---
title: "Dense coding: three messages and a resource ledger"
description: "What did the 1996 three-message experiment demonstrate beyond an ideal four-state protocol?"
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[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# Dense coding: three messages and a resource ledger

Sandbox Physics · 2026-10-18

What did the 1996 three-message experiment demonstrate beyond an ideal four-state protocol?

## Count the messages actually resolved

Mattle and colleagues’ 1996 demonstration used polarization-entangled photons, local encoding on one photon and a joint optical analyzer. It transmitted a three-symbol alphabet. Its selected character examples used five trits per character, rather than demonstrating four reliably resolved Bell messages. Five trits represent 243 words, fewer than the 256 possible eight-bit patterns; the illustrated characters fit, but this is not a universal fixed-length conversion of every eight-bit value.

The distinction matters because a theoretical encoding space can exceed an analyzer’s resolved outcomes. The original detector arrangement distinguished two Bell states individually while grouping the other two. An additional splitting arrangement registered a third category with a lower detection rate.

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 require a shared pair

In an ideal teaching protocol, start with the positive Bell superposition of both-horizontal and both-vertical polarization. Applying identity, bit flip, phase flip, or both to one member produces four mutually orthogonal Bell states, up to irrelevant global phases.

The sender must then physically send that member to the receiver, who already holds the other member and performs a joint measurement. The receiver’s local photon alone is maximally mixed for every encoding. Pre-shared entanglement and a transmitted qubit are both resources; neither can be omitted from the claimed communication advantage.

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 abstract K. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger (1996), Dense Coding in Experimental Quantum Communication, PRL 76, 4656–4659 — Illinois copy

## Alphabet size is not delivered information

For an ideal error-free receiver, three equally likely messages carry the base-two logarithm of three bits per accepted symbol, about 1.585. Four perfectly resolved, equally likely messages carry two bits. Unequal use of the letters reduces the input entropy.

The general quantity below is mutual information between the chosen message and registered output, including the declared probability distribution. Its logarithm uses base two and omits zero-probability terms. A nonzero alphabet size does not guarantee that the detector output identifies the message.

$$
I(M;D)=\sum_{m,d}p(m,d)\log_2\!\frac{p(m,d)}{p(m)p(d)}
$$

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 abstract K. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger (1996), Dense Coding in Experimental Quantum Communication, PRL 76, 4656–4659 — Illinois copy

## Try two different channel ledgers

For our own three-letter symmetric channel, let the correct output occur with probability 0.90 and either wrong letter with probability 0.05. With uniform inputs the output remains uniform, but mutual information is only about 1.016 bits per trial. Label count alone would have overstated the delivered information.

In a different invented channel, detection is perfect when it succeeds but has a message-independent erasure probability of one half. Keeping erasure as an output gives about 0.792 bits per attempted symbol, while the accepted subset still carries 1.585 bits per accepted symbol. These are different denominators; the examples are not reconstructions of the 1996 apparatus.

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

## Normalized traces can hide unequal acceptance

The historical character plot normalized each registered category to its own maximum. That makes the message sequence legible, but it is not an equal-efficiency count ledger. The paper also explains the reduced registration probability of its added third-category detector configuration.

To evaluate an actual channel, keep attempted encodings, all registered outputs, erasures and acquisition time. When acceptance depends on the message, even the accepted input distribution changes. A confusion matrix and a resource budget answer questions that three clean normalized peaks cannot settle.

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

## Use the ingredients without merging the claims

Dense coding does not send a chosen message by entanglement alone. The travelling member and joint readout make the message accessible. Its advantage compares specified communication resources; counting every photon used to distribute, stabilize and detect a pair gives a broader engineering budget.

The qubit and Hong–Ou–Mandel Labs let readers inspect local operations and two-photon interference separately. They do not implement this dense-coding channel or reproduce its measured efficiency. Read the three-message result as a concrete optical demonstration with a declared analyzer, then compare it with the ideal four-state resource contract.

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 abstract K. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger (1996), Dense Coding in Experimental Quantum Communication, PRL 76, 4656–4659 — Illinois copy

## Continue with the models

These are existing teaching Labs. Their simulated records are separate from the historical observations discussed above.

- [Inspect single-qubit rotations](https://sandboxphysics.com/labs/qubit-state/)
- [Compare the analyzer’s two-photon interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

## Sources and reading notes

We read Mattle et al.’s four-page 1996 PRL, especially Table I and Figures 4–5, from an Illinois-hosted copy. Bennett and Wiesner’s 1992 proposal is cited through the IBM author record and abstract; its full text was not accessible here. The orthogonality, entropy and erasure examples are independent teaching calculations.

1. [K. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger (1996), Dense Coding in Experimental Quantum Communication, PRL 76, 4656–4659 — Illinois copy](https://courses.grainger.illinois.edu/phys513/sp2019/reading/week9/Dense_coding_Zeilinger1996.pdf)
2. [C. H. Bennett and S. J. Wiesner (1992), Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states — IBM author record and abstract](https://research.ibm.com/publications/communication-via-one-and-two-particle-operators-on-einstein-podolsky-rosen-states)

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