ONLINE COURSE

Sample Intermediate Lab and Assignment Showcase

Welcome Quantum Dawn Commander, to your Intermediate Level. Here, you’ll move from single-qubit effects to protocol analysis: BB84/decoy-state reasoning, error budgets, hypothesis tests, parameter estimation, and basic finite-key intuition. You’ll complete 20 labs, each with the same structure—problem → stepwise solution → matching Jupyter lab—now with more open code sections and emphasis on defensible claims (confidence intervals, model checks, plots). The sample lab below previews the format and level of rigor. Remember, this is not a certificate mill—mastery is demonstrated through mathematically sound reasoning and working code.

Problem

Intermediate — Night Watch 2031 — QKD Alarm Triage (Student Problem)

Intermediate — Night Watch 2031 — QKD Alarm Triage

Student Problem • Scenario triage • Discrete‑variable QKD (decoy‑state BB84) • Signal anomaly vs noise

Mission Brief

03:13:00 Z, 13 September 2031. Far East Operations, Signals Bay 3. The ops floor is quiet in the way hospitals are quiet: a kind of humming focus. You’re the midnight quantum security analyst on duty, eyes cycling between two monitors—telemetry on the left, event triage on the right. A soft chime breaks the rhythm. Then another. Your triage screen blooms amber: QKD EARLY‑ALERT: ANOMALOUS PATTERN DETECTED.

On your left display, an allied sea‑to‑shore discrete‑variable QKD (DV‑QKD) link—call sign SPRITE‑19—streams status from a USN picket ship to an island relay. The link runs a decoy‑state BB84 protocol with three intensities: signal $\mu_s$, weak decoy $\mu_w$, and vacuum $\mu_0$. It has been clean for months: stable channel loss, boring background counts, and a crisp quantum bit error rate (QBER) hovering around 1.7–2.1%. Tonight, however, traffic elsewhere crackles. Satellite chatters. AIS tracks do little dances in the South China Sea. Intelligence has warned of “heightened military activity” with unclear intent. It’s a night for paying attention.

The alert expands. Over the last 90 seconds, the system flagged three things at once: (1) a QBER bump—still below panic thresholds but notably above baseline; (2) a decoy‑mismatch signature—weak‑decoy detection rates diverging from expected Poisson predictions by several standard deviations; and (3) a mild but coordinated time‑bin asymmetry in the detector gates (early vs late) that didn’t show up on yesterday’s calibration.

The ops SOPs are clear but leave judgement to the analyst. A QKD glitch could be benign: sea spray, pointing jitter, micro‑weather, or a wandering clock skew. But patterns matter. Certain blends of QBER lift + decoy yield distortion + early/late skew whisper about classic eavesdropping tactics: photon‑number‑splitting (PNS) probing, time‑shift attacks that exploit detector efficiency mismatch, or brightly lit nudges at the receiver (though that’s less common on this platform).

Your watch lead doesn’t want speculation—only a defensible call in under five minutes: “material” (raise readiness, move to alternate keys, tighten link policy) or “noise” (re‑sync, keep calm, continue session). If you call it material and it’s nothing, you slow everyone down on a tense night. If you call it noise and it’s the start of a real probing run, you risk feeding suspect keys into downstream crypto and triggering a cascade of bad decisions.

You breathe, snapshot the last ten minutes of telemetry, and lock in. The dataset is small—counts and rates—but the math is sharp enough to separate smoke from fog. Your goal: run a fast, defensible triage using decoy‑state consistency, binomial tests on the time‑bin split, and QBER context against historical baselines. If the combination looks like a basis‑dependent manipulation or a multi‑sigma decoy distortion that normal channel noise can’t explain, you label it MATERIAL. If signatures line up with a global timing drift or benign channel wobble, you label it NOISE with a recommendation to re‑sync gates and continue.

The South China Sea doesn’t pause while you think. Make the call.

Background & Technical Context

The monitored link is a DV‑QKD decoy‑state BB84 implementation. The transmitter randomly interleaves three intensities: a signal mean‑photon number $\mu_s$ (e.g., around 0.5–0.6), a weak decoy $\mu_w$ (e.g., $\approx 0.1\!-\!0.2$), and vacuum $\mu_0$ (ideally 0). In honest operation under Poissonian photon statistics and a channel with overall transmissivity $\eta$, the per‑pulse detection probabilities follow

\[ Q_{\mu} \approx Y_0 + 1 - e^{-\eta \mu}, \]

where $Y_0$ is the background (dark + stray) yield. Comparing $Q_{\mu_s}, Q_{\mu_w}, Q_{\mu_0}$ lets an analyst estimate single‑photon yields and bound an eavesdropper’s advantage (e.g., from PNS). If an adversary tries to split multi‑photon pulses, the observed decoy statistics become inconsistent with the Poisson model in a way that can be caught via simple hypothesis tests or bounds used in decoy‑state analysis.

In parallel, the QBER (quantum bit error rate) serves as a coarse integrity signal:

\[ \mathrm{QBER} = \frac{E}{E + C}, \]

where $E$ is the number of erroneous sifted bits and $C$ the correct ones (post‑basis sifting). Benign channel changes (fog, sea haze, pointing jitter) typically push both correct and error counts down proportionally, often raising QBER gently. Actively induced, basis‑dependent distortions (e.g., timing nudges at one detector) can produce tell‑tale asymmetries that are statistically “too sharp” to be mere weather.

A rapid check for time‑shift tricks leverages early/late gate counts per basis (Z and X). Let $E_b$ and $L_b$ be early/late counts for basis $b\in\{Z, X\}$ over a short window, and define an asymmetry ratio

\[ \rho_b = \frac{\max(E_b, L_b)}{\min(E_b, L_b)}. \]

A modest $\rho_b>1$ is expected due to noise and finite sampling. However, elevated $\rho_b$ combined with a sign flip across bases (e.g., $E_Z>L_Z$ while \(E_X

For quick significance, treat early/late as $ \text{Binom}(N_b, 0.5) $ and compute

\[ z_b = \frac{E_b - 0.5 N_b}{\sqrt{0.25 N_b}} = \frac{2E_b - N_b}{\sqrt{N_b}}. \]

With large $N_b$ (tens of thousands), even small offsets yield high $|z_b|$, so you don’t decision purely on $p$-values—you combine the pattern (sign comparison across bases) and a practical tolerance for $\rho_b$ (e.g., 1.10–1.15 depending on platform calibration).

On decoys, define observed detection fractions over a fixed window:

\[ \hat{Q}_{\mu} = \frac{\text{clicks at intensity }\mu}{\text{pulses at intensity }\mu}. \]

Fit a simple Poisson‑loss model with estimated $\hat{\eta}$ and $\hat{Y}_0$ from vacuum and weak decoy, then predict $Q_{\mu_s}^{\text{pred}}$. A multi‑sigma deviation between $\hat{Q}_{\mu_s}$ and $Q_{\mu_s}^{\text{pred}}$, while $\hat{Q}_{\mu_w}, \hat{Q}_{\mu_0}$ also drift in the “PNS‑consistent” direction, pushes you toward a material finding. If, instead, all three move together in a way that’s consistent with a small $\eta$ wobble or $Y_0$ bump (e.g., temporary background light), you downgrade to noise—especially if the time‑bin pattern screams global skew.

