Mach–Zehnder Interferometry

Mach–Zehnder interferometry estimates a relative optical phase by placing the phase in one path of a coherent two-mode experiment and recombining the paths before measurement. In quantum metrology, however, the optical layout is only the beginning. A defensible sensitivity statement must identify the encoded phase, the available phase reference, every counted photon resource, the loss channel, the measured likelihood, the estimator, and whether the claim is local or globally valid.

Required background. Parameter estimation, Fisher information, and Cramér–Rao bounds connect a detector likelihood to precision. The standard quantum limit provides the independent-photon comparison; optical interferometers supplies the beam-splitter transformations.

Relative phase and physical response

A known small phase near quadrature and an arbitrary phase on the circle are different estimation problems. Likewise, information per detected photon and information per photon absorbed by a sample answer different experimental questions. A bright local oscillator can improve the available measurement without traversing the sample; its phase stability still matters.

If $x$ is the physical quantity of interest and the calibrated transduction is $\phi=\phi(x)$, Fisher information transforms as

$$F_x = F_\phi \left(\frac{\partial\phi}{\partial x}\right)^2.$$

This relation separates the optical estimation problem from the sensor’s mechanical, electromagnetic, or material response. Improving the phase estimator does not improve a poorly known transduction coefficient.

Two-Mode Description

Let $a$ and $b$ annihilate photons in the two arms. The Schwinger operators are

$$\begin{aligned} J_x &= \frac{a^\dagger b+b^\dagger a}{2}, & J_y &= \frac{a^\dagger b-b^\dagger a}{2i}, \\ J_z &= \frac{a^\dagger a-b^\dagger b}{2}, & \hat N &= a^\dagger a+b^\dagger b. \end{aligned}$$

A symmetric differential phase convention assigns phase delays $+\phi/2$ and $-\phi/2$ to the arms, giving

$$U_\phi=e^{-i\phi J_z}.$$

The first beam splitter prepares coherence between eigenstates of $J_z$; the phase rotates that coherence; the second beam splitter and detector choose a measurement basis. This is the same prepare–encode–measure structure used in Ramsey Interferometry, but the bosonic modes make photon-number fluctuations, optical references, and loss accounting especially important.

A one-arm convention, $e^{-i\phi a^\dagger a}$, is also common. Since

$$a^\dagger a=\frac{\hat N}{2}+J_z,$$

it differs from the symmetric convention by a total-number phase. That extra phase is operational only if the experiment contains a reference capable of comparing different total-number sectors. Consequently, QFI values quoted under the two conventions need not agree for indefinite-photon-number states. The apparent discrepancy is a reference-frame question, not a contradiction.

From a Fringe to Fisher Information

For one detected photon, a common effective two-port likelihood is

$$p_\pm(\phi) = \frac{1\pm V\cos(\phi-\theta)}{2},$$

where $V$ is visibility and $\theta$ is a controllable analysis phase. Its single-photon Fisher information is

$$F_1(\phi) = \frac{ V^2\sin^2(\phi-\theta) }{ 1-V^2\cos^2(\phi-\theta) }.$$

At quadrature, $\phi-\theta=\pi/2$, this becomes $F_1=V^2$. If $n$ independent photons are actually detected and the likelihood model is valid, then $F=nV^2$. This familiar result is useful, but it is conditional on detection. It says nothing yet about photons lost before the detector, energy absorbed by a sample, phase-reference resources, or information sacrificed during postselection.

For a pure probe $|\psi\rangle$ undergoing the differential unitary, the QFI is

$$F_Q = 4\operatorname{Var}_\psi(J_z).$$

For fixed total photon number $N$, the eigenvalues of $J_z$ lie between $-N/2$ and $N/2$. The maximum-variance bound therefore gives

$$F_Q\leq N^2.$$

Independent balanced-path photons have $\operatorname{Var}(J_z)=N/4$ and hence $F_Q=N$. An ideal NOON state,

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

has $\operatorname{Var}(J_z)=N^2/4$ and reaches $F_Q=N^2$. These are local, lossless statements. They do not yet certify an attainable estimator, resolve the $N$ phase aliases of a NOON fringe, or include preparation failures.

