channel parameters. The asymptotic error then has SQL-like scaling,

$$\Delta\theta \gtrsim \frac{1}{ \sqrt{c(\eta)N} },$$

although a valuable constant-factor quantum gain may remain. This conclusion is powerful but not universal: correlated noise, non-Markovian dynamics, environment monitoring, noiseless subsystems, control, and quantum error correction define different channel classes.

Markovian dephasing example

Consider frequency estimation with independent Markovian dephasing, no preparation/readout dead time, $\gamma>0$, and total laboratory time $T\gg1/\gamma$. Let one qubit interrogate for time $t$ while its coherence decays as $e^{-\gamma t}$. For $N$ product probes per round,

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

With approximately $T/t$ rounds,

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

An $N$-qubit GHZ coherence decays as $e^{-N\gamma t}$, giving

$$F_{Q,\mathrm{GHZ}}^{\mathrm{tot}} = TN^2t e^{-2N\gamma t}.$$

The product strategy is optimized at $t=1/(2\gamma)$, whereas the GHZ strategy must shorten its interrogation to $t=1/(2N\gamma)$. Both maxima are

$$F_Q^{\mathrm{tot,max}} = \frac{TN}{2e\gamma}$$

under this idealized dephasing convention. The entangled state acquires phase faster but loses coherence faster by the same factor. Other noise models can leave a constant advantage, but this calculation shows why noiseless scaling cannot simply be extrapolated.

Loss and postselection

Postselecting no-loss events can restore high conditional visibility while making success exponentially rare. If $p_{\mathrm s}$ is independent of the unknown parameter and failures carry no information, the Fisher information per attempted round is

$$F_{\mathrm{attempt}} = p_{\mathrm s}F_{\mathrm{success}}.$$

For an intact $N$-particle branch with $p_{\mathrm s}\sim\eta^N$, the product $\eta^N N^2$ eventually decreases with $N$. Conditional precision alone is not an unconditional metrological gain.

When error correction can help

For a finite-dimensional Markovian model with signal Hamiltonian $H_\theta=\theta G$ and parameter-independent Lindblad operators $L_k$, define the Hermitian Lindblad span

$$\mathcal S=\operatorname{span}_{\mathbb R} \left\{I,L_k^{\mathrm H},L_k^{\mathrm A}, (L_j^\dagger L_k)^{\mathrm H},(L_j^\dagger L_k)^{\mathrm A}\right\},$$

where $A^{\mathrm H}=(A+A^\dagger)/2$ and $A^{\mathrm A}=(A-A^\dagger)/(2i)$. The identity and the Hermitian and anti-Hermitian parts are essential: a real span of unspecified non-Hermitian products is not the same space.

Under sufficiently fast accurate controls and suitable noiseless ancillas, $G\notin\mathcal S$ allows error-corrected QFI proportional to total interrogation time squared. If $G\in\mathcal S$, the QFI is at most linear in time within this model and fixed-resource setting (Zhou and colleagues, 2018).

For dephasing $L\propto Z$, $\mathcal S=\operatorname{span}_{\mathbb R}\{I,Z\}$. A $Z$ signal cannot be separated from that noise by perfect correction, whereas an $X$ signal satisfies the condition. Extra noise channels or imperfect ancillas can change this conclusion. The theorem’s time scaling is not automatically a particle-number scaling law.

Finite-Resource Crossovers

Asymptotic exponents do not determine which sensor is best at accessible resources. A useful schematic model is

$$(\Delta\theta)^2 = \frac{a}{N^2} + \frac{b}{N} + c,$$

where the terms may represent ideal coherent scaling, independent noise, and a technical floor. The logarithmic slope of the root error is

$$\alpha(N) = -\frac{d\log\Delta\theta}{d\log N}.$$

It can pass from nearly $1$ to $1/2$ and finally to $0$. A short data range can resemble any intermediate power. Fitting a straight line on a log–log plot without a mechanistic model, uncertainty on the exponent, or tests for correlated residuals does not establish asymptotic scaling.

Conversely, loss of the asymptotic exponent does not make quantum enhancement useless. A constant gain can reduce averaging time, sample exposure, or probe power substantially. The physically relevant comparison may be the minimum achievable error at a fixed finite budget, not the limiting slope.

Exercises

1. Spectral-width bound

Let $G$ have eigenvalues in $[g_{\min},g_{\max}]$. Show that a pure-state unitary family generated by $G$ has $F_Q\le(g_{\max}-g_{\min})^2$, and identify a state that attains equality.

Solution

For a pure unitary family,

$$F_Q = 4(\Delta G)^2.$$

The variance of a random variable supported on an interval of width $g_{\max}-g_{\min}$ is at most one quarter of the squared width. Measurement of $G$ in any state produces such a random variable, so

$$(\Delta G)^2 \le \frac{(g_{\max}-g_{\min})^2}{4}.$$

Therefore $F_Q\le(g_{\max}-g_{\min})^2$. Equality is reached by

$$\frac{ |g_{\min}\rangle +e^{i\alpha}|g_{\max}\rangle }{\sqrt2},$$

which assigns equal probabilities to the interval endpoints.

2. GHZ fringe information and ambiguity

For $p_\pm=(1\pm\cos N\theta)/2$, compute the classical Fisher information. Then give two distinct phases in $[0,2\pi)$ that cannot be distinguished by this likelihood.