Your triage packet for this exercise includes (synthetic but realistic) counts for a 90‑second window: per‑basis early/late gate counts, total sifted errors/corrects, and per‑intensity pulse and click totals. Your task is not to run a full finite‑key proof; it’s to execute a crisp, auditable triage:

Step 1: Compare observed decoy detection fractions $\hat{Q}_{\mu}$ against a quick Poisson fit.
Step 2: Compute $\rho_Z,\rho_X$ and note sign (early vs late) per basis; check for cross‑basis sign flip.
Step 3: Put the QBER bump in context versus the last 24h baseline.
Step 4: Render a MATERIAL vs NOISE decision and a one‑line action.

Your Deliverable

Deliver a one‑paragraph decision (MATERIAL or NOISE) with 3 bullets justifying the call:

Decoy consistency (numbers and direction),
Time‑bin asymmetry pattern ($\rho_Z,\rho_X$ and sign behavior),
QBER context versus baseline.

Then propose the immediate action (re‑sync & continue vs quarantine & escalate).

Notation Cheatsheet

$Q_{\mu}$: detection probability at intensity $\mu$.
$Y_0$: background yield.
$\eta$: overall transmissivity.
$\rho_b$: early/late asymmetry ratio in basis $b$.
$\mathrm{QBER} = E/(E+C)$.

Student Edition

Intermediate — Glassmaker Mirage: DV/CV Drift vs. Spoof

1.0 Mission Brief (Scenario)

03:13 local, 13 September 2031. The ops floor is the kind of quiet that amplifies tiny sounds: HVAC breath, the faint chirp of optics alarms two rows over, status LEDs pulsing in a slow constellation. You are the on‑call analyst for a multinational’s quantum backbone when the triage daemon Lighthouse blinks amber: ALERT — TIME‑BIN/NOISE ANOMALY DETECTED. The alert packet is intentionally compact—one short watch window (about two minutes), just enough to decide whether to keep keys flowing or freeze the spur. The map overlay highlights the South China Sea segment; the event console unfurls the snapshot. You take one slow breath and triage.

Your job is not to prove a theorem. It is to make a minutes‑scale, auditable call that an on‑call manager can understand at 03:13. The packet gives you two complementary lenses. First, discrete‑variable (DV) time‑bin balance: for each basis (Z and X) the system counted early and late detections in a detection gate; totals are large enough to steady the fractions. Healthy links wobble with temperature, pointing, and load, but that wobble tends to nudge both bases in the same direction. The troubling signature is an opposite‑sign skew—say, Z leans early while X leans late—which hints at basis‑selective manipulation such as a time‑shift tactic.

Second, a quick continuous‑variable (CV) shot‑noise audit: the system estimates shot noise with the quantum signal present and again with the signal blocked. In a healthy configuration, the ratio of these estimates rides near unity and does not co‑move with phase‑noise or pilot‑tone SNR spikes. If the yardstick itself is biased—say by bright‑light tricks or LO drift—the ratio lifts and traces misbehave. The pair of checks (DV pattern and CV yardstick) is deliberately simple: each can be computed on a napkin but together they are surprisingly discriminative in noisy field conditions.

Policy removes improvisation. Tonight’s guardrails are pinned in the commissioning doc and are not to be tuned mid‑incident: a DV magnitude tolerance of ρ_tol ≈ 1.12 (max‑over‑min time‑bin asymmetry) and a CV yardstick tolerance of R_tol ≈ 1.05 (shot‑noise ratio with‑signal vs. blocked). Your steps are therefore linear and testable: compute per‑basis early fractions and asymmetry ratios; check whether Z and X share the same sign or flip; audit the CV ratio and scan for correlated phase/SNR spikes; compare to thresholds; then decide PASS (re‑sync and continue) or QUARANTINE (halt generation, capture diagnostics, raise posture). The entire choreography fits on a single line of your scratchpad because real alerts arrive amid paging storms and coffee that has gone cold.

Outside the glass, forklifts ping a loading dock. Inside, your HUD ticks through counts. You have seen nights like this—gusts, HVAC steps, subtle clock drift—and you have also seen the mirage that attackers try to paint: patterns that masquerade as weather but tilt opposite across bases or quietly skew the noise yardstick. The discipline that keeps you honest is the one baked into your training: clear denominators, explicit thresholds, one‑paragraph briefs. You scroll again: early/late counts for Z and X, totals comfortably large, a shot‑noise audit line with “on” and “off,” and a note: “no correlated phase/SNR spikes observed.” The decision is yours, but the tools have pared the problem to its bones. If the pattern is same‑sign, within ρ_tol, and the CV ratio rides ≤ R_tol without pathology, you will reset gates, PASS, and keep the session alive; if not—if signs flip or the ratio lifts—you will QUARANTINE, seal logs, and escalate. The sea does not pause while you think. Neither should you. Make the call.

2.0 Historical & Conceptual Background

Quantum key distribution—two families, one operational goal. Since the earliest BB84 proposals, production QKD has bifurcated into DV schemes (with discrete detection events in gates) and CV schemes (with analog quadrature samples measured against a noise yardstick). In both, secrecy rests on disturbance: an adversary trying to glean information inevitably leaves a statistical fingerprint. Practical deployments, however, rarely resemble textbook, noiseless channels. Temperature gradients move alignment; ferries or cranes at a harbor nudge poles; a bright LO drifts; polarization rotates slowly in fiber. Operators therefore lean on patterns rather than ornate proofs in the heat of an incident. If the pattern “fits benign drift” and stays within guardrails, production continues. If not—if it looks basis‑selective or the yardstick is dubious—the line gets quarantined while deeper forensics run offline.

Why time‑bin balance is such a good canary. DV receivers carve time into narrow gates aligned to expected arrival. A small global offset—say a few picoseconds of clock skew—shifts both Z and X in the same direction (earlier or later). A manipulator seeking advantage, by contrast, pushes the two bases in opposite directions: one detector window is favored while its conjugate suffers. The sign of the skew across bases thus carries more meaning than its raw magnitude. A mildly imbalanced, same‑sign pattern is almost certainly drift; a modest but opposite‑sign pattern merits suspicion even before p‑values are invoked. That is Glassmaker’s first lesson: pattern outranks drama.

Shot‑noise as a yardstick—and how it can lie. In CV, all secrecy accounting flows through the shot‑noise unit (SNU). If an adversary can inflate the measured shot noise during “signal‑present” moments—by adding classical light or nudging phase—then subsequent reconciliation and parameter estimation think the channel is cleaner than it is. A robust audit blocks the signal briefly, re‑measures, and compares the yardsticks. The expected ratio is unity within a small tolerance. Persistent elevation of the ratio or correlated spikes in phase/SNR traces point to mis‑calibration or spoofing. This is Glassmaker’s second lesson: always check the yardstick before you trust the ruler.

Guardrails over free‑hand thresholds. Production environments pick tolerances before the night gets weird. Tighter values catch more subtle issues at the cost of frequent false alarms; looser values keep wheels turning but increase risk. The commissioning document for this theater pegs DV magnitude at ρ_tol ≈ 1.12 and CV ratio at R_tol ≈ 1.05—settings that assume maritime vibration and daily thermal cycles. These are policy choices: a junior analyst at 03:13 doesn’t edit them; they merely apply them and leave a crisp audit trail.

