Photonic Entanglement Creation, Photon Measurements, and Statistical Assurance: A Defence‑Relevant Study
Gerard King
www.gerardking.dev
September 2025

Abstract
In the realm of quantum technologies, photonic entanglement offers foundational advantages for secure communications, sensing, and networking. For Canadian defence applications, robust entangled‑photon sources, precise measurement, and rigorous statistical validation are essential. This paper surveys state‑of‑the‑art techniques in entanglement generation (bulk SPDC, integrated sources, quantum dots), reviews photon measurement modalities, examines statistical frameworks for validating entanglement and Bell inequalities under realistic noise, and presents a defence use‑case: field deployment of an entanglement‑based quantum key distribution (QKD) link between remote DND/CAF installations. Recommendations and technical benchmarks are provided.

Keywords: photonic entanglement, quantum key distribution, photon measurement, Bell inequality, defence communications, quantum dots, statistical hypothesis testing

1. Introduction

Quantum entanglement of photons is a resource that enables unconditionally secure communications (e.g., entanglement‑based QKD), high‑precision sensing, and distributed quantum networks. For defence agencies, these properties are of strategic relevance: secure transmission that is resistant to interception, the ability to detect tampering by eavesdroppers, and robustness in varied environmental conditions.

This paper addresses three technical pillars:

1. Methods of creating entangled photon pairs or multiphoton entangled states, with an eye toward brightness, fidelity, and integration.
   
   

2. Measurement of photons: detection technologies, coincidence counting, tomography, and loophole closure.
   
   

3. Statistical assurance: how to assert that entanglement is present, how to quantify its quality, and how to report confidence in a defence‑grade environment.
   
   

Finally, a use‑case scenario is detailed: a QKD link between two remote DND/CAF facilities (e.g., between bases separated by tens of kilometers), including design parameters, expected performance, and risk analysis.

2. Photonic Entanglement Generation

2.1 Bulk SPDC (Spontaneous Parametric Down‑Conversion) in Nonlinear Crystals

Bulk SPDC remains a mature and reliable method for generating entangled photon pairs. Nonlinear crystals (e.g., β‑Barium Borate, periodically poled KTP, etc.) pumped by continuous wave (CW) or pulsed lasers produce photon pairs under conservation of energy and momentum (phase matching). The widely used configurations include Type‑II (orthogonally polarized signal and idler) and Type‑0/Type‐I with specialized phase matching to optimize efficiency.

Li, Zhou, Xu, Shi, & Guo (2016) demonstrated multiplexed polarization‑ and time‑bin entangled photon‑pair sources using dispersion‑shifted fiber and dense wavelength division multiplexing (DWDM), yielding entanglement suitable for all‑fiber quantum networks. Their design shows how SPDC techniques can be scaled in multiple channels to support many simultaneous users. (Li et al., 2016)

A comprehensive review of SPDC sources in bulk optics, including continuous‐wave pumping, highlights trade‑offs: brightness vs multi‑pair emissions, crystal length vs group‑velocity mismatch and spectral correlation, collection optics vs coupling losses. (Entangled Photon‑Pair Sources Based on Three‑Wave Mixing in Bulk Crystals, 2022)

2.2 Integrated Photonic Circuits and On‑Chip Sources

To improve stability, size, and integration, on‐chip entangled sources are increasingly important. Femtosecond‑laser‑written circuits, waveguides, microring resonators, and periodically poled waveguide crystals permit smaller form factors and multiplexed output. Atzeni et al. (2017) demonstrated integrated sources at telecom wavelengths, with switching between path‑entangled and polarization‑entangled output and visibilities above 0.92. (Atzeni, Rab, Corrielli, Polino, Valeri, & Spagnolo, 2017)

A micrometer‑scale silicon microring resonator emitting time‑energy entangled photon pairs with >10⁷ Hz per nm pair generation rate, in the telecom band, operating at sub‑milliwatt pump power, was demonstrated by Grassani et al. (2014). (Grassani et al., 2014)

2.3 Quantum Dots and On‑Demand Entangled Photon Sources

Quantum dot sources have the advantage of deterministic emission (or close to), potentially reducing multi‑pair emission noise.
Zeuner, Jöns, Schweickert, et al. (2021) reported on‑demand entangled photon pairs in the telecom C‑band from InAs/GaAs quantum dots, achieving high concurrence (~91.4 ± 3.8%) and fidelity to the maximally entangled Bell state Φ⁺ (~95.2 ± 1.1%) at those wavelengths. (Zeuner et al., 2021)

