Q039 · Budget / count / infer

Quantum Phase Metrology

Give single photons and NOON probes the same photon budget. Count every loss, keep every possible phase, and spend a few ordinary photons to break the ambiguity.

Recorded sample $\text{—}$

Measurement result $\text{—}$

01 / SPEND THE SAME NUMBER OF PHOTONS.

Count the cost of every fringe.

Give the three protocols the same input photon budget. Compare the possible phases, then lower 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. The controls never acquire photons by themselves.

Ready to record.

MAKE A COMPARISON

Input photon budget

B

600

Photons per NOON probe

N

4

Photon survival

\eta

0.85

Prepare the unknown & sampling seed

Hidden phase to prepare

\phi/{}^\circ

67

Sampling seed

s_0

37

Each button starts a fresh experiment with the selected budget. Save a result before trying the next protocol. Source preparation overhead is excluded.

Ideal prepared path states with two to four photons, equal independent arm loss, a balanced recombiner and ideal number-resolving detection. Preparation overhead, dark counts and imperfect mode overlap are omitted. The posterior uses recorded counts and a uniform phase prior, not the hidden phase.

02 / WHAT DID THE DETECTORS SAY?

Keep every possible answer.

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 COUNTS

\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: single photons in gray, chosen NOON size in blue. This local bound does not resolve separated global aliases.

\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 / KEEP THE EVIDENCE

Save a result. Make one change.

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: Sharper fringes. Better measurement?
One-sentence intuition: Count all input photons and all allowed phases before claiming an advantage.

Core mathematical model

Surviving NOON probe

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

Every photon must survive. Partial detections remain in the budget.

Local information per input photon

F/B=N\eta^N

An ideal equal-loss model with known photon number and number-resolving detection; this local quantity does not identify the correct fringe globally.

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 is uniform around the full circle. Recorded detector counts and analyzer settings are the only phase evidence.

Common difficulties

A plot is not a certificate

Typical misconceptionMore rapid fringes always give a better measurement.

Better mental modelLoss, source costs and global aliases matter. This Lab excludes preparation overhead and compares only its declared ideal protocols.

Model versus records

Typical misconceptionChanging a slider improves data already recorded.

Better mental modelPreparation changes start a new sample. Frozen samples preserve their original settings and raw readings.

Run the experiment

01
Predict
With no loss, compare singles and four-photon probes. Predict the number of possible phase peaks.
What to observe: Keep the optional model reference hidden.
02
Collect and save
Give each protocol 600 photons and save each result. The mixed protocol must spend part of its budget on ordinary singles.
What to observe: Incomplete pulses lose phase information but do not disappear from accounting.
03
Find the failure
Lower survival to 40 percent. Repeat the same budgets and compare both local information and the full credible region.
What to observe: A narrower chosen fringe is not the same as a globally unambiguous phase estimate.