A quantum sensor is useful when a small change in the quantity being measured produces a distinguishable change in its outcome statistics. Fisher information measures that sensitivity; a precision bound states what an estimator can achieve under specified assumptions.
Suppose an unknown phase changes a quantum state. Looking at the state equation may suggest that the phase is present, but that is not enough. A measurement must turn the change into data, and an estimator must turn the data into a numerical answer. The whole chain matters: preparation, encoding, noise, measurement and inference.
Quantum metrology studies this chain and the precision it can support. Classical Fisher information describes a chosen measurement’s statistics. Quantum Fisher information describes the locally available information after optimising over measurements. These quantities connect statistical distinguishability with the geometry of quantum states. [1,2]
This is Guide 15 of the BTT Quantum Mathematics series. It is university-level enrichment built from probability, differentiation, linear algebra and quantum measurement. We will calculate an elementary phase experiment, derive its precision scaling, compare independent and entangled probes, and show why ideal entanglement gains can disappear under a stated noise model.
Reading route: likelihood and Fisher information → a worked phase experiment → quantum Fisher information → independent and entangled probes → noise and interrogation time → worked practice.
1. Begin with a parameter, a model and a likelihood
Let φ be the unknown parameter. A measurement produces an outcome x with probability p(x|φ). After independent repetitions with outcomes x₁,…,xν, the likelihood is the product of those probabilities, viewed as a function of φ with the data held fixed.
An estimator φ̂ is a rule that maps the observed data to an estimated parameter. The estimator is not the quantum state, the measurement operator or the likelihood itself. It is the final inference rule.
This distinction prevents a common overclaim. A state may contain a rapidly varying phase while the chosen measurement produces probabilities independent of that phase. In that case the measurement gives no information, however elaborate the state preparation may look.
2. Classical Fisher information
For a differentiable discrete probability model, the classical Fisher information is
FC(φ) = Σx p(x|φ)[∂φ ln p(x|φ)]².
Where probabilities are positive, this is equivalently Σ [∂φp(x|φ)]²/p(x|φ). The derivative asks how quickly the probability changes; the denominator weights that change relative to the probability of the outcome. The logarithm here is the natural logarithm.
For a binary outcome with probability p(φ) and complementary probability 1−p(φ), the formula simplifies to FC = p′(φ)²/[p(φ)(1−p(φ))]. This is often the easiest starting point for a worked sensor model. Boundary cases with zero probabilities require limiting arguments and attention to regularity rather than division by zero. [1]
3. Why independent information adds
The logarithm of an independent-data likelihood is a sum. Its derivative, called the score, is therefore a sum of individual scores. Under the standard regularity conditions, each score has mean zero.
For ν independent repetitions of the same experiment, cross terms vanish when the squared total score is averaged. The total Fisher information is νFC. Repetition improves precision because evidence accumulates.
Independence is a physical and statistical assumption. Drift, shared calibration fluctuations or correlated noise can invalidate this simple multiplication. A thousand correlated readings are not automatically worth a thousand independent readings.
4. Derive the classical Cramér–Rao bound
For an unbiased estimator, its mean is φ. Differentiating that mean with respect to φ gives one. Under conditions permitting differentiation inside the expectation, this implies that the covariance between the estimator and the total score is one.
Cauchy–Schwarz bounds the square of that covariance by the product of the two variances. The score variance is the total Fisher information νFC, so
Var(φ̂) ≥ 1/[νFC(φ)].
The corresponding standard-deviation bound is 1/√(νFC). It applies to unbiased, or appropriately locally unbiased, estimation under regularity assumptions. It is not an unconditional claim about every estimator, every parameter point or every finite data set. In many regular models, efficient estimators approach the bound asymptotically. [1]
5. A phase can be present but invisible to the wrong measurement
Prepare ψφ = (|0⟩+exp(iφ)|1⟩)/√2. A computational-basis measurement gives 0 and 1 with probability one half, independently of φ. Both probability derivatives are zero, so FC = 0.
The state genuinely changes with φ, but this measurement discards the relative phase. Repeating the same measurement cannot recover information that its probability model never contains. More data do not solve an identifiability failure.
Now measure in the X basis, whose states are |+⟩ and |−⟩. The plus amplitude is (1+exp(iφ))/2, giving p(+) = (1+cos φ)/2. The measurement has converted a phase change into a probability change.
6. Calculate the information in the phase fringe
For the X measurement, p′(φ) = −sin φ/2. Also p(1−p) = sin²φ/4. At interior regular points with sin φ ≠ 0, substitution into the binary formula gives FC = 1.
