PPM-4 Error Probability — Classical Photon Counting vs Quantum SRM

This project studies symbol-error probability (P_e) for 4-slot pulse-position modulation (PPM-4) under two receivers:

• Classical photon counting (choose the slot with the largest photon count)
• Quantum square-root measurement (SRM) (the optimal symmetric POVM for these states)

The implementation and plots are in Final_Code.ipynb, and the derivations/background are in Report.pdf.

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What we did (method in brief)

• Channel & symbols. One “on” slot (signal) and three “off” slots per 4-PPM symbol; background light modeled as displaced-thermal noise with mean photons/slot (N). Signal energy (N_s) is the mean photons in the on slot.
• Classical receiver. Counts in each slot are modeled as independent Poisson RVs: on-slot (\mathrm{Pois}(N_s+N)), off-slots (\mathrm{Pois}(N)). The correct decision occurs when the on-slot count exceeds all three off-slot counts. We compute the exact (P_e) from these order statistics.
• Quantum SRM receiver. We construct the 4 symmetric signal states and apply the square-root measurement (SRM), which maximizes average correct detection for symmetric ensembles. Practically, we build the Gram matrix of state overlaps and evaluate (P_e = 1-P_c) from the SRM POVM elements.
• Sweep. We sweep (N_s) over a practical range (photons) at several background levels (N) and plot (P_e) vs (N_s) (log scale).

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Results (figures)

1) Classical photon-counting receiver

Figure 1. Error probability \(P_e\) for the classical receiver vs signal energy \(N_s\), with multiple curves for different background levels \(N\). As expected, higher \(N_s\) lowers \(P_e\); more background \(N\) shifts the curves upward. At very low \(N_s\), the error approaches the random-guessing limit (3/4 for 4-PPM).

2) Quantum SRM receiver

Figure 2. \(P_e\) for the quantum SRM. The SRM uniformly outperforms photon counting: for the same \(N\) and \(N_s\), the SRM curve sits markedly lower, indicating higher detection reliability especially in the low-photon regime.

3) Side-by-side comparison

Figure 3. Overlay of classical vs SRM curves. The **gap** between the black/gray (SRM) and green (classical) curves quantifies the quantum advantage. Across the sweep, SRM achieves **orders-of-magnitude** lower error probability—often \(10^2\!-\!10^3\)× better—most prominently at moderate \(N_s\) and non-zero background.

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Key takeaways

• SRM dominates photon counting for 4-PPM across practical signal and background levels.
• The advantage is largest where classical counting struggles: low-to-moderate (N_s) and non-negligible background (N).
• As (N_s) grows large, both receivers approach very low error, but SRM reaches that regime sooner (lower (N_s) for the same (P_e)).

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Files in this repo

• Final_Code.ipynb — end-to-end notebook to compute curves and generate figures.
• Report.pdf — project report with derivations and references.
• plot_classical.png, Plot_SRM.png, plot_output.png — the figures described above.