Humility about “significance.” With tens of thousands of counts per basis, almost any nonzero offset can look statistically significant. That is why Glassmaker pairs the binomial z‑score with pattern and guardrails. A large |z| on a same‑sign pattern is still drift‑colored; a moderate |z| on an opposite‑sign pattern is worrisome. Put differently, “significant” is not the same as “suspicious.” Operational judgment depends on what is deviant, not merely how many sigma away it sits.

3.0 Technical Foundations

3.1 Discrete‑Variable (DV) Time‑Bin Balance

For each basis b ∈ {Z, X} the receiver reports early and late counts (E_b, L_b). We use totals N_b = E_b + L_b, the early fraction p_b, and an asymmetry magnitude ρ_b (max‑over‑min) together with a sign check across bases.

p_b = \frac{E}{N}

Early fraction in basis b: counts falling in the early gate divided by total counts.

ρ_b = \frac{max (E, L)}{min (E, L)}

Asymmetry magnitude in basis b: the larger time‑bin count divided by the smaller (always ≥ 1).

A light‑weight context score treats early/late as a Binomial(N_b, 0.5). The z‑score below rises with both imbalance and sample size; we use it for color, not as a binary switch.

z_b = \frac{2 E - N}{\sqrt{N}}

Binomial context score: deviations of early counts from half the total, scaled by √N.

Pattern test. Compare the signs (E_b − L_b) across Z and X. If both are positive or both negative, we say the skews are same‑sign (drift‑colored). If one is positive and the other negative, they are opposite‑sign (suspect). The numerical guardrail then asks whether any ρ_b exceeds ρ_tol.

3.2 Continuous‑Variable (CV) Shot‑Noise Audit

Let N₀^(on) and N₀^(off) denote shot‑noise estimates with signal present vs. blocked. The ratio is our yardstick integrity check:

R = \frac{N^{0 (on)}}{N^{0 (off)}}

Shot‑noise yardstick ratio: unity (within tolerance) indicates an honest yardstick.

Interpretation. If R sits within tolerance—here, ≤ R_tol ≈ 1.05—and phase/SNR traces are calm, we treat the CV yardstick as sound. If R lifts beyond tolerance or co‑moves with traces, suspect bright‑light mis‑calibration or LO abuse. Combine this with the DV pattern to make the call.

Decision rule (operational): QUARANTINE if DV skews are opposite‑sign across bases, or if any ρ_b > ρ_tol, or if R > R_tol with phase/SNR anomalies. Otherwise PASS (re‑sync/re‑gating + monitor).

4.0 Problem Setup & Data

Spur: Offshore trunk, South China Sea segment. Window size: ≈120 seconds (single alert window). Policy guardrails: ρ_tol = 1.12 (DV magnitude), R_tol = 1.05 (CV yardstick). Decision logic: as stated in the callout above.

4.1 DV — Time‑Bin Balance (per basis)

Z basis: E_Z = 26,350, L_Z = 23,650 ⇒ N_Z = 50,000.

X basis: E_X = 25,500, L_X = 24,500 ⇒ N_X = 50,000.

4.2 CV — Shot‑Noise Audit

With signal present: N₀^(on) = 1.002 SNU. With signal blocked: N₀^(off) = 0.998 SNU. Traces note: no correlated phase/SNR spikes observed during the audit.

All values above are the alert packet for this exercise. Your outputs will not be raw code; they will be a short table, a one‑line CV summary, and a one‑sentence operational brief suitable for a manager.

5.0 Student Tasks & Expected Outputs

Compute DV early fractions p_Z, p_X. Report each as both a reduced fraction and a decimal to three places.
Compute DV asymmetry magnitudes ρ_Z, ρ_X. State whether each basis leans early or late.
Determine the DV sign pattern across bases. Are the skews same‑sign or opposite‑sign?
Compute the CV yardstick ratio R = N₀^(on) / N₀^(off). Note any phase/SNR anomalies if present.
Apply guardrails. Compare max{ρ_Z, ρ_X} to ρ_tol and R to R_tol. Combine with the DV sign pattern to make the call.
Write a one‑sentence console brief that a manager can read without your notebook. It must cite denominators, thresholds, and the pattern that mattered (e.g., “same‑sign DV within ρ_tol, R ≤ R_tol”).

Deliverables: (a) A compact DV table with (E_b, L_b, N_b, p_b, ρ_b, sign) for b ∈ {Z, X}; (b) a one‑line CV summary; (c) the single‑sentence brief (PASS or QUARANTINE + reason).

6.0 Step‑By‑Step Guidance (Non‑Solution Scaffolding)

6.1 Read the Packet Once, End‑to‑End

Before computing, read the entire alert block like a weather map. Confirm that the DV section lists counts for Z and X and that the CV section lists N₀^(on), N₀^(off), and any qualitative notes about phase/SNR. Verify that the window duration is what you expect (≈120 s) and that the totals are plausibly large (tens of thousands per basis). If anything looks malformed, you are allowed to say “insufficient packet” and request a fresh pull—triage prefers clean inputs to heroic inference.

6.2 Start with Fractions, Not z‑Scores

Compute p_Z and p_X. Your first description of the skew should be in plain language: “Z skews early, X also skews early,” or “Z early, X late.” If you open with a statistic (“z_Z = 3.1”), you risk substituting a number for a concept. Pattern first; numbers second.

6.3 Add Magnitude with ρ

Compute ρ_Z and ρ_X. Values slightly larger than 1 are routine; the guardrail ρ_tol = 1.12 is deliberately tolerant for this spur. Do not conflate “bigger than you expected” with “beyond policy.” If both bases are same‑sign and each ρ ≤ ρ_tol, you are already leaning toward a PASS—unless the CV yardstick misbehaves.

6.4 Sanity‑Check with z

Compute the optional z scores if you want color. With N = 50,000, even a 1% offset produces a noticeable |z|, so treat these as supporting actors. If you do compute them, record the sign: a positive z implies early gating has the edge.

6.5 Audit the Yardstick

Calculate R. If it sits inside tolerance (≤ 1.05) and the traces are calm, you have no evidence of LO spoofing or calibration drift in this short window. If R lifts beyond tolerance or the traces jump in lockstep, call that out explicitly. Students sometimes forget that a clean CV yardstick can exonerate a scary‑looking DV magnitude when signs are same‑direction.

6.6 State the Pattern & Decide

Now synthesize: is the DV pattern same‑sign or opposite‑sign? Is the magnitude within tolerance? Is the CV yardstick honest? You should be able to produce the one‑sentence brief in under a minute. Triage is not a courtroom; it is a gate on a live system.

7.0 Worked Reading Prompts

7.1 Why same‑sign implies drift

Imagine two synchronized scoreboards for two courts that ought to behave alike. A global clock running fast nudges both boards in the same direction; one might log points a sliver before whistles, the other likewise. That is the same‑sign signature of benign timing skew. An attacker who can touch one court differently from the other leaves a tell more reminiscent of mirrors: one scoreboard leans early while the other leans late. This is the essence of Glassmaker’s qualitative test.

7.2 Why R ≈ 1 is comforting

Shot noise is the whisper of light itself. If the microphone (LO) and room (detector chain) are stable, measured noise with or without a performer onstage should be the same. A bright‑light trick that floods the stage would make the “with performer” measurement noisier, shifting R upward; a saggy calibration might do the same. When R is close to 1 and other traces are calm, you are justified in treating the yardstick as honest.

7.3 Why not tune guardrails mid‑alert

Thresholds are policy. If analysts tune them mid‑alert, two bad things happen: first, the audit trail becomes ambiguous (“why did you move the bar?”); second, the system incentivizes optimism during busy nights. The discipline of fixed guardrails protects you from both hindsight bias and incident drift.