Incident photons, repeated interactions, and absorption

Distinguish source photons $N_{\mathrm{src}}$, photons entering the sensing region $N_{\mathrm{in}}$, photon–sample interactions $N_{\mathrm{int}}$, absorbed photons $N_{\mathrm{abs}}$, and detected photons $N_{\mathrm{det}}$. Repeated passes can make the interaction count exceed the incident count; loss makes the detected count smaller. For continuous operation, an integrated circulating population $\int n(t)\,dt$ has units of photon-time, rather than being another photon count.

The next formulas use a symmetric two-arm model: each pass encodes a differential phase and both arms have the same independent transmissivity $\eta_s$. Every lost photon is charged to the stated absorption cost, as when both arms interrogate equally absorbing samples. A sample in only one arm requires different probabilities and a different dose calculation; Exercise 8 derives that case.

For $k$ passes in this model, the expected counts are

$$\begin{aligned} N_{\mathrm{int}} &= N_{\mathrm{in}} \sum_{j=0}^{k-1}\eta_s^j = N_{\mathrm{in}} \frac{1-\eta_s^k}{1-\eta_s}, \\ N_{\mathrm{abs}} &= N_{\mathrm{in}}(1-\eta_s^k) = (1-\eta_s)N_{\mathrm{int}}. \end{aligned}$$

The model assumes that a photon surviving one pass is available for the next, that absorption events are independent, and that $\eta_s$ does not depend on the unknown phase. Saturable, nonlinear, bleaching, heating, or backaction- limited samples require a more detailed channel and damage functional.

The Phase Reference Is Part of the Experiment

Optical phase is relational. Without an external phase reference, coherences between different total-photon-number sectors are not operationally accessible. The appropriate state is then the globally phase-averaged state

$$\mathcal G(\rho) = \frac{1}{2\pi} \int_0^{2\pi} d\vartheta\, e^{-i\vartheta\hat N} \rho e^{i\vartheta\hat N}.$$

This $U(1)$ twirl removes matrix elements connecting different total-number sectors while preserving coherence between the two paths within each sector. The operational QFI must be calculated from $\mathcal G(\rho)$ if no suitable reference is present.

Consider a coherent state injected into one input port and vacuum into the other. After a balanced first beam splitter, the mean internal photon number is $\bar N$. For the differential generator $J_z$,

$$F_Q=4\operatorname{Var}(J_z)=\bar N.$$

The same value survives global phase averaging because every fixed-$N$ sector contains the relevant path coherence. In contrast, treating a one-arm phase as generated by $a^\dagger a$ before declaring a reference can yield an additional QFI contribution from number coherence. That contribution is usable only if a phase standard, such as a local oscillator, makes the corresponding common phase observable.

This distinction matters in two ways:

• a local oscillator need not be counted as probe dose if it never touches the sample, but its energy, stability, and mode quality remain instrument resources;
• a comparison between reference-free counting and reference-assisted homodyne detection must state which reference operations are free.

There is no universal rule that every reference photon must be added to the sample photon budget. Use the same definition of sample exposure and allowed reference in both protocols.

Local Precision and Global Phase Risk

The Cramér–Rao framework describes local curvature of a likelihood near a specified phase. An optical phase is periodic, so a globally meaningful protocol must also resolve aliases. An ideal NOON measurement contains terms such as

$$p(x|\phi,\theta) \propto 1+(-1)^x\cos[N(\phi-\theta)],$$

which repeat every $2\pi/N$. The local slope is steep, but one observation does not identify which of the $N$ branches contains the true phase.

