Count the messages the receiver could tell apart
In Mattle and colleagues’ 1996 experiment, the sender chose one of three messages by changing the polarization of one photon. The receiver read the message by measuring that photon together with a partner already held at the receiving station. The pair had been entangled in advance. This shared pair let the experiment go beyond the two alternatives available from an isolated polarization qubit.
The demonstration used three reliably distinguishable message categories. The initial optical arrangement identified two Bell states separately and grouped the other two together; extra beam splitting detected a third category with lower efficiency. The textbook protocol allows four distinct Bell messages, so the receiver’s optical design determined how much of that alphabet the experiment could use.
The example text used five trits per character. A trit has three possible values, so five trits give 243 code words. That was enough for the displayed characters, but falls short of all 256 eight-bit patterns; it was not a general fixed-length replacement for every eight-bit value.
Read the evidence: K. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger (1996), Dense Coding in Experimental Quantum Communication, PRL 76, 4656–4659 — Illinois copy
Four encodings need a shared entangled pair
In the ideal protocol, the shared pair is a coherent superposition of both photons horizontal and both vertical. A superposition keeps a relative phase between its alternatives. The sender can do nothing, flip horizontal and vertical, reverse that relative phase, or do both. These four operations produce four orthogonal Bell states: four pair states that an ideal joint measurement can distinguish perfectly.
The sender must then physically send that photon to the receiver, who already holds the other one and measures the two jointly. On its own, the receiver’s photon is maximally mixed whatever the encoding. The advantage needs both resources, the entanglement shared in advance and the qubit that travels, and neither can be left out of the count.
Read the evidence: C. H. Bennett and S. J. Wiesner (1992), Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states — IBM author record and abstractK. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger (1996), Dense Coding in Experimental Quantum Communication, PRL 76, 4656–4659 — Illinois copy
The number of letters does not measure the information delivered
With an ideal error-free receiver, three equally likely messages carry the base-two logarithm of three, about 1.585 bits, per accepted symbol. Four perfectly resolved, equally likely messages carry two bits. Using some letters more often than others lowers the input entropy.
Mutual information measures how much a detected output tells the receiver about the chosen message. In the expression, M is the message and D the output; the probability with both labels is their joint probability, and the two single-label probabilities describe each separately. The logarithm is base two, and zero-probability terms are omitted. The input probabilities must be specified, because the information delivered depends on which messages are sent and how often.
Take a symmetric three-letter channel of our own design: the correct letter comes out with probability 0.90 and each wrong letter with probability 0.05. With uniform inputs the output stays uniform, yet the mutual information is only about 1.016 bits per trial. Counting letters alone would overstate the information delivered.
In a second invented channel, every detection that succeeds is correct, but each symbol is erased with probability one half, whatever the message. Counting erasure as an output gives about 0.792 bits per symbol sent, while the accepted symbols still carry 1.585 bits each. The two numbers divide by different totals. Neither example reconstructs the 1996 apparatus.
Read the evidence: C. H. Bennett and S. J. Wiesner (1992), Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states — IBM author record and abstractK. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger (1996), Dense Coding in Experimental Quantum Communication, PRL 76, 4656–4659 — Illinois copy
Normalized plots can hide unequal detection rates
The paper’s character plot scales each detected category to its own maximum. That makes the message sequence easy to read, but the peak heights are not counts at equal efficiency. The paper also explains why the added detector arrangement for the third category registered events with lower probability.
To judge a real channel, keep every attempted encoding, every detected output, the erasures and the acquisition time. If the chance of acceptance depends on the message, even the input distribution among accepted symbols changes. A confusion matrix and a full count of resources answer questions that three clean normalized peaks cannot.
Entanglement alone cannot send a chosen message. The message becomes readable only when the encoded photon travels to the receiver and the pair is measured jointly. The advantage is defined against specific communication resources; counting every photon used to distribute, stabilize and detect a pair gives a broader engineering cost.
A known trial clock lets the receiver recognize an erasure, a trial with no registered message. If different messages are erased at different rates, that silence can itself carry information. Evaluating a channel therefore means keeping the complete output categories and the resources used, alongside the detected symbols. The examples give information per attempt for the specified ideal channels; a historical transmission rate and channel capacity require their own measurement or optimization.
Read the evidence: K. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger (1996), Dense Coding in Experimental Quantum Communication, PRL 76, 4656–4659 — Illinois copyC. H. Bennett and S. J. Wiesner (1992), Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states — IBM author record and abstract