The ideal locally unbiased standard-deviation bound after ν independent shots is therefore 1/√ν radians. Four times as many independent shots halve this standard uncertainty. That square-root behaviour is a statement about sampling, not about a changing value of Planck’s constant.
At φ = π/2, the outcomes are maximally random but their probabilities have a large local slope. Randomness does not make the measurement useless. At the same time, the Z measurement also gives maximally random outcomes and provides zero phase information. Outcome entropy and parameter information answer different questions.
7. A synthetic data example
Suppose an illustrative experiment has 400 independent shots and 220 plus outcomes. These are invented teaching counts, not laboratory results. The estimated plus probability is 220/400 = 0.55.
Assuming the phase is known to lie in the identifiable interval 0 < φ < π, the maximum-likelihood relation gives cos φ̂ = 2(0.55)−1 = 0.10. Therefore φ̂ = arccos(0.10), approximately 1.47062891 radians.
The local asymptotic standard uncertainty from FC = 1 is about 1/√400 = 0.05 radians. This is not an exact finite-sample confidence interval, and the finite-sample estimator need not be perfectly unbiased. The stated phase interval is essential: without it, cosine alone gives multiple compatible phases.
8. Quantum Fisher information asks for the best local measurement
A parameterised density matrix ρφ can be measured in many ways. Each measurement creates its own classical probability model and Fisher information. Quantum Fisher information, FQ, is the supremum of that classical information over allowed quantum measurements at the parameter point.
Thus FC ≤ FQ. The inequality separates the information available in the quantum state from the information extracted by a chosen measurement. The geometric connection was developed through statistical distinguishability of neighbouring states. [1,2]
The optimiser over measurements can depend on the unknown parameter. Local optimality is therefore not automatically a ready-to-use global measurement prescription. A coarse preliminary estimate, adaptive control or an independently justified working region may be needed.
9. The symmetric logarithmic derivative
A standard way to calculate FQ introduces a Hermitian operator Lφ, the symmetric logarithmic derivative, satisfying ∂φρ = (Lρ+ρL)/2. Then FQ = Tr(ρL²).
In an eigenbasis of ρ with eigenvalues λa, a useful formula is
FQ = 2Σa,b |⟨a|∂φρ|b⟩|²/(λa+λb),
where terms with zero denominator are omitted under the corresponding support convention. These formulas connect parameter sensitivity to a matrix equation. Rank-changing boundaries can require additional care. [1]
For a single parameter, an appropriate measurement in an SLD eigenbasis can attain the local quantum information at the given parameter value. That still leaves estimator design, finite-sample behaviour and unknown-parameter dependence to be addressed.
10. Pure-state information has a simpler expression
For a differentiable normalised pure state ψφ,
FQ = 4(⟨∂φψ|∂φψ⟩ − |⟨ψ|∂φψ⟩|²).
The first term measures how fast the coordinate vector changes. The subtracted term removes the component corresponding to a physically irrelevant global phase. Only movement that changes the physical state contributes useful information. [1]
For ψφ = (|0⟩+exp(iφ)|1⟩)/√2, the derivative has squared norm one half, and the squared overlap with the state is one quarter. Therefore FQ = 4(1/2−1/4) = 1. Our X measurement reaches this value at regular points; the Z measurement wastes it.
11. Unitary encoding connects information to generator variance
Suppose ψφ = exp(−iφG)ψ₀, with a parameter-independent Hermitian generator G. Differentiation gives ∂φψ = −iGψ. Substitution into the pure-state formula yields
FQ = 4 Var(G).
For G = Z/2 and initial |+⟩, the variance is one quarter, so FQ = 1. For initial |0⟩, the variance is zero because |0⟩ is a generator eigenstate. Its evolution changes only global phase, so FQ = 0.
A large generator expectation is not the relevant resource here. The variance matters because it measures the spread of eigenvalues across which a relative phase can develop. For mixed states, the simple equality with four times the variance does not generally hold.
12. The quantum Cramér–Rao bound
Combining the classical estimation bound with FC ≤ FQ gives Var(φ̂) ≥ 1/(νFQ) under the corresponding locally unbiased regular estimation assumptions.
This is a lower bound on achievable variance, not the variance of every procedure. A poor measurement or estimator can perform much worse. Even the optimal measurement can require enough repetitions before an asymptotic estimator approaches the bound.