For a circular parameter, ordinary mean-squared error can depend on an arbitrary branch cut. Two useful alternatives are the periodic cost

$$c(\hat\phi,\phi) = 4\sin^2\!\left( \frac{\hat\phi-\phi}{2} \right)$$

and the Holevo variance of the circular error

$$V_H = \left| \left\langle e^{i(\hat\phi-\phi)} \right\rangle \right|^{-2}-1.$$

The Holevo variance measures concentration and is insensitive to a constant angular bias: an estimator always wrong by the same angle has $V_H=0$. The periodic cost above detects that bias. A global analysis specifies a prior on the circle and a Bayesian or minimax risk, and includes every coarse measurement or adaptive step used to select the final branch. Common strategies combine low-frequency and high-frequency fringes, vary the interrogation number $k$, or adapt $\theta$ from the current posterior. Bayes’ Rule provides the probability update; Cramér–Rao Bounds explains why a local bound alone does not settle this global problem.

Benchmark Probe Families

The following idealized families illuminate different tradeoffs. Their QFI values are not interchangeable performance claims.

Probe family	Ideal local information	Main strength	Main qualification
Independent balanced-path photons or coherent light	$F_Q=N$ or $\bar N$	Simple preparation and robust readout	Establishes the matched independent-photon baseline only after losses and references are fixed
Coherent and squeezed-vacuum inputs, with a balanced first splitter	$F_Q=N_c e^{2r}+\sinh^2r$ at the optimal relative squeezing phase	Practical constant-factor noise reduction	Loss and squeezing-angle error mix anti-squeezed or vacuum noise into readout
NOON state	$F_Q=N^2$	Maximum fixed-$N$ generator variance	$N$-fold phase ambiguity and exponential fragility of its path coherence under loss
Twin-Fock input and Holland–Burnett state	$F_Q=N(N+2)/2$ for total even $N$	High ideal information with better loss behavior than NOON states in many regimes	Preparation and near-optimal number-resolving readout are demanding
A probe making $k$ coherent passes	$F_Q\propto k^2$ per surviving probe	Reuses preparation and detection hardware	Uses $k$ phase queries, narrows global range, takes time, and incurs compounded loss

Here $N_c$ is the bright coherent photon number and $r$ is the squeezing parameter. The squeezed-input entry assumes pure lossless inputs and matched squeezing phase; its total mean probe number is $N_c+\sinh^2r$. The bright-displacement term dominates only in its stated regime. Optical preparation and readout are developed in Squeezed Light and Interferometers.

The Holland–Burnett entry begins with equal number states $|n,n\rangle$ at the input of a balanced beam splitter, with total photon number $N=2n$. The state inside the arms has

$$F_Q = 2n(n+1) = \frac{N(N+2)}{2}.$$

This is below the NOON value $N^2$ at large $N$ by a factor approaching two, but a lower ideal QFI can be a good trade when the resulting likelihood is more robust or experimentally accessible.

Loss Changes the Asymptotic Limit

Model pure loss in an arm by coupling the optical mode to an inaccessible environmental vacuum mode. If the environment can carry information about which path lost a photon, it destroys some of the coherence that encoded $\phi$. Tracing out that environment turns a pure probe into a mixture and reduces its QFI.

Loss location matters:

• source and preparation loss changes the state and $N_{\mathrm{in}}$;
• internal arm loss can reveal path information and changes the encoded state;
• sample absorption is both a noise mechanism and often the relevant cost;
• output and detector loss discards outcomes after encoding;
• reference-path loss degrades mode matching and phase readout.

One cannot in general combine these into a single efficiency and retain the same physical interpretation, even when a fitted visibility happens to agree.

For an ideal NOON state under equal independent transmissivity $\eta$ in both arms, the no-loss block retains the original normalized state with probability $\eta^N$. Every block with a lost photon is independent of the encoded phase after tracing the environment. Therefore the exact output-state QFI in this model is

$$F_Q^{\mathrm{NOON}}=N^2\eta^N.$$

An optimal local measurement can attain this information, but extra preparation or detection imperfections can reduce it. Unequal-arm loss, access to the lost field, and phase-dependent loss are different models.