8.0 Real‑World Variations & What‑Ifs

8.1 What if totals are small?

If N_b dips into a few thousand, ρ becomes jittery and z becomes tiny. You should still check signs, but lean more on repeated windows or a short re‑calibration run. Small samples magnify randomness; your narrative should say so.

8.2 What if signs are same‑sign but ρ is large?

That can happen if a global drift grows big enough. A sustained same‑sign skew beyond ρ_tol is a policy quarantine even if it is not an adversary. Keys minted during large drifts are often lower quality, and safing the system lets ops re‑gate calmly.

8.3 What if R is slightly high but traces are clean?

Corroboration matters. An R of 1.052 with pristine traces on a bouncy spur could be a measurement hiccup. A modest re‑audit may be justified. But do not hand‑wave a high R when phase/SNR are also dancing; that combination is the yardstick screaming.

8.4 What if signs flip across consecutive windows?

Rapid flips often indicate marginal gate alignment plus gusty conditions. Re‑sync and monitor for stability rather than escalate immediately. Attackers prefer persistence; weather prefers whimsy.

8.5 What if only one basis has very low counts?

If N_Z or N_X collapses, your sign test still holds but you must treat ρ with caution. A starving detector can produce extreme ratios for trivial reasons. In such cases, your brief should lead with count health before pattern.

9.0 Assessment Rubric & Self‑Check

Correctness (40%). Fractions and ratios computed accurately; pattern classification correct; guardrails applied without modification; decision aligns with policy.
Clarity (25%). Outputs are compact: a two‑row DV table, a one‑line CV summary, and a one‑sentence brief. No meandering, no hidden assumptions.
Defensibility (20%). The brief names denominators and thresholds; a peer could re‑create the decision with only your outputs.
Discipline (15%). No threshold tuning; optional statistics flagged as “context only”; language distinguishes drift vs. spoof.

10.0 Frequently Asked Questions

10.1 Do I need to compute confidence intervals?

No. You may compute them for personal comfort, but the decision rests on the sign pattern, ρ vs. ρ_tol, and R vs. R_tol, with trace context. Intervals are fine as long as they do not delay the call.

10.2 Why use max‑over‑min instead of a difference?

Ratios present a scale‑free magnitude and are less sensitive to total counts. Ops can compare ρ across links with different rates without mentally rescaling differences.

10.3 Could an attacker fake same‑sign skews?

Yes, but it is harder: they must tilt both bases in the same direction without inflating other signals. That kind of manipulation typically damages throughput or leaves CV fingerprints. The pairing of DV pattern with CV audit reduces risk.

10.4 What if I disagree with the policy thresholds?

Capture your rationale in the ticket and propose an update at the next review. In the middle of the night, follow policy. The artifact’s purpose is crisp, repeatable action, not threshold debates during an incident.

11.0 Notation & Math Glossary

E_b, L_b — early/late counts in basis b.
N_b — total counts in basis b (E_b + L_b).
p_b — early fraction (E_b/N_b).
ρ_b — asymmetry magnitude (max/min).
z_b — binomial context score ((2E − N)/√N).
N₀^(on), N₀^(off) — shot‑noise estimates with signal present vs. blocked.
R — shot‑noise yardstick ratio N₀^(on)/N₀^(off).
ρ_tol, R_tol — policy guardrails for DV magnitude and CV yardstick.

References

Bennett, C. H., & Brassard, G. (1984). Quantum cryptography: Public key distribution and coin tossing. In Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (pp. 175–179). Bangalore, India. https://arxiv.org/abs/2003.06557

Hwang, W.-Y. (2003). Quantum key distribution with high loss: Toward global secure communication. Physical Review Letters, 91(5), 057901. https://doi.org/10.1103/PhysRevLett.91.057901

Lo, H.-K., Ma, X., & Chen, K. (2005). Decoy state quantum key distribution. Physical Review Letters, 94(23), 230504. https://doi.org/10.1103/PhysRevLett.94.230504

Wang, X.-B. (2005). Beating the photon‑number‑splitting attack in practical quantum cryptography. Physical Review Letters, 94(23), 230503. https://doi.org/10.1103/PhysRevLett.94.230503

Qi, B., Fung, C.-H. F., Lo, H.-K., & Ma, X. (2007). Time‑shift attack in practical quantum cryptosystems. Quantum Information & Computation, 7(1), 73–82. https://arxiv.org/abs/quant-ph/0512080

Grosshans, F., & Grangier, P. (2002). Continuous variable quantum cryptography using coherent states. Physical Review Letters, 88(5), 057902. https://doi.org/10.1103/PhysRevLett.88.057902

Jouguet, P., Kunz‑Jacques, S., & Diamanti, E. (2013). Preventing calibration attacks on the local oscillator in continuous‑variable quantum key distribution. Physical Review A, 87(6), 062313. https://doi.org/10.1103/PhysRevA.87.062313

Sajeed, S., Huang, A., Sun, S.-H., Makarov, V., Chaiwongkhot, P., Gisin, N., & Curty, M. (2015). Insecurity of detector‑device‑independent quantum key distribution. Physical Review A, 91(6), 062301. (Background on detector attacks and calibration pitfalls relevant to practical triage.) https://doi.org/10.1103/PhysRevA.91.062301

Pirandola, S., Andersen, U. L., Banchi, L., et al. (2020). Advances in quantum cryptography: Security of quantum key distribution. Advances in Optics and Photonics, 12(4), 1012–1236. (Survey with DV/CV operational considerations.) https://doi.org/10.1364/AOP.361502

Solution

Intermediate — Glassmaker Mirage — Solution (South China Sea 2031)

Intermediate — Glassmaker Mirage — Solution

Triage playbook for a 03:13 local alert from a quantum link during a South China Sea watch window (Sep 13, 2031). You are the on‑call analyst; this is the “how to decide in minutes” guide.

Situation recap. A QKD span reports unusual statistics over a short interval (Δt ≈ 120 s). Your task is to decide whether to PASS (re‑sync and continue) or QUARANTINE (halt generation / raise readiness).

Quick Model Recap (what the system should look like)

Signals and observables. Two common regimes: Discrete‑variable (DV) QKD (e.g., BB84/decoy) gives you counts per time bin and basis. We denote early/late bins in basis $b\in\{Z,X\}$ by $E_b, L_b$ with $N_b = E_b + L_b$. Continuous‑variable (CV) QKD gives you quadrature samples $(x, p)$ measured by homodyne/heterodyne; you monitor the shot‑noise level $N_0$ and the total measured variance $V_{\text{meas}}$. In both cases, a healthy link looks “balanced”: no persistent time‑bin bias in DV; stable $N_0$ and phase in CV.
What an active manipulation looks like. Adversaries exploit hardware asymmetries. In DV, a classic time‑shift attack nudges detection windows so one detector is subtly favored in one basis and disfavored in the orthogonal basis. The hallmark pattern: opposite signs of skew across bases (e.g., $E_Z > L_Z$ but $E_X < L_X$). In CV, a local‑oscillator (LO) spoof adds bright classical light or phase‑drifts to bias your shot‑noise calibration so the channel appears cleaner than it is. Tell‑tales include inflated “shot‑noise with signal present,” phase wander, and excess‑noise under‑reporting.
What normal drift looks like. Production links do wobble. Clock offsets, temperature drifts, and slow polarization rotation can produce mild, same‑sign skews across both bases (DV) or gentle $N_0$ breathing and correlated phase noise (CV). Normal drift tends to be global and consistent, not basis‑dependent or mode‑selective.
Your control knobs. You can (1) re‑center detection gates / re‑sync clocks; (2) tighten/loosen per‑window tolerances; (3) run an on‑the‑fly shot‑noise audit (CV), i.e., block the quantum signal and re‑estimate $N_0$; and (4) flip to conservative sifting (e.g., discard suspect windows) while you decide. Your decision is a function of simple statistics + pattern tests + these knobs.