The bound is local. It concerns nearby parameter values in an identifiable model. It does not automatically resolve a phase that has many separated, equally plausible likelihood peaks. Local sensitivity and global ambiguity must both be checked before claiming a precision advantage. [1,3]
13. Independent probes: the standard square-root scaling
Take N independent qubits, each initially |+⟩, each encoded by exp(−iφZ/2). Each contributes unit QFI, and independent product-state information adds, so the N-probe state has FQ = N.
If the whole N-probe experiment is repeated ν times, the standard-deviation bound is 1/√(νN). This is the familiar shot-noise or standard-quantum-limit scaling for this particular independent-probe model.
State the resource convention. Here N counts probes per experiment and ν counts independent repetitions of that experiment. Treating both as the same symbol can accidentally count the same resource twice. Different sensing problems can use time, photon number, channel queries or energy as the relevant resource.
14. Entangled probes: calculate the ideal GHZ gain
Use the GHZ state (|0⟩⊗N+|1⟩⊗N)/√2 and collective generator G = Σ Zj/2. The two components have generator eigenvalues +N/2 and −N/2.
The mean is zero and the variance is N²/4. Therefore FQ = N². The ideal local standard-deviation bound after ν repetitions is 1/(N√ν), compared with 1/√(Nν) for independent probes. The gain is a factor √N at the same N and ν.
This is Heisenberg scaling for the stated noiseless linear-encoding resource model. It does not violate the quantum bound; it uses a different input state with a larger generator variance. The distinction between independent and entangled strategies is central to quantum-metrology theory. [3,4]
15. A narrower fringe creates a larger ambiguity problem
The GHZ relative phase is Nφ. A suitable parity readout produces a fringe proportional to cos(Nφ). Its slope can be N times steeper than the single-probe fringe, but it also repeats N times across a full 2π phase interval.
A highly precise local estimate can therefore identify the wrong fringe if no coarse localisation is available. The experiment needs a strategy for resolving which neighbourhood contains the true phase. That strategy consumes resources too.
This connects metrology to quantum phase estimation. Controlled powers create finer phase structure, but greater formal resolution also requires accounting for interrogation strength, repeated operations and phase wrapping. A fine grid is not free information.
16. Entanglement is useful only relative to the encoding
An entangled state can be insensitive to a particular parameter if that parameter’s generator acts only as a global phase on the state. Entanglement is therefore not a universal sensitivity score independent of the sensing task.
Likewise, replace the GHZ superposition by the equal classical mixture of its two basis components. That mixed state commutes with the collective Z generator and does not change under the encoding. Its QFI for φ is zero, even though the basis outcomes are strongly correlated.
For the specified sum of unit-range local generators, separable probe states have QFI at most N. Exceeding that bound witnesses useful entanglement for this encoding, but the generator normalisation and resource assumptions are part of the statement. [4]
17. Finite visibility reduces phase information
Model a dephased equatorial qubit by ρφ = [I+v cos φ X+v sin φ Y]/2, where 0 ≤ v ≤ 1 is a known, parameter-independent visibility. Its off-diagonal coherence has magnitude v/2.
An X measurement now has p(+) = (1+v cos φ)/2. Differentiating and using the binary formula gives
FC(φ) = v²sin²φ/(1−v²cos²φ).
At the working point φ = π/2, this becomes v². The state’s QFI is also v², which can be checked using the SLD or spectral formula. Thus the chosen readout reaches the local optimum there. For v = 0.6, each shot contributes 0.36 rather than one unit of information.
18. Independent dephasing can remove the large-GHZ advantage
Assume each of N probes suffers an independent phase-damping process that multiplies its single-qubit coherence by v. The GHZ coherence between all zeros and all ones is then multiplied by vN.
Within its two-dimensional support, the noisy GHZ state has a phase Nφ and visibility vN. Its QFI is therefore N²v2N. N independent noisy |+⟩ probes instead have total QFI Nv².
For v = 0.9 and N = 4, the GHZ value is approximately 6.88747536, compared with 3.24 for independent probes. At N = 20, the GHZ value is approximately 5.91235318, compared with 16.2 for independent probes. In this explicit model, making the GHZ group larger eventually loses its advantage.
These are calculations for a stated noise model, not universal experimental crossover numbers. More general metrological noise bounds explain why ideal scaling advantages can be severely restricted under decoherence. [5]
19. Estimating frequency changes the units and the time budget
If the phase is φ = ωt with known interrogation time t, the chain rule gives F(ω) = t²F(φ). The Fisher information for frequency therefore has squared-time units, and the resulting standard uncertainty has inverse-time units.