More generally, in the standard symmetric pure-loss model with fixed transmissivity $0<\eta<1$, the channel bound $F_Q\le\eta N/(1-\eta)$ gives, for a locally unbiased estimator,

$$\Delta\phi \geq \sqrt{ \frac{1-\eta}{\eta N} }.$$

Here $N$ is the incident mean photon number; $\nu$ independent repetitions replace it by $\nu N$ in this bound. Fixed-number probes also satisfy the lossless $F_Q\le N^2$ bound. The finite-$N$ optimum need not attain either bound, but the asymptotic conclusion under these assumptions is clear: fixed nonzero loss restores a $1/\sqrt N$ asymptote. Quantum probes may retain a valuable constant advantage, yet the ideal $1/N$ root-mean-square scaling does not persist to arbitrarily large $N$ through a fixed-loss channel. This is one reason bright coherent light plus squeezed vacuum can approach practically relevant optical bounds without requiring very large path-entangled number states.

For coherent light with mean photon number $N$ incident before a phase-independent net efficiency $\eta$, a corresponding shot-noise benchmark is

$$\Delta\phi_{\mathrm{coh}} = \frac{1}{\sqrt{\eta N}}.$$

Comparing the two formulas shows that the asymptotic quantum improvement is a constant in $N$ under the stated channel model. It does not show that the bound is attained by a particular source, detector, and estimator.

Multipass Interferometry

In the symmetric two-arm model above, $k$ coherent traversals accumulate differential phase $k\phi$. With ideal output counting at quadrature, the FI per launched photon is

$$f_{\mathrm{in}}(k) = k^2\eta_s^k.$$

The factor $k^2$ is the squared phase lever; $\eta_s^k$ is the survival probability. Calling this a free $k^2$ gain would omit both channel uses and loss. With $N_{\mathrm{in}}$ launched photons, the total number of phase queries in the lossless case is

$$Q=kN_{\mathrm{in}}.$$

Sequential queries and simultaneous entanglement can realize related generator variances, so a fair Heisenberg-scaling statement counts every use of the unknown phase channel. Heisenberg Scaling develops this query-complexity viewpoint.

For a sample-limited comparison, divide the information by absorbed photons:

$$\frac{F_\phi}{N_{\mathrm{abs}}} = \frac{k^2\eta_s^k}{1-\eta_s^k}$$

for the simple coherent multipass model. Treating $k$ continuously, write $x=-k\ln\eta_s$. Maximizing $x^2/(e^x-1)$ gives

$$2(1-e^{-x})=x, \qquad x\simeq1.5936,$$

provided the continuous optimum permits $k\ge1$. Its surviving fraction is approximately

$$\eta_s^k=e^{-x}\simeq0.2032.$$

If the continuous optimum is below one pass, the constrained optimum is $k=1$; otherwise compare neighboring integer pass counts. This result is specific to the symmetric loss and absorption model. Additional circulation loss, finite coherence time, detector saturation, sample dynamics, and a maximum allowed measurement time can move the optimum. It also shows why a single-pass coherent comparator can be too weak for a sample-damage claim: a well-designed classical multipass strategy already uses the sample much more efficiently.

Sample Dose and Disturbance

In microscopy, spectroscopy, and material characterization, available source power may be cheap while absorbed photons alter or destroy the sample. The relevant optimization is then information per disturbance, not information per detected photon.

For independent absorption with energy $\hbar\omega$ per lost photon, the expected deposited energy is approximately

$$E_{\mathrm{dep}} = \hbar\omega N_{\mathrm{abs}}.$$

Real samples may instead be limited by peak intensity, nonlinear absorption, heating rate, photochemical pathways, or the probability of any destructive event. A general task should therefore introduce a disturbance observable or cost $D$ and optimize a risk such as

$$\mathcal R = \mathbb E[c(\hat\phi,\phi)] + \lambda\mathbb E[D],$$

where $\lambda$ expresses the application’s tradeoff. This formulation prevents “dose” from becoming a vague synonym for whichever photon count makes a protocol look best.