Step‑by‑Step Triage (do this in order)

Aggregate and sanity‑check the window. For DV, compute early/late fractions $p_b = E_b/N_b$ and asymmetry ratios \[ \rho_b = \frac{\max(E_b, L_b)}{\min(E_b, L_b)}. \] Compute a binomial z‑score \[ z_b = \frac{2E_b - N_b}{\sqrt{N_b}}, \] under $H_0: p_b = 0.5$. Expect nonzero $z_b$ at large $N_b$—significance alone isn’t the decision lever; the pattern is. For CV, log $V_{\text{meas}}$, the current shot‑noise estimate $N_0$, instantaneous phase‑error, and the pilot tone SNR.
Run the cross‑basis sign‑pattern test (DV) or the shot‑noise audit (CV).
- DV: Check whether $\operatorname{sign}(E_Z - L_Z)$ and $\operatorname{sign}(E_X - L_X)$ are opposite. Opposite signs → basis‑dependent skew → suspect. Same sign → global drift → likely benign.
- CV: Temporarily block the signal and re‑estimate shot noise: $N_0^{(\text{off})}$. Compare with the “with‑signal” estimate $N_0^{(\text{on})}$ via \[ R \equiv \frac{N_0^{(\text{on})}}{N_0^{(\text{off})}}. \] In a healthy system you expect $R \approx 1\pm \epsilon$. A bright‑LO spoof often yields $R \gg 1$ or phase‑noise spikes co‑moving with LO power.
Apply guardrails and thresholds. Example tolerances that work well in noisy environments: DV: $\rho_{\text{tol}} \approx 1.12$ and opposite‑sign skew to flag; CV: $R_{\text{tol}} \approx 1.05$ with corroborating phase‑noise or pilot SNR dips. These are policy setpoints—operators can tighten for crisis ops.
Decide and act.
- PASS (re‑sync & continue): DV skews are same‑sign across bases and $\max \rho_b \le \rho_{\text{tol}}$; CV shot‑noise ratio $R \le R_{\text{tol}}$ with no excess‑noise spike. Perform a quick re‑gating/re‑sync and keep the session alive.
- QUARANTINE (halt & escalate): DV shows opposite‑sign skew and $\rho_b$ beyond tolerance, or CV’s $R \gg R_{\text{tol}}$ with phase/SNR anomalies. Freeze keys, capture diagnostics, and raise posture.

Worked Example (numbers from the alert window)

DV check — time‑bin balance

Counts in the window (per basis):

Z-basis: E_Z = 26,350,  L_Z = 23,650,  N_Z = 50,000
X-basis: E_X = 25,500,  L_X = 24,500,  N_X = 50,000

Fractions and ratios:

$p_Z = E_Z/N_Z = 0.527$, $\rho_Z = 26{,}350/23{,}650 \approx 1.115$.
$p_X = E_X/N_X = 0.510$, $\rho_X = 25{,}500/24{,}500 \approx 1.041$.

Both skews are statistically non‑zero at this $N$ (for context): \[ z_Z = \frac{2E_Z - N_Z}{\sqrt{N_Z}} \approx \frac{700}{223.61} \approx 3.13,\quad z_X \approx 4.47. \] Importantly, the signs match ($E_b > L_b$ in both $Z$ and $X$), which is the signature of a global timing skew, not a basis‑dependent manipulation.

CV check — shot‑noise audit (if applicable to this link)

Audit results over the same window (illustrative values):

With signal: estimated shot noise $N_0^{(\text{on})} = 1.002\,\mathrm{SNU}$.
Signal blocked: $N_0^{(\text{off})} = 0.998\,\mathrm{SNU}$.

Ratio $R = N_0^{(\text{on})}/N_0^{(\text{off})} \approx 1.004 \le R_{\text{tol}} = 1.05$. Pilot tone SNR and phase error show no correlated spikes. That’s consistent with no LO spoof.

Decision

DV: $\rho_Z \approx 1.115$, $\rho_X \approx 1.041$, both $\le \rho_{\text{tol}} = 1.12$; signs are the same across bases → global drift.
CV: $R \approx 1.004 \le 1.05$; no phase/SNR pathology → no LO spoof.
Outcome: PASS. Perform re‑sync/re‑gating, continue session, and schedule a follow‑up calibration window.

Why this is the right call

Imagine two basketball scoreboards (the $Z$ and $X$ bases) that should stay roughly tied. An attacker who can tilt the floor only when you switch courts would make one scoreboard favor “early” while the other favors “late.” That’s the opposite‑sign pattern we hunt. Here, both scoreboards lean the same way by a small margin—like a single gym clock running a tad fast. In CV terms, your “microphone hiss” (shot noise) is the same whether the singer is on stage (signal on) or not (signal off). If a fake LO flooded the mic, the hiss measurement would be off when the singer stands there. It isn’t.

The pass decision isn’t “shrug and move on.” You act: slide detection windows by a few picoseconds, re‑center the phase‑tracker, and keep auditing. If the pattern ever flips to the tell‑tale of targeted tampering (opposite signs, big $\rho$, or $R$ spikes), you already know the QUARANTINE play: freeze keys, capture diagnostics, and escalate readiness for the theater.

Operator Checklist (fast path)

✅ Compute $p_b, \rho_b, z_b$ for DV; log $N_0^{(\text{on})}, N_0^{(\text{off})}, R$ for CV.
✅ Check sign pattern (DV) or $R$ and phase/SNR (CV).
✅ Compare to policy thresholds ($\rho_{\text{tol}}$, $R_{\text{tol}}$).
✅ PASS → re‑gating/re‑sync + monitor; QUARANTINE → halt, seal logs, escalate.

Notes: numeric tolerances shown (e.g., $\rho_{\text{tol}}=1.12$, $R_{\text{tol}}=1.05$) are representative and can be tuned by your ops policy. All math is rendered client‑side via MathJax.

Instructor Edition

Glassmaker Mirage

1.0 Historical Background & Conceptual Deep Dive

Operations teams who live with production quantum links learn quickly that the most useful playbooks are those you can run at 03:13 on a Tuesday. The physics is profound; the decision must be simple. Glassmaker Mirage sits in that philosophy: take two families of practical signals seen in the field—discrete‑variable (DV) time‑bin counts and continuous‑variable (CV) shot‑noise measurements—and convert them into a short, auditable decision tree. The “mirage” in the title hints at a known adversarial trick: create patterns that resemble benign drift (e.g., global timing skew) so that a tired operator shrugs and passes. Your task is to build a chapter that makes that shrug unlikely by pairing simple statistics with pattern logic and a small number of guardrails that can be tuned in policy.