Longer interrogation can improve sensitivity by accumulating more phase, but it may also allow more decoherence and reduce the number of independent repetitions available in a fixed laboratory duration. Comparing two protocols at equal shot count while ignoring unequal interrogation times can be misleading.
Every precision claim needs a resource ledger: probes per interrogation, interrogation time, repetitions, preparation and readout overhead, discarded trials and any auxiliary measurements used for phase localisation or calibration.
20. Worked trade-off: an optimal interrogation time
Consider a single-probe model with visibility v(t) = exp(−γt), where γ > 0 is known. The frequency QFI per interrogation is t²exp(−2γt). Suppose a fixed total duration T allows approximately T/t repetitions and preparation/readout overhead is neglected.
The total information is then Tt exp(−2γt). Differentiating its logarithm gives 1/t−2γ, so the optimum is t = 1/(2γ). Extremely short interrogations accumulate too little phase; extremely long ones lose too much coherence.
The result is conditional on the model. Non-negligible dead time changes the repetition count. Non-exponential dephasing changes the optimum. An unknown γ creates a calibration or joint-estimation problem. The useful lesson is the method of accounting, not memorising one time as universally optimal.
21. Bias, calibration and nuisance parameters
An estimator can have low variance and still be wrong because of bias. If its bias is b(φ) = E[φ̂]−φ, then mean-squared error is Var(φ̂)+b(φ)². Under the corresponding regularity assumptions, the variance bound contains [1+b′(φ)]² in the numerator rather than simply one.
A fixed phase offset, uncertain contrast or miscalibrated interrogation time can all shift the inferred answer. Repeating measurements may produce a very narrow distribution around that shifted answer. Precision is not the same as accuracy.
If several parameters must be inferred together, use a Fisher-information matrix. An unknown nuisance parameter can reduce effective information about the target. In quantum multiparameter estimation, optimal measurements for different parameters can also be incompatible, so single-parameter bounds need not all be simultaneously attainable. [1]
22. Why post-processing cannot create missing information
If the raw outcome probabilities do not depend on φ, no classical calculation applied to those outcomes can make them informative about φ. The Z measurement of our phase state is the simplest example.
More generally, parameter-independent processing cannot increase the accessible statistical information beyond that present in the original experiment. Discarding outcomes or selecting only favourable events can make a conditional signal look steeper, but success probability and discarded resources must be included in the complete accounting.
This is closely related to the data-processing ideas in the earlier guides on state discrimination and fidelity and quantum information and correlations. A processing step can reorganise evidence; it cannot manufacture sensitivity from a parameter-independent distribution.
23. Worked practice: twelve questions with explanations
1. A measurement has p(0|φ) = 1/2 for every φ. Find its information
Both outcome probabilities have zero derivative, so FC = 0. Repeating the measurement does not identify φ. The problem is not a small sample but a probability model that is insensitive to the parameter.
2. Calculate the Fisher information of a Bernoulli model whose parameter is p itself
For 0 < p < 1, the derivative of the success probability with respect to p is one. Therefore F(p) = 1/[p(1−p)]. For ν independent samples, the sample proportion has variance p(1−p)/ν, matching the classical bound in this regular model.
3. How many ideal phase shots are needed for a local standard uncertainty of 0.02?
With F = 1, the lower-bound scale is 1/√ν. Set 1/√ν = 0.02 to obtain ν = 2,500. This is the information-theoretic sampling scale; an actual estimator still needs an appropriate measurement, identifiability and control of systematic error.
4. What changes when the visibility is 0.5 at the optimal working point?
The information per shot becomes v² = 0.25. With 400 shots, the standard-deviation bound is 1/√(400×0.25) = 0.1 radians. This is twice the ideal-visibility uncertainty at the same shot count.
5. Find the QFI for |0⟩ under exp(−iφZ/2)
The state is an eigenstate of Z/2, so Var(Z/2) = 0 and FQ = 0. The evolving ket acquires only global phase. No measurement on that state alone can reveal φ through this encoding.
6. Find the QFI for |+⟩ under the same encoding
The generator mean is zero and its squared expectation is one quarter. Hence FQ = 4×1/4 = 1. Unlike the eigenstate input, |+⟩ has support on two generator eigenvalues, allowing a measurable relative phase to develop.
