Superdense Coding

A qubit sent down a wire can carry, on its own, exactly one bit of classical information. That is a theorem, not a limitation of the hardware. Superdense coding does not break it — it changes what “on its own” means.

Alice and Bob share an entangled pair, prepared long before Alice knows what she wants to say. When she does know, she applies one of four operations to her half — nothing, X, Z, or both — and posts that single qubit. Bob undoes the entanglement and reads two bits, with no uncertainty at all. Step through it and watch which bar is standing at the end.

Watch the Alice’s |r| readout as you switch between the four messages. It stays at 0.000. Her qubit’s Bloch arrow has no length whatever she sends, which means the qubit crossing the channel is statistically identical in all four cases — that is why the packet in the drawing is a featureless dot rather than four different-looking ones.

The message lives in how her qubit is correlated with Bob’s, and Bob’s never moved. Intercept the qubit in flight and you learn nothing; you need both halves, which is the same reason this cannot be eavesdropped and the reason it needs the pair.

With the pair shared, every overlap off the diagonal is 0.000: the four states are mutually orthogonal, and orthogonal states can be told apart with certainty. That single fact is the whole protocol. Turn the pair off and two of those zeros become 1.000 — two of the four messages become literally the same state, and no measurement in the universe separates them.

Untick Share an entangled pair first. Alice applies exactly the same gates, sends exactly the same qubit, and Bob runs exactly the same decoder. The message now comes back 50.0% of the time instead of 100.0%.

The bit that survives is the one Alice wrote with X. The one she wrote with Z is gone, because Z on an un-entangled |0⟩ does nothing at all — it changes a phase that no measurement on that qubit alone can see. Entanglement is what turns that invisible phase into a bit Bob can read, and that is the concrete sense in which it is a resource: it is consumed, it cannot be copied, and one pair buys exactly one extra bit.

Put the two protocols side by side and they are one idea. Teleportation spends one entangled pair plus two classical bits to move one qubit. Superdense coding spends one entangled pair plus one qubit to move two classical bits.

Same ledger, read in opposite directions — and in both cases the pair has to be there first, shared in advance, and is used up in the process. Neither protocol sends anything faster than light, and neither works without the classical or quantum message that actually crosses the gap.

The circuit simulator has this as the preset Superdense Coding (message 11). It is only six gates: H, CX, then Alice’s Z and X, then CX and H again. Change the middle two and the answer changes with them, every shot, with no spread at all.