In DV QKD deployments, a widely used health indicator is the balance of early/late detections within a time‑bin basis. A well‑behaved link can tolerate small imbalances—temperature drift, clock wander, and stress on fibers will do that. But basis‑selective imbalances, where one basis leans early while the orthogonal basis leans late, are operationally suspicious. They suggest a time‑shift or related manipulation that nudges detection windows unevenly across bases. That is the first lens taught in this lab: a sign pattern across bases is as important as the magnitude.

In CV QKD, the local oscillator (LO) provides phase reference and sets the yardstick for noise. If an adversary biases your shot‑noise calibration—say by bright‑light tricks co‑moving with LO power—the “with‑signal” shot‑noise estimate is no longer commensurate with the genuine vacuum noise. The fix is to perform a short shot‑noise audit that re‑estimates the noise with the quantum signal blocked, then compare the two. The audit collapses a potentially complex story into a single ratio that is easy to track under pressure.

The point of pairing DV and CV views is not to multiply work; it is to reduce ambiguity. DV tells you whether the link shows a basis‑dependent skew; CV tells you whether your noise yardstick is honest. Together they support a conservative, documented call: PASS (re‑sync and continue) or QUARANTINE (halt generation and raise posture). The remainder of this chapter turns that philosophy into an instructor‑ready, fully auditable path using minute‑scale windows, display‑quality fractions, clear thresholds, and short English glosses after every display equation so that students see mathematics as words, not incantations.

2.0 Executive Summary

We triage a short alert window using two complementary checks. DV check: compute early/late fractions per basis and an asymmetry ratio; evaluate the sign pattern across bases (opposite‑sign skew ⇒ suspect; same‑sign ⇒ likely global drift). CV check: if applicable to the link, temporarily block the quantum signal, re‑estimate shot noise, and compare to the with‑signal estimate via a ratio. Guardrails: a policy tolerance on the DV asymmetry (example: ρ_tol ≈ 1.12) and on the CV ratio (example: R_tol ≈ 1.05) bound acceptable drift. A runbook then translates outcomes into PASS or QUARANTINE, with immediate next steps (re‑gating/re‑sync vs. freeze keys, capture diagnostics, escalate). Worked numbers below mirror the alert window pictured in the solution PDF for this intermediate splash lab.

3.0 Learning Objectives

Compute DV time‑bin balance metrics per basis and interpret the sign pattern across orthogonal bases.
Perform a CV shot‑noise audit and summarize it with a single ratio that is easy to brief.
Apply guardrails consistently via fixed thresholds carried in policy, not ad‑hoc adjustments.
Write a one‑paragraph, audit‑ready brief that a non‑specialist can read without the notebook.
Explain why the decision is conservative, how it fails safe, and what next actions follow immediately.

4.0 What “Healthy” Looks Like (and What Manipulation Looks Like)

4.1 Signals & observables

DV regime. For each basis b ∈ {Z, X} we record counts in early/late bins: E_b, L_b, with N_b = E_b + L_b. A healthy link has no persistent, basis‑dependent sign flip in early‑minus‑late, and asymmetry magnitudes that live within policy tolerances.

CV regime. We log the shot‑noise level N₀ and the total measured variance V_meas under homodyne/heterodyne. A healthy link keeps N₀ stable and free of correlations with LO power in the audit; phase and pilot‑tone SNR should show no co‑moving spikes.

4.2 Manipulation signatures vs. benign drift

Time‑shift style manipulation (DV): opposite‑sign skew across bases (E_Z > L_Z but E_X < L_X, or vice‑versa) with large asymmetry.
Bright‑LO or yardstick bias (CV): shot‑noise ratio far from unity (with‑signal vs. blocked‑signal), and anomalies co‑moving with LO power.
Benign drift: same‑sign skew across bases (global timing skew), small asymmetries, stable noise yardstick.

5.0 Givens & Notation (per one alert window)

The solution packet provides a worked alert window with counts (per basis) and a CV audit. We adopt the notation below and refer back to those numbers in §9.0.

Symbol	Meaning	Remarks
E_b, L_b	DV early and late counts (basis b)	Per basis b ∈ {Z, X}.
N_b	Total DV counts (basis b)	N_b = E_b + L_b.
p_b	Early fraction	p_b = E_b/N_b.
ρ_b	Asymmetry ratio	ρ_b = max(E_b, L_b)/min(E_b, L_b).
z_b	Binomial z‑score (context only)	z_b = (2E_b − N_b)/√N_b.
N₀^(on), N₀^(off)	CV shot‑noise with signal on / blocked	Re‑estimate “yardstick” with signal blocked.
R	CV shot‑noise ratio	R ≡ N₀^(on)/N₀^(off).
ρ_tol, R_tol	Policy tolerances	Example bands: ρ_tol ≈ 1.12; R_tol ≈ 1.05 (tunable).

6.0 Estimators & Tests (with English Gloss)

6.1 DV early fraction

p_{b} = \frac{E_{b}}{N_{b}} .

English gloss: within a basis, the early fraction is early counts divided by the total counts.

6.2 DV asymmetry (magnitude) via ratio

ρ_{b} = \frac{max (E_{b}, L_{b})}{min (E_{b}, L_{b})} .

English gloss: ρ measures how unbalanced early vs. late are, regardless of which side is bigger.

6.3 DV sign pattern (across bases)

Define the sign per basis by comparing E_b and L_b. Opposite signs across Z and X suggest basis‑selective manipulation; same signs suggest global drift. Always report the pattern explicitly in your brief.

6.4 DV z‑score (context only)

z_{b} = \frac{2 E_{b} - N_{b}}{\sqrt{N_{b}}} .

English gloss: the z‑score quantifies how far the early fraction is from 50% at this denominator.

6.5 CV shot‑noise audit ratio

R = \frac{N^{(on)}}{N^{(off)}} .

English gloss: compare the shot‑noise yardstick with signal present to the yardstick with signal blocked; unity means the yardstick is honest.

7.0 Guardrails & Policy Thresholds

Your commissioning document pins numeric tolerances for the environment. For noisy maritime links, a useful starting point is ρ_tol ≈ 1.12 (DV asymmetry) and R_tol ≈ 1.05 (CV ratio). Teams can tighten or loosen these with experience. The essential rule is consistency: do not adjust thresholds mid‑incident. Document a revision later if seasonality or hardware revisions warrant new bands.

8.0 Step‑by‑Step Triage (Run This in Order)

Aggregate & sanity‑check the window (DV): compute p_b, ρ_b, and z_b per basis; verify denominators are plausible.
Check the cross‑basis sign pattern (DV): are the signs opposite (suspect) or the same (likely drift)?
Run the shot‑noise audit (CV, if applicable): block the signal briefly; estimate N₀^(off); compute R.
Apply guardrails: ensure max ρ_b ≤ ρ_tol and R ≤ R_tol for a PASS; any violation or opposite‑sign pattern → QUARANTINE.
Decide & act: PASS → re‑gating/re‑sync + monitor; QUARANTINE → freeze keys, capture diagnostics, escalate posture.

9.0 Worked Example (Alert Window Numbers)

The solution packet displays a DV time‑bin balance table and a CV audit over the same window. The DV counts (per basis) read approximately: Z basis: E_Z = 26,350; L_Z = 23,650; N_Z = 50,000. X basis: E_X = 25,500; L_X = 24,500; N_X = 50,000. From these we compute:

Early fractions: p_Z = 26,350/50,000 ≈ 0.527; p_X = 25,500/50,000 = 0.510.
Asymmetry ratios: ρ_Z = 26,350/23,650 ≈ 1.115; ρ_X = 25,500/24,500 ≈ 1.041.
Sign pattern: E>L in both bases → same‑sign → global drift signature.
Contextual z‑scores (not a guardrail): z_Z ≈ (2·26,350 − 50,000)/√50,000 ≈ 2,700/223.61 ≈ 12.1; z_X ≈ 1,000/223.61 ≈ 4.47.