The comparison must use the same disturbance measure, reference operations, acquisition time, and estimation risk. Absorption in an auxiliary reference path is not sample damage merely because it contributes to total optical loss.

Common Phase and Other Nuisance Parameters

For arm phase delays $\phi_a$ and $\phi_b$, define the sum $\phi_c=\phi_a+\phi_b$ and difference $\phi_d=\phi_a-\phi_b$. The joint encoding is

$$U(\phi_c,\phi_d) = \exp\!\left[ -i\left( \frac{\phi_c\hat N}{2} + \phi_d J_z \right) \right].$$

Without an external reference, $\phi_c$ is unobservable. With one, common-mode phase noise, local-oscillator drift, and path mismatch can couple into the readout. Other nuisance parameters include visibility, both arm transmissivities, detector gain and dark counts, mode overlap, background light, and the phase-to-signal calibration coefficient.

For parameters $\boldsymbol\theta=(\phi,\boldsymbol\lambda)$, the relevant classical Fisher information is a matrix. If the nuisance block is invertible, the information remaining for $\phi$ after jointly fitting $\boldsymbol\lambda$ is the Schur complement

$$F_{\mathrm{eff}} = F_{\phi\phi} - F_{\phi\lambda} F_{\lambda\lambda}^{-1} F_{\lambda\phi}.$$

For $F_{\mathrm{eff}}>0$ and local unbiasedness in signal and nuisance directions,

$$\operatorname{Var}(\hat\phi) \geq \frac{1}{F_{\mathrm{eff}}} \geq \frac{1}{F_{\phi\phi}}.$$

Calibrating a nuisance parameter in a separate data set does not make its uncertainty vanish; the calibration information or prior must enter the joint model. A phase sensitivity quoted after fixing visibility, loss, and reference phase at their fitted values is usually conditional and should be labeled as such.

Readout, Postselection, and Attainability

QFI is an upper bound over measurements. The achieved FI belongs to the actual record, including detector response and data rejection. Number-resolving detection can preserve information hidden by threshold detectors; homodyne detection supplies a quadrature reference; parity can be optimal for some ideal states but difficult at large photon number. Photon Counting and Homodyne and Heterodyne Detection develop those measurement models.

Suppose a trial is accepted with probability $q(\phi)$ and its conditional accepted-data FI is $F_{\mathrm{acc}}$. If a rejected trial carries no further record, the full per-launch FI is

$$F_{\mathrm{full}} = qF_{\mathrm{acc}} + \frac{[\partial_\phi q]^2}{q(1-q)}.$$

When $q$ is phase independent, this reduces to $qF_{\mathrm{acc}}$. Reporting only $F_{\mathrm{acc}}$ treats failed trials as free and can manufacture an apparent advantage. When $q$ depends on phase, the accept/reject flag itself carries information and must remain in the likelihood.

Attainability also depends on the operating regime. A POVM that saturates QFI may depend on the unknown phase; adaptive feedback or a preliminary coarse estimate may be needed. Finite data can produce biased, multimodal, or boundary-limited estimates even when an asymptotic local bound is tight.

Worked three-pass comparison

Suppose a source emits $10^6$ photons per trial. Preparation transmission is $\eta_{\mathrm{pre}}=0.8$, each arm in the symmetric model has per-pass transmission $\eta_s=0.9$, each surviving photon makes $k=3$ passes, and output plus detector efficiency is $\eta_{\mathrm{post}}=0.7$. Then

$$N_{\mathrm{in}} = 0.8\times10^6 = 8.0\times10^5.$$

The expected interaction, absorption, and detection counts are

$$\begin{aligned} N_{\mathrm{int}} &= (8.0\times10^5)(1+0.9+0.9^2) \\ &= 2.168\times10^6, \\ N_{\mathrm{abs}} &= (8.0\times10^5)(1-0.9^3) \\ &= 2.168\times10^5, \\ N_{\mathrm{det}} &= (8.0\times10^5)(0.9^3)(0.7) \\ &= 4.0824\times10^5. \end{aligned}$$