Solution

The derivatives are

$$\partial_\theta p_\pm = \mp\frac N2\sin(N\theta).$$

Thus

$$\begin{aligned} F_C &= \sum_{s=\pm} \frac{(\partial_\theta p_s)^2}{p_s} \\ &= \frac{N^2\sin^2(N\theta)}{4} \left( \frac1{p_+}+\frac1{p_-} \right) \\ &= N^2, \end{aligned}$$

at regular points. The nearby limit has that value, but an exactly zero-probability outcome is nonregular. Yet $\theta$ and $\theta+2\pi/N$ give identical probabilities. For example, $0$ and $2\pi/N$ are indistinguishable when $N>1$.

3. Parallel versus sequential queries

A single qubit in $|+\rangle$ experiences $U_\theta=e^{-i\theta\sigma_z/2}$ $m$ times coherently. Find its QFI with respect to $\theta$. Compare it with an $m$-qubit GHZ state and state the matched resource.

Solution

The total unitary is

$$U_\theta^m = e^{-im\theta\sigma_z/2}.$$

The effective generator is $m\sigma_z/2$, whose variance in $|+\rangle$ is $m^2/4$. Hence

$$F_Q = 4\frac{m^2}{4} = m^2.$$

An $m$-qubit GHZ state under one parallel use per qubit also has $F_Q=m^2$. The matched ideal resource is $m$ elementary channel uses or the same total generator exposure, not the number of simultaneously present carriers.

4. Optimize the dephasing time

Maximize

$$F(t) = TN^2t e^{-2N\gamma t}$$

over $t>0$. What scaling remains after optimization?

Solution

Differentiate the logarithm:

$$\frac{d}{dt}\log F = \frac1t-2N\gamma.$$

The maximum occurs at

$$t_* = \frac{1}{2N\gamma}.$$

Substitution gives

$$F(t_*) = \frac{TN}{2e\gamma}.$$

The optimized QFI is linear in $N$, so the root error scales as $N^{-1/2}$. The shorter optimal coherence interval removes the noiseless GHZ exponent.

5. Information per attempted preparation

An $N$-photon protocol has conditional Fisher information $N^2$ when every photon survives, and this happens with probability $\eta^N$. Find the Fisher information per attempt if failures carry no information. For fixed $0<\eta<1$, does increasing $N$ indefinitely help?

Solution

Including the success flag, and assuming its probability is parameter-independent, gives

$$F_{\mathrm{attempt}} = \eta^N N^2.$$

Its logarithm is

$$\log F_{\mathrm{attempt}} = N\log\eta+2\log N.$$

Because $\log\eta<0$, the linear negative term dominates the logarithm for large $N$. There is a finite optimum; arbitrarily large NOON states become worse per attempt in this simplified loss model.

6. Sine-state asymptotics

Starting from

$$V_H = \tan^2 \left( \frac{\pi}{Q+2} \right),$$

derive its large-$Q$ scaling and compare the root Holevo variance with the unit-prefactor local bound.

Solution

For small $x$, $\tan x=x+O(x^3)$. Therefore

$$V_H = \frac{\pi^2}{(Q+2)^2} + O(Q^{-4}),$$

and

$$\sqrt{V_H} \sim \frac{\pi}{Q}.$$

Both the global result and the local bound have exponent $-1$, but the global uniform-prior task carries a factor approaching $\pi$. Constants cannot be compared without matching the loss and prior.

7. Spectral width of a pair interaction

A protocol estimates the coefficient of

$$H_\theta = \theta \sum_{i<j} \sigma_z^{(i)}\sigma_z^{(j)}$$

for a fixed evolution time. Compute the generator width for even and odd $N$, and explain why an $N^{-2}$ local precision law would not violate a linear-query Heisenberg limit.

Solution

Let $J_z=\sum_i Z_i/2$. The generator is

$$G=\sum_{i<j}Z_iZ_j=2J_z^2-\frac N2.$$

Its maximum is $N(N-1)/2$. The minimum is $-N/2$ for even $N$ and $(1-N)/2$ for odd $N$. Thus

$$\operatorname{width}(G)= \begin{cases}N^2/2,&N\ \text{even},\\(N^2-1)/2,&N\ \text{odd}.\end{cases}$$

The experiment couples the unknown parameter to $O(N^2)$ pair terms, rather than to $N$ fixed-strength one-body terms. The quadratic width allows a local $N^{-2}$ bound at fixed time without violating the linear-query theorem. Coupling strength, evolution time, and the cost of implementing the pair interaction remain part of the comparison.

8. A changing finite-resource exponent

For $\Delta\theta=\sqrt{a/N^2+b/N+c}$ with positive constants, derive $\alpha=-d\log\Delta\theta/d\log N$. Identify the regimes when the three terms dominate.

Solution

Differentiating the squared uncertainty gives

$$\alpha(N)=\frac{a/N^2+b/(2N)}{a/N^2+b/N+c}.$$

The limits are $1$ when the coherent term dominates, $1/2$ when the independent-noise term dominates, and $0$ at the floor. Pairwise crossover estimates are $N\sim a/b$ and $N\sim b/c$. A clear intermediate $1/2$ regime requires $a/b\ll b/c$; it need not occur for arbitrary coefficients.

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