The CV audit shows N₀^(on) ≈ 1.002 SNU and N₀^(off) ≈ 0.998 SNU over the same window, giving R = N₀^(on)/N₀^(off) ≈ 1.004, with no phase‑noise or pilot‑tone SNR anomalies. These values sit safely at or below the example tolerance R_tol ≈ 1.05.

10.0 Decision & Rationale

DV: ρ_Z ≈ 1.115, ρ_X ≈ 1.041, both ≤ ρ_tol = 1.12; signs are the same across bases → global drift. CV: R ≈ 1.004 ≤ R_tol = 1.05; no phase/SNR pathology → no LO spoof. Outcome: PASS. Perform re‑sync/re‑gating, continue the session, and schedule a follow‑up calibration window. Capture counts, ratios, and thresholds in the ticket for audit.

11.0 Why This Is the Right Call

In DV, imagine two scoreboards—one per basis—that should stay roughly tied. An attacker who can tilt the floor only when you switch courts would make one scoreboard favor “early” while the other favors “late.” That opposite‑sign pattern is the red flag we hunt. The alert window does not show it: both bases lean the same way by small margins within tolerance. That is what global drift looks like (e.g., slow timing skew).

In CV, the “microphone hiss” (shot noise) should measure the same whether the singer is on stage (signal present) or not (signal blocked). If a fake LO flooded the mic, the hiss measurement would swing. Here it does not; the ratio rides near unity, and phase/SNR traces are calm. Pairing these observations gives a defensible PASS that also comes with action: re‑center detection windows by a few picoseconds, re‑align clocks, and keep auditing the strip for any sign‑flip or ratio jump.

12.0 Operator Checklist (Fast Path)

Compute p_b, ρ_b, z_b for DV; log N₀^(on), N₀^(off), and R for CV.
Check sign pattern across bases (DV) or R and phase/SNR (CV).
Compare against policy thresholds (ρ_tol, R_tol).
PASS → re‑gating/re‑sync + monitor; QUARANTINE → halt, seal logs, escalate readiness.

13.0 Pitfalls, Edge Cases, and Seasonality

Denominator hygiene: Always publish N_b next to p_b. The same asymmetry looks more/less alarming as counts change.
Basis mis‑mapping: Ensure the mapping of physical channels to logical bases is stable; swaps can mimic sign flips.
Maintenance pings: Documented scans can nudge timing; annotate the window and correlate with change records.
Phase wander (CV): Slow LO drift can broaden R slightly; look for co‑moving phase‑noise spikes before calling spoof.
HVAC/diurnal patterns: Repeating time‑of‑day signatures should be captured in policy notes so night shifts aren’t surprised.
Threshold drift: Never “nudge” thresholds mid‑incident. If you learn better bands, record the evidence and revise at the next review.

14.0 Audit‑Ready Math (Compact Derivations)

14.1 DV: early fraction and asymmetry

The early fraction p_b is a binomial proportion with denominator N_b. For large N_b, the standard error of p̂ is

SE (\hat{p}) = \sqrt{\frac{\hat{p} (1 - \hat{p})}{N_{b}}} .

English gloss: sampling spread shrinks as counts grow and is largest near 50–50.

The asymmetry ratio ρ_b is a simple magnitude measure. We pair it with the cross‑basis sign test to avoid mis‑classifying benign global drift.

14.2 DV: z‑score for context

The z‑score compares early bias to the null of 50–50. We present z_b in the instructor note for context; decisions do not rest on it alone.

14.3 CV: shot‑noise audit ratio

With the quantum signal present, the shot‑noise proxy is N₀^(on). With the signal blocked, it is N₀^(off). The audit ratio R should remain near unity in a healthy system. Values far above unity or co‑moving with LO power indicate a biased yardstick and trigger QUARANTINE.

15.0 Implementation Quality (Autograders & Dashboards)

Print an audit line per basis: {'basis': 'Z'|'X', 'E': ..., 'L': ..., 'N': ..., 'p_hat': ..., 'rho': ..., 'sign': 'E>L'|'L>E'}.
For CV links, add {'N0_on': ..., 'N0_off': ..., 'R': ...} and include phase/SNR markers in the same line.
Store thresholds and a configuration hash; emit the hash in the audit line so policy drift is detectable after the fact.
Plot p̂ per basis with shaded tolerance bands; add a panel for R; annotate maintenance markers.

16.0 What‑Ifs (Sensitivity & Scenario Variations)

Scenario	Effect	Likely Outcome	Instructor Talking Point
Denominators drop by half	SE doubles; ratio ρ unchanged	DV confidence weakens; still check sign pattern	Confidence and magnitude are distinct; publish both.
Opposite‑sign skew appears (same ρ)	Basis‑selective signature	QUARANTINE even if ρ close to ρ_tol	Pattern outranks magnitude when consistent with attack.
R drifts to 1.06 without phase/SNR spikes	Slight yardstick bias	QUARANTINE per policy (R > R_tol)	Do not “explain away” a ratio beyond tolerance mid‑incident.
R ≈ 1.02 but phase noise spikes with LO power	Correlated anomaly	Investigate aggressively; consider temporary quarantine	Ratios summarize; auxiliary traces corroborate.
ρ_Z=1.14, ρ_X=1.01 (both E>L)	Same‑sign with mild excess on Z	Borderline; policy threshold decides	Write the rationale either way; tie to denominators and history.

17.0 Instructor Prompts (For Discussion)

Ask students to explain why opposite signs across bases hint at targeted manipulation, while same signs suggest global drift.
Have them compute what ρ_tol they would choose for a calmer metro link vs. a windy harbor spur—and defend the choice.
Show a CV trace with subtle LO‑power co‑movement and let the class debate whether to quarantine.
Compare the DV z‑scores to the ratio‑based guardrail; discuss why z alone is not the decision lever.

References

Bennett, C. H., & Brassard, G. (1984). Quantum cryptography: Public key distribution and coin tossing. In Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (pp. 175–179). IEEE. https://arxiv.org/abs/2003.06557

Scarani, V., Bechmann‑Pasquinucci, H., Cerf, N. J., Dušek, M., Lütkenhaus, N., & Peev, M. (2009). The security of practical quantum key distribution. Reviews of Modern Physics, 81(3), 1301–1350. https://doi.org/10.1103/RevModPhys.81.1301

Weedbrook, C., Pirandola, S., García‑Patrón, R., Cerf, N. J., Ralph, T. C., Shapiro, J. H., & Lloyd, S. (2012). Gaussian quantum information. Reviews of Modern Physics, 84(2), 621–669. https://doi.org/10.1103/RevModPhys.84.621

Jouguet, P., Kunz‑Jacques, S., Leverrier, A., Grangier, P., & Diamanti, E. (2013). Experimental demonstration of long‑distance CV‑QKD. Physical Review A, 87, 062313. https://doi.org/10.1103/PhysRevA.87.062313

Lydersen, L., Wiechers, C., Wittmann, C., Elser, D., Skaar, J., & Makarov, V. (2010). Hacking commercial QKD systems by bright illumination. Nature Photonics, 4, 686–689. https://doi.org/10.1038/nphoton.2010.214

Nielsen, M. A., & Chuang, I. L. (2010). Quantum computation and quantum information (10th anniversary ed.). Cambridge University Press. https://doi.org/10.1017/CBO9780511976663

Lab

I-06_Glassmaker_Mirage_no_readme

Objectives¶

Shot‑noise calibration test: Fit the vacuum quadrature variance $\sigma^2_{\text{vac}}$ versus local‑oscillator power $P_{\mathrm{LO}}$ and check linearity: $$\sigma^2_{\text{vac}} \;\approx\; a\,P_{\mathrm{LO}} + b.$$
SNR drift: Track quadrature variance across time windows to detect anomalous shifts.
LO reference‑tap consistency: Validate the ratio between LO at the reference tap and at the signal path.
Decision rule: Combine the tests into a single PASS / QUARANTINE outcome for triage.