For ideal independent-photon readout at quadrature, each detected photon sees phase $3\phi$, so

$$F_\phi = k^2N_{\mathrm{det}} = 3.67416\times10^6.$$

The local shot-noise uncertainty is therefore

$$\Delta\phi \geq \frac{1}{\sqrt{F_\phi}} \simeq 5.22\times10^{-4}\,\mathrm{rad}.$$

This is a conditional model result, not yet an instrument specification. It assumes known efficiencies, unit visibility, no phase drift, no detector background, a locally known branch of a $2\pi/3$-periodic likelihood, and no extra circulation time or loss.

Exercises

1. Independent photons and a NOON state

For fixed total photon number $N$, calculate the QFI for (a) $N$ independent photons, each in an equal path superposition, and (b) a NOON state. State the local uncertainty scaling after $\nu$ independent repetitions.

Solution

For one balanced-path photon, $\operatorname{Var}(J_z)=1/4$, so $N$ independent photons have

$$F_Q^{\mathrm{ind}}=N.$$

The NOON state is an equal superposition of the extremal eigenvalues $J_z=\pm N/2$, hence

$$\operatorname{Var}(J_z)=\frac{N^2}{4}, \qquad F_Q^{\mathrm{NOON}}=N^2.$$

For $\nu$ independent repetitions, QFI adds. The local quantum Cramér–Rao bounds are

$$\Delta\phi_{\mathrm{ind}} \geq \frac{1}{\sqrt{\nu N}}, \qquad \Delta\phi_{\mathrm{NOON}} \geq \frac{1}{N\sqrt\nu}.$$

The second result presumes a known local branch and an attainable measurement.

2. Visibility and the operating point

Starting from

$$p_\pm = \frac{1\pm V\cos\delta}{2}, \qquad \delta=\phi-\theta,$$

derive the single-photon FI and find its value at quadrature.

Solution

The derivatives are

$$\partial_\phi p_\pm = \mp\frac{V\sin\delta}{2}.$$

Therefore

$$\begin{aligned} F_1 &= \sum_{s=\pm} \frac{(\partial_\phi p_s)^2}{p_s} \\ &= \frac{V^2\sin^2\delta} {1-V^2\cos^2\delta}. \end{aligned}$$

At $\delta=\pi/2$, $F_1=V^2$. Visibility reduces the best local FI quadratically in this effective likelihood.

3. Loss-limited NOON size

Use the equal-arm pure-loss result $F_Q^{\mathrm{NOON}}=N^2\eta^N$. Determine the continuous $N$ that maximizes FI per incident photon, $F/N$, for $\eta=0.9$.

Solution

The information per incident photon scales as

$$g(N)=N\eta^N.$$

Taking a logarithm,

$$\frac{d}{dN}\ln g = \frac1N+\ln\eta.$$

The continuous optimum is

$$N_* = -\frac{1}{\ln\eta}.$$

For $\eta=0.9$,

$$N_* \simeq 9.49.$$

An actual fixed-number protocol compares nearby integers and includes all other efficiencies. The calculation illustrates that loss creates a finite optimal entangled size; larger is not automatically better.

4. Optimal symmetric multipass survival

For the symmetric two-arm model, maximize

$$h(k) = \frac{k^2\eta^k}{1-\eta^k}$$

with respect to continuous $k$. Show that the optimal survival probability is independent of $\eta$.

Solution

Set $x=-k\ln\eta$, so $\eta^k=e^{-x}$. Apart from the constant $1/(\ln\eta)^2$,

$$h(x) \propto \frac{x^2}{e^x-1}.$$

The stationary condition is

$$2(e^x-1)-xe^x=0,$$

or

$$2(1-e^{-x})=x.$$

The nonzero solution is $x\simeq1.5936$, giving

$$\eta^{k_*} = e^{-x} \simeq 0.2032.$$

Thus the simple ideal model chooses $k$ so roughly $20\%$ of launched photons survive the sample, regardless of the per-pass transmission. If the resulting $k_*$ is smaller than one, only the boundary $k=1$ is allowed. Otherwise compare neighboring integers; extra apparatus losses also modify the choice.