Another recent work embeds quantum dots in circular Bragg resonators, strain‑tuned to optimize both brightness and degree of entanglement, enabling improvements in photon extraction, emission rate, and entanglement purity. (A source of entangled photons based on a cavity‑enhanced and strain‑tuned GaAs quantum dot, 2024)

2.4 High‑Dimensional & Novel Material Entanglement

Emerging research into novel 2D materials (e.g., 3R‑WS₂) has yielded ultracompact polarization‑entangled photon sources, providing high purity and efficiency in miniature devices. (Feng et al., eLight, 2024)

High‑order entangled states (multi‑photon, high‑dimensional OAM, time‑bin encoding) provide richer resources for security and channel capacity but impose more stringent demands on indistinguishability, synchronization, and system stability.

3. Photon Measurement and Validation

3.1 Detectors, Timing, and Efficiency
Key parameters: detection quantum efficiency, timing jitter, dark count rates, afterpulsing, dead time.

Superconducting nanowire single‑photon detectors (SNSPDs) currently lead in terms of efficiency (> 80‑90%), low dark count rates, and timing jitter on the order of tens of picoseconds. Commercially available APDs offer lower cost but inferior performance in many respects.

3.2 Coincidence Counting, Accidental Coincidences, and Background Noise

Coincidences are inferred by detecting photons at two (or more) detectors within a coincidence window τ_c. True coincidences derive from entangled pair emission; accidental coincidences result from background/single‑photon noise, dark counts, or multiple pair emissions.

If R_A and R_B are singles count rates at each detector, then accidental coincidence rate C_acc ≈ R_A R_B τ_c for small τ_c. Adjustments are made for detection probability, losses, and background. Correct estimation and subtraction of accidentals is essential for accurate entanglement metrics.

3.3 Quantum State Tomography & Fidelity Metrics

Reconstruction of the two‑photon density matrix ρ from measured projection outcomes in multiple bases (e.g., H/V, D/A, R/L for polarization) using linear inversion, maximum‑likelihood estimation (MLE), or Bayesian estimation. From ρ, one computes:

• Fidelity F(ρ, |Φ⁺⟩), the overlap with the ideal Bell state
  
  

• Concurrence or Negativity, measuring entanglement strength
  
  

• Purity, visibility, correlation contrast
  
  

3.4 Bell Inequalities, Loophole Closure, and Statistical Significance

Testing nonlocality via inequalities such as CHSH (Clauser‑Horne‑Shimony‑Holt) requires ensuring that classical (local hidden variable) models cannot explain the observations. Key practical considerations:

• Detection efficiency loophole: ensuring enough photons are detected so that fair sampling assumption is valid. Eberhard (1993) showed certain detection thresholds (~2/3) under which loophole‑free violation is possible given well‑chosen states.
  
  

• Locality (space‑like separation): measurement settings chosen sufficiently late so that no causal influence can propagate between detectors.
  
  

• Coincidence window and timing jitter: must be small relative to photon arrival spread and coherent time.
  
  

Statistical hypothesis testing:

• Null hypothesis H₀: outcomes generated by local realistic theory (e.g., S ≤ 2 in CHSH)
  
  

• Alternative H₁: quantum mechanics, entanglement, yielding S > 2
  
  

• Compute standard error via Poisson (or Binomial) approximations of counts
  
  

• Use Monte Carlo resampling/bootstrap to validate confidence intervals
  
  

• Report p‑values, number of standard deviations (σ) above classical bound
  
  

Gill (2014) offers rigorous treatment of statistical causality, post‑selection, missing data, and how to align experimental design with statistical inference in Bell‑test scenarios. (Gill, 2014)

4. Use‑Case: Defence Quantum Key Distribution (QKD) Link

4.1 Scenario Description

Suppose DND/CAF wishes to deploy a secure QKD link between two base installations, Base A and Base B, separated by 50 km fiber, or potentially 20 km free‑space (line of sight). The goal is to deliver symmetric key material at a secure key rate (SKR) sufficient for operational command and control messaging, e.g., on the order of 10² to 10³ bits per second, with quantum bit error rate (QBER) ≤ 5%, under adverse environmental conditions (daylight, temperature variation, vibration).

4.2 Design Parameters

• Source: Entangled photon pairs in the telecom C‑band (≈1550 nm) to use existing fiber infrastructure, or free‑space optical link with atmospheric compensation if needed.
  