7. Compare ideal N = 16 product and GHZ probes at equal repetition count
The product QFI is 16 and the GHZ QFI is 256. The ratio of information is sixteen, so the ratio of local standard-deviation bounds is four. This calculation excludes state-preparation cost, decoherence and global phase ambiguity; those must be considered before interpreting it as an experimental advantage.
8. Why does a cos(8φ) fringe not uniquely determine φ over 0 to 2π?
The fringe repeats eight times and cosine also has reflection symmetry within each period. Multiple phases produce identical probabilities. A coarse estimate, additional settings or an adaptive protocol is needed to identify the correct neighbourhood before a narrow local precision claim is meaningful.
9. Convert phase information F(φ) = 0.64 into frequency information at t = 3
For φ = ωt, F(ω) = t²F(φ) = 9×0.64 = 5.76. The units are squared time. The calculation assumes t is known; uncertainty in t would have to be included in the inference model.
10. Find the optimal t in the exponential-dephasing toy model when γ = 0.2 per unit time
Using t = 1/(2γ) gives t = 2.5 time units. This follows from maximising total information Tt exp(−2γt) with negligible overhead, not from maximising information per shot alone. Those are different optimisation objectives.
11. Does smaller variance always mean a better estimate?
No. A biased estimator can produce tightly clustered but systematically wrong values. Mean-squared error includes both variance and squared bias. A precision report should identify calibration assumptions and distinguish random uncertainty from systematic displacement.
12. What must accompany a claim of Heisenberg scaling?
Specify the parameter encoding, generator normalisation, counted resource, repetition convention, noise assumptions, measurement attainability and treatment of global ambiguity. Without those details, the phrase can compare inequivalent experiments or hide resources that supply the apparent gain.
24. A complete estimation report
A useful report begins with the parameter and its units, the allowed parameter interval and the preparation and measurement model. It records how probabilities depend on the parameter, how many independent samples were obtained and what estimator was applied.
It then separates an observed estimate from a theoretical bound. A lower bound is not an observed standard deviation; an asymptotic approximation is not an exact finite-sample interval. Calibration, drift, loss and nuisance parameters belong beside the statistical calculation, not outside the final conclusion.
The final verification question is whether nearby parameter values really produce distinguishable data under the implemented procedure. A complicated state preparation is not sufficient. The mathematical chain must return all the way to an interpretable estimate with an honest uncertainty statement.
25. The bridge back to school Mathematics
Probability supplies the outcome model. Differentiation measures sensitivity. Variance quantifies sampling fluctuation. Cauchy–Schwarz produces the estimation bound. Trigonometry describes the phase fringe. Linear algebra shows which state directions can encode the parameter.
A strong learning sequence therefore starts with a two-outcome experiment rather than an abstract promise of quantum advantage. Calculate its probabilities, differentiate them, compare two measurement bases and only then ask what entanglement changes.
The central habit is to keep four questions separate: does the state change, does the measurement detect that change, can the estimator identify the correct parameter region, and what resources support the claimed precision? Quantum metrology becomes understandable when those questions are answered in order.
Sources and further study
[1] Matteo G. A. Paris, Quantum estimation for quantum technology. A treatment of local quantum estimation, symmetric logarithmic derivatives, Fisher information and the assumptions behind precision bounds.
[2] Samuel L. Braunstein and Carlton M. Caves, Statistical distance and the geometry of quantum states. The foundational connection between optimal statistical distinguishability and quantum-state geometry.
[3] Vittorio Giovannetti, Seth Lloyd and Lorenzo Maccone, Quantum metrology. A framework for comparing classical and quantum strategies under explicitly counted resources.
[4] Géza Tóth and Iagoba Apellaniz, Quantum metrology from a quantum information science perspective. This review connects GHZ and other probe states, QFI, entanglement criteria and precision scaling.
[5] Rafał Demkowicz-Dobrzański, Jan Kołodyński and Mădălin Guţă, The elusive Heisenberg limit in quantum enhanced metrology. Bounds for noisy parameter-estimation models explain why noiseless asymptotic gains do not automatically survive decoherence.
Continue through Quantum Mathematics
Guide 13: Hamiltonian Simulation, Trotter–Suzuki Product Formulas and Error Bounds examines errors in the encoding dynamics. Guide 14: Variational Quantum Eigensolvers, Rayleigh Quotients and Optimisation uses repeated quantum measurements inside an optimisation loop. Guide 16: Tensor Networks, Matrix Product States and Entanglement Compression asks how structured states can be represented and approximated efficiently.