Measurement and messages

Question

How does a limited polarization measurement model predict errors in a transmitted message?

The rectilinear basis uses 0° for bit 0 and 90° for bit 1. The diagonal basis uses 45° for bit 0 and 135° for bit 1. Sender and receiver independently choose either basis with probability 1/2. Matching bases give the same bit. Different bases give equally likely outcomes, with probability 1/2 of bit disagreement. The final receiver detects the photon; no resending role is included.

Model and variables

B is the event that receiver and sender use different bases. P(B) = 1/2. Given B, the error probability is 1/2. Given matching bases, it is 0. I model ideal single photon states, ideal analyzers, no detector dark counts, no loss, no eavesdropper strategy, and comparison across all trials before separately evaluating retained matching basis positions.

Derivation

Analytical result

Numerical example. For a 20 trial illustration, the expected counts are 10 matching bases, 10 different bases and 5 disagreements across all trials. These counts are not guaranteed in a random sample. The matching ten should agree in the ideal model. Of the different ten, about five agree and five disagree on average. If the protocol checks only matching positions, the expected observed error is 0%. If it checks all positions against the original encoding, five of twenty are wrong, or 25%. The distinction is a protocol definition, not a contradiction.

Reproducible method

I calculated expected disagreement by weighting each conditional probability by its basis probability. The reproducible simulation below accompanies that analytical result, with separate denominators for all trials and retained matching bases. It is a computational illustration, not a claim of physical photon measurements or a prior simulation campaign. The final receiver detects; it does not resend.

Reading the plot

The cumulative plot shows the disagreement fraction across all simulated trials alongside its 25% expectation. Fluctuations typically decrease as the sample grows, but a particular run need not produce the expected count. The table and counters use the same generated trial record.

Computational illustration

One reproducible run

Seed 37; 400 trials. Deterministic LCG: state = (1664525 × state + 1013904223) modulo 2³². Each draw below 0.5 selects one of two equally likely outcomes. Mismatched bases receive a fair random bit; the final receiver detects and does not resend.

All trial disagreements

100 / 400 (25.00%)

Retained matching bases

197 / 400 (49.25%)

Retained errors

0 / 197 (0.00%)

Expectations: 25% disagreement across all trials, 50% retained, 0% errors among retained matching bases in this ideal baseline. Expected counts are not guaranteed counts in one run.

This simulation compares every bit internally for evaluation. Real key distribution does not publicly disclose the entire retained secret key. Basis comparison and testing selected retained bits are separate steps. Security needs further protocol assumptions and authenticated classical communication.

Uncertainty and limits

This is a teaching model, not a complete security proof or a claim about a physical cryptographic implementation. Losses, detector efficiency, basis leakage, finite sample effects, and an adversary’s strategy are omitted. A zero error rate in a small sample does not prove the channel is secure.

Further question

Conditioning and finite samples

The denominator changes when matching bases are retained. A finite sample can differ from its expected count even when every trial follows the ideal model.

Next question

How many trials are needed before an observed error rate becomes informative? Connect the result to binomial uncertainty and finite sample reasoning.