Q039 · Phase estimation with equal photon numbers

Quantum Phase Metrology

Do sharper interference fringes always mean a better phase measurement? Not automatically. NOON states do make the fringes sharper, but a fair comparison counts every photon put in and every photon lost. The extra fringes also repeat sooner, so you cannot tell which fringe you are on without help from ordinary photons.

Give single photons and NOON probes the same number of photons. Lost photons still count toward that total, and every phase that fits the counts is kept. A few ordinary photons then decide which repeated NOON fringe is the true one.

Recorded sample $\text{—}$

Measurement result $\text{—}$

01 / Use the same number of photons

Every fringe costs photons.

Give all three protocols the same number of input photons. Compare which phases remain possible, then lower the survival and repeat.

An ideal path state enters two arms, accumulates relative phase and recombines. Arm colors identify paths, not measured which-path information.

Reference phase offset

\theta_0/{}^\circ

0

Each protocol cycles four analyzer settings relative to this offset. Changing a control does not record any photons by itself.

Ready to record.

Make a comparison

Input photon budget

B

600

Photons per NOON probe

N

4

Photon survival

\eta

0.85

Set the unknown phase and sampling seed

Hidden phase to prepare

\phi/{}^\circ

67

Sampling seed

s_0

37

Each button starts a fresh experiment with the selected number of photons. Save the result before trying the next protocol. The cost of preparing the source is not counted.

The path states are prepared ideally, and each NOON probe holds two to four photons. Both arms lose photons equally and independently, the recombiner is balanced, and the detectors resolve photon number perfectly. The cost of preparing the states, dark counts and imperfect mode overlap are left out. The posterior is computed only from the recorded counts and a uniform prior over phase.

02 / What did the detectors say?

Keep every phase that fits the data.

Reveal hidden phase

$\phi/{}^\circ$ · Blue: current posterior. Orange: last saved other sample. Shaded: highest-density 95% credible grid set. Separate peaks are separate possible phases.

Every input photon is counted

\text{—}

\text{—}

Latest records

03 / Where does the advantage go?

One lost photon can erase the phase information.

$F/B=\eta\quad\text{or}\quad N\eta^N$ · Model reference only: gray for single photons, blue for the chosen NOON size. This local bound says nothing about telling apart widely separated phases that fit the data equally well.

\text{—}

The mixed protocol spends one fifth of the budget on singles and the rest on NOON probes. The singles locate the correct fringe; all input photons count, even when detectors see nothing.

The phase prior is uniform around the full circle. The 0.25-degree grid credible set is Bayesian, conditional on this calibrated ideal model; it is not a frequentist coverage guarantee or a hardware advantage certificate.

04 / Save the evidence

Save a result, then change one thing

Eight frozen samples. Restore for inspection; further acquisition starts a new run. Every saved raw record is exported once. Reusing a seed reproduces draws, not independent evidence.

Physical model, limits & primary sources

$|\psi_N\rangle=(|N,0\rangle+|0,N\rangle)/\sqrt2$

$P(k,N-k)=\eta^N\binom Nk2^{-N}[1+(-1)^k\cos N(\phi-\theta)]$

All partial counts are retained. With equal loss, any incomplete NOON pulse loses its phase coherence; its surviving photons split binomially. The preparation box supplies the stated ideal path state directly. Guide rays do not assign a path to an individual photon. Geometric dimensions are schematic; phase is an optical phase delay, not a solved refractive plate angle.

Rubin & Kaushik · loss and maximally entangled phase probes
Dorner et al. · lossy interferometry and optimized alternatives

Start with the essentials

Focus question: Do sharper interference fringes always mean a better phase measurement?
One-sentence intuition: Before you claim an advantage, count every photon sent in and every phase the data still allow.

Core mathematical model

Chance that the whole NOON probe survives

P_{\mathrm{complete}}=\eta^N

Every photon must survive. Pulses with only some photons detected still count toward the photon total.

Local Fisher information per input photon

F/B=N\eta^N

This holds for ideal equal loss, a known photon number and number-resolving detectors. It describes sensitivity near one phase and cannot tell which fringe is the correct one.

Posterior from all recorded pulses

p(\phi\mid D)\propto p(\phi)\prod_j p(d_j\mid\phi,N_j,\theta_j)

The prior gives every phase around the circle equal weight. Only the recorded detector counts and analyzer settings carry information about the phase.

Common difficulties

What a plot alone cannot prove

Typical misconceptionFaster fringes always give a better measurement.

Better mental modelLoss, the cost of the source and the repeated phase solutions all matter. This Lab leaves out preparation cost and compares only its ideal protocols.

Settings do not change saved data

Typical misconceptionMoving a slider improves data that was already recorded.

Better mental modelChanging the prepared state starts a new sample. Saved samples keep their original settings and raw readings.

Run the experiment

01
Predict
With no loss, compare single photons with four-photon probes. Predict how many possible phase peaks each one gives.
What to observe: Leave the optional model reference turned off for now.
02
Collect and save
Give each protocol 600 photons and save each result. The mixed protocol has to spend some of those photons on ordinary single photons.
What to observe: A pulse that loses a photon no longer carries phase information, but it still counts toward the photon total.
03
Find where it fails
Lower survival to 40 percent and repeat with the same photon numbers. Compare the local information with the full credible region.
What to observe: Narrowing one fringe does not tell you which fringe holds the true phase.