  

• Source type: On‑demand quantum dot in a cavity‑enhanced, strain‑tuned configuration (targeting fidelity ≥ 95%, concurrence ≥ 90%) to minimize multi‑pair noise.
  
  

• Detectors: SNSPDs with detection efficiency ≥ 85%, timing jitter ≤ 50 ps, dark count rate ≤ 100 cps.
  
  

• Coincidence window τ_c ~100 ps to 1 ns depending on link dispersion and timing jitter.
  
  

• Tomography: measurement in at least 3 mutually unbiased bases, MLE or Bayesian reconstruction.
  
  

4.3 Expected Performance & Statistical Assurance

Assuming source brightness yields 10⁶ entangled pair emissions per second, coupling and losses (fiber + connectors, detector inefficiency) reduce detected pair rate to ~10⁴ per second at each end. With background/dark counts and accidental coincidences, true coincidence rate might be ~5000/s, accidental background ~200/s, giving a signal‑to‑noise ratio (SNR) of 25:1.

Under those conditions, expected CHSH S‑value might be ~2.6 ‑ 2.7. Performing repeated sampling over, say, 10⁴ coincidence counts yields standard error δS ~ O(0.01), enabling ≈ 50σ violation above classical bound (S = 2).

Secure key extraction: with QBER ≤ 5%, after error correction and privacy amplification, SKR might be in the hundreds of bits per second, sufficient for high‑priority message encryption.

4.4 Risks, Environmental & Adversarial Challenges

• Fiber attenuation (0.2‑0.35 dB/km) accumulates; connector/splice losses.
  
  

• In free‑space: atmospheric absorption, scattering, turbulence, pointing error.
  
  

• Background optical noise (sunlight, stray light), thermal noise in detectors.
  
  

• Timing drift, polarization drift (for polarization encoded schemes).
  
  

• Potential adversarial interference: injection of noise, jamming, spoofing, side‑channel attacks.
  
  

4.5 Robustness Measures

• Active stabilization of alignment, polarization, and phase.
  
  

• Narrow spectral filtering to reduce background.
  
  

• Redundant paths or backup classical encryption link in failover.
  
  

• Real‑time monitoring of QBER, visibility, singles/coincidence ratios to detect anomalies.
  
  

• Statistical thresholds for key acceptance: e.g., only accept keys if CHSH violation ≥ some predefined margin, QBER below threshold, and statistical confidence ≥ 5σ or p ≤ 10⁻⁷.
  
  

5. Statistical Frameworks & Standards for Defence Validation

5.1 Hypothesis Testing Model

Let A and B denote measurement settings on sides 1 and 2. Outcomes are ±1 (or equivalent). For four combinations (A, B), (A, B′), (A′, B), (A′, B′), record counts CijxyC_{ij}^{xy}, where i,j ∈ {+,−} under setting x,y. Then estimate correlation functions:

E(x,y)=C+++C−−−C+−−C−+C+++C−−+C+−+C−+E(x,y) = \frac{C_{++} + C_{--} - C_{+-} - C_{-+}}{C_{++} + C_{--} + C_{+-} + C_{-+}}

CHSH statistic:

S=E(A,B)+E(A,B′)+E(A′,B)−E(A′,B′)S = E(A,B) + E(A,B′) + E(A′,B) - E(A′,B′)

Null hypothesis: H₀: S ≤ 2 (local realism). Alternative: H₁: S > 2.

5.2 Error and Variance Estimation

Assuming Poisson statistics for photon counts, approximate variance of each E(x,y)E(x,y) via propagation of errors; variance of S is sum of variances of individual E terms (assuming independence or adjusting for correlated uncertainties).

Confidence interval on S: compute standard error δS and find number of σs that S exceeds 2. Alternatively, compute exact p‑value by considering the joint multinomial or Poisson model via Monte Carlo or bootstrap.

5.3 Loophole Metrics

Define:

• Detection efficiency η: fraction of entangled photons successfully detected
  
  

• Locality separation d and timing delays such that settings are space‑like separated.
  
  

Include measurement drift, dead time, and fair‑sampling assumptions in uncertainty model rather than assuming they are negligible.

5.4 Reporting Standards

For defence grading, each experimental run/report should include:

• Source parameters: brightness (pairs/s), multi‑pair emission probability, fidelity, concurrence.
  
  

• Detector stats: efficiency, jitter, dark count, dead time.
  
  

• Environmental conditions: temperature, propagation medium (fiber attenuation or free‑space conditions).
  