5. Expected counts after three passes

Let $N_{\mathrm{src}}=2.0\times10^5$, $\eta_{\mathrm{pre}}=0.75$, $\eta_s=0.8$, $k=3$, and $\eta_{\mathrm{post}}=0.6$. Find $N_{\mathrm{in}}$, $N_{\mathrm{int}}$, $N_{\mathrm{abs}}$, and $N_{\mathrm{det}}$.

Solution

First,

$$N_{\mathrm{in}} = (2.0\times10^5)(0.75) = 1.5\times10^5.$$

Then

$$\begin{aligned} N_{\mathrm{int}} &= (1.5\times10^5)(1+0.8+0.8^2) \\ &= 3.66\times10^5, \\ N_{\mathrm{abs}} &= (1.5\times10^5)(1-0.8^3) \\ &= 7.32\times10^4, \\ N_{\mathrm{det}} &= (1.5\times10^5)(0.8^3)(0.6) \\ &= 4.608\times10^4. \end{aligned}$$

Notice that the interaction count exceeds the launched count, whereas the detected count is much smaller. Each answers a different resource question.

6. Postselection accounting

A heralded protocol accepts a phase-independent fraction $q=0.04$ of trials. Accepted records have $F_{\mathrm{acc}}=100$ per accepted trial. Find the FI per launched trial. Compare it with a deterministic protocol having FI $5$ per trial.

Solution

Because $q$ is phase independent,

$$F_{\mathrm{full}} = qF_{\mathrm{acc}} = (0.04)(100) = 4.$$

The accepted subset looks twenty times more informative than the deterministic record, but the complete heralded protocol has less FI per launch: $4<5$. This does not make heralding useless; it shows that its success probability belongs in the chosen resource normalization.

7. Nuisance-parameter penalty

Suppose the Fisher matrix for phase $\phi$ and visibility $V$ is

$$F = \begin{pmatrix} 100 & 18\\ 18 & 9 \end{pmatrix}.$$

Find the effective phase information when $V$ is estimated jointly, and compare the resulting bound with the bound obtained by incorrectly treating $V$ as known.

Solution

The Schur complement is

$$F_{\mathrm{eff}} = 100-18(9^{-1})18 = 64.$$

Thus

$$\Delta\phi \geq \frac1{\sqrt{64}} = 0.125.$$

Treating $V$ as known would give $1/\sqrt{100}=0.1$. Correlation with the uncertain visibility weakens the attainable phase precision.

8. What changes when only one arm absorbs?

A balanced single-photon path state experiences transmission $t=\eta_s^k$ and phase $k\phi$ in arm $a$, while arm $b$ is lossless. After a balanced recombiner, derive the two count probabilities and the no-count probability per launched photon. Find the FI at quadrature and divide by the expected absorbed photon number.

Solution

The unnormalized surviving path amplitudes are $(\sqrt t\,e^{-ik\phi},1)/\sqrt2$. A balanced recombiner gives

$$q_\pm=\frac{1+t\pm2\sqrt t\cos(k\phi)}4, \qquad q_0=\frac{1-t}{2}.$$

At quadrature, $q_+=q_-=(1+t)/4$ and $|\partial_\phi q_\pm|=k\sqrt t/2$. The no-count flag has no phase information for fixed $t$. Thus

$$F_\phi=\frac{2k^2t}{1+t},\qquad \frac{F_\phi}{N_{\mathrm{abs}}} =\frac{4k^2t}{1-t^2}.$$

These differ from the symmetric-arm formulas. The lossless reference arm still contributes detected photons, and only half the incident path population initially encounters the sample. A coherent input with mean photon number $\bar N$ has the same FI per mean incident photon under two-port counting, with Poisson output statistics.

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