Inputs¶

A short vacuum run: several LO power points $P_{\mathrm{LO}}$ with measured vacuum quadrature variance $\sigma^2_{\text{vac}}$.
A live time series of variance estimates to monitor SNR drift.
Two LO power taps per window: a reference tap (near the laser) and a signal‑path tap (near the homodyne), used to check ratio consistency.

You can replace the synthetic example below with your own CSV later.

In [ ]:

# Your Code Starts Here
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

np.random.seed(7)  # reproducibility

In [ ]:

# --- Synthetic data (you can swap with your CSV later) ---
# Calibration: two clusters to mimic mild nonlinearity + arbitrary units
size_lo = 36
plo = np.linspace(0.1, 1.8, size_lo)           # LO sweep (arbitrary units)
a_true, b_true = 0.89, 0.012                    # true line model for sigma^2
sigma2_vac = a_true * plo + b_true + np.random.normal(0, 0.005, size_lo)

calib_df = pd.DataFrame({"P_LO": plo, "sigma2_vac": sigma2_vac})

# Time windows for drift & taps
T = 180
base = np.linspace(0, 1, T)
baseline = 0.38 + 0.03*np.sin(2*np.pi*base)     # small benign drift
# Introduce a mild anomaly around t~90
drift_profile = baseline.copy()
drift_profile[85:100] += 0.10

# LO taps (reference vs signal path), nominal ratio ~0.72
lo_ref = 1.0 + 0.02*np.random.randn(T)
lo_sig = 0.72*lo_ref + 0.02*np.random.randn(T)

var_series = drift_profile + 0.02*np.random.randn(T)

data_df = pd.DataFrame({
    "t": np.arange(T),
    "var": var_series,
    "lo_ref": lo_ref,
    "lo_sig": lo_sig,
})
display(calib_df.head(), data_df.head())

In [ ]:

# --- Linear fit:  sigma^2_vac ≈ a * P_LO + b ---
A = np.vstack([calib_df["P_LO"].values, np.ones_like(calib_df["P_LO"].values)]).T
y = calib_df["sigma2_vac"].values
a_hat, b_hat = np.linalg.lstsq(A, y, rcond=None)[0]

# R^2
y_hat = a_hat*calib_df["P_LO"].values + b_hat
ss_res = np.sum((y - y_hat)**2)
ss_tot = np.sum((y - y.mean())**2)
r2 = 1 - ss_res/ss_tot

# Plot
plt.figure(figsize=(6,4))
plt.scatter(calib_df["P_LO"], calib_df["sigma2_vac"], label='vacuum data', s=18)
xline = np.linspace(calib_df["P_LO"].min(), calib_df["P_LO"].max(), 200)
plt.plot(xline, a_hat*xline + b_hat, label=f'fit: a={a_hat:.3f}, b={b_hat:.3f}\\n$R^2$={r2:.3f}')
plt.xlabel('P_LO')
plt.ylabel('sigma2_vac')
plt.title('Shot‑noise calibration')
plt.legend()
plt.tight_layout()
plt.show()

In [ ]:

# --- SNR drift monitoring ---
window = 9
baseline_roll = pd.Series(var_series).rolling(window, min_periods=window).median()
drift_ratio = (data_df["var"] / baseline_roll).to_numpy()

# flag if drift exceeds 8% above rolling baseline for >= 3 consecutive windows
excess = np.where(drift_ratio > 1.08, 1, 0)
runs = []
count = 0
for i, e in enumerate(excess):
    if e == 1:
        count += 1
        if count == 3: runs.append(i)  # just mark the first index of a 3‑run
    else:
        count = 0

print(f"Drift flags at t~: {runs} (first run >=3 above +8%)")

plt.figure(figsize=(6,3.5))
plt.plot(data_df["t"], data_df["var"], marker='o', markersize=2, lw=1, label='var')
plt.plot(data_df["t"], baseline_roll, lw=2, label='rolling baseline')
plt.plot(data_df["t"], drift_ratio, lw=1, label='variance proxy / baseline')
plt.axhline(1.08, color='k', linestyle='--', lw=1, label='1.08× band')
plt.title('SNR drift monitor')
plt.legend()
plt.tight_layout()
plt.show()

In [ ]:

# --- LO reference‑tap consistency ---
ratio = data_df["lo_sig"] / data_df["lo_ref"]
mean = ratio.mean()
std = ratio.std()

# band: ±6% relative band around nominal
nominal = 0.72
band = 0.06 * nominal
lo_ok = np.all((ratio > (nominal - band)) & (ratio < (nominal + band)))

plt.figure(figsize=(6,3.5))
plt.plot(data_df["t"], ratio, marker='o', markersize=2, lw=1, label='ratio')
plt.axhline(nominal, lw=2, label='nominal')
plt.axhline(nominal + band, lw=1, linestyle='--', label='+6%')
plt.axhline(nominal - band, lw=1, linestyle='--', label='-6%')
plt.title('LO reference‑tap consistency')
plt.legend()
plt.tight_layout()
plt.show()

Decision rule¶

We PASS if all hold:

Linearity looks good (e.g., $R^2 \ge 0.95$).
No multi‑window SNR drift spike (e.g., no $\ge 8\%$ above rolling baseline for $\ge 3$ consecutive windows).
LO tap ratio near nominal (mean within $\pm 6\%$ of $\sim 0.72$).

Deliverables¶

Calibration fit: $(\hat a, \hat b, R^2)$ for $\sigma^2_{\text{vac}} = \hat a\,P_{\mathrm{LO}} + \hat b$.
SNR drift report: plot + flag.
LO tap ratio check: mean ratio + guard‑band result.
Final triage: PASS / QUARANTINE.

In [ ]:

# --- Decision rule ---
# PASS if all hold:
# 1) Linearity looks good (e.g., R^2 >= 0.95)
# 2) No multi‑window SNR drift spike (e.g., no >=8% above rolling baseline for ≥3 consecutive windows)
# 3) LO tap ratio near nominal (mean within 6% of ~0.72)

r2_ok = r2 >= 0.95
drift_flag = (len(runs) > 0)  # True means we saw a concerning drift
lo_ratio_ok = lo_ok

decision = "PASS" if (r2_ok and (not drift_flag) and lo_ratio_ok) else "QUARANTINE"

print(f"R^2 ok? {r2_ok}  |  drift_flag? {drift_flag}  |  lo_consistent? {lo_ratio_ok}")
print(f"Decision = **{decision}**")

# Your Code Ends Here

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