  

• Statistical metrics: counts, coincidence rates, accidental rates, E(x, y) values, S, standard errors, p‑values, confidence margins.
  
  

• Error rates: QBER, drift over time, stability metrics.
  
  

6. Recommendations & Benchmarks for Canadian Defence R&D

To ensure that Canada fields credible entanglement‑based quantum communication and sensing systems, the following benchmarks and investments are recommended.

Metric

Minimum Acceptable

Stretch Goal

Entangled photon detection rate at link ends

≥ 10³ true coincidences/s under realistic loss

≥ 10⁴ true coincidences/s

Fidelity to Bell state (Φ⁺ etc.)

≥ 90%

≥ 97%

CHSH S‑value

≥ 2.5 with ≥ 5σ significance

≥ 2.7 with ≥ 10σ significance

QBER

≤ 5%

≤ 2%

Detector efficiency

≥ 80%

≥ 90%

System logistics

Operable in field conditions (temp, vibration)

Integration into portable/ruggedized units

Investments should be made in:

• Quantum dot fabrication and packaging (strain tuning, cavity coupling)
  
  

• On‑chip photonic circuits, particularly telecom band sources compatible with existing optical fiber infrastructure
  
  

• High‑efficiency, low‑jitter single photon detectors (SNSPDs) and timing electronics
  
  

• Field testbeds (fiber spans, free‑space links, satellite or airborne platforms)
  
  

7. Conclusion

Photonic entanglement is no longer merely a laboratory curiosity; recent progress has brought on‑demand sources, integrated photonic platforms, and high‑fidelity performance to levels that make field deployment viable. For Canadian defence interests, building secure quantum communication links (QKD), distributed sensing, and eventual quantum networks depends critically on mastering entanglement generation, photon measurement, and statistical validation under real‑world conditions.

The proposed use case shows that with currently achievable parameters, a 50 km fiber or 20 km free‑space link providing secure key rates of hundreds of bits per second is feasible with strong statistical assurance. To ensure this becomes reality, policy, funding, standards, and industrial‐academic partnerships must align.

References

Atzeni, S., Rab, A. S., Corrielli, G., Polino, E., Valeri, M., & Spagnolo, N. (2017). Integrated sources of entangled photons at telecom wavelength in femtosecond‑laser‑written circuits. arXiv. https://arxiv.org/abs/1710.09618

“Entangled Photon‑Pair Sources Based on Three‑Wave Mixing in Bulk Crystals” (2022). Review Article. PubMed.

Feng, J., et al. (2024). Polarization‑entangled photon‑pair source with van der Waals 3R‑WS₂ crystal. eLight, 4, Article 13. https://doi.org/10.1186/s43593‑024‑00074‑6

Gill, R. D. (2014). Statistics, causality and Bell’s theorem. Statistical Science, 29(4), 512–528. https://doi.org/10.1214/14‑STS490

Grassani, D., Azzini, S., Liscidini, M., Galli, M., Strain, M. J., Sorel, M., Sipe, J. E., & Bajoni, D. (2014). A micrometer‑scale integrated silicon source of time‑energy entangled photons. arXiv. https://arxiv.org/abs/1409.4881

Li, Y.‑H., Zhou, Z.‑Y., Xu, Z.‑H., Shi, B.‑S., & Guo, G.‑C. (2016). Multiplexed entangled photon‑pair sources for all‑fiber quantum networks. Physical Review A, 94(4), 043810. https://doi.org/10.1103/PhysRevA.94.043810

“Quantum Key Distribution with Entangled Photon Sources” (Ma, X., Fung, C.‑H., & Lo, H.‑K.) (2007). Physical Review A, 76(1), 012307. https://doi.org/10.1103/PhysRevA.76.012307

Zeuner, K. D., Jöns, K. D., Schweickert, L., Lettner, T., Nunez Lobato, C., Gyger, S., Zwiller, V., & colleagues. (2021). On‑demand generation of entangled photon pairs in the telecom C‑band with InAs quantum dots. ACS Photonics, 8(8), 2337–2344. https://doi.org/10.1021/acsphotonics.1c00504

“A Source of Entangled Photons Based on a Cavity‑Enhanced and Strain‑Tuned GaAs Quantum Dot” (2024). eLight, 4, Article 13.

Li, Y.‑H., Zhou, Z.‑Y., Xu, Z.‑H., Shi, B.‑S., & Guo, G.‑C. (2016). Multiplexed entangled photon‑pair sources for all‑fiber quantum networks. Physical Review A, 94(4), 043810.