Superdense coding

In quantum information theory, superdense coding is a technique used to send two bits of classical information using only one qubit. It is the inverse of quantum teleportation, which sends one qubit with two classical bits. Both superdense coding and quantum teleportation require, and use up, entanglement between the sender and receiver in the form of Bell pairs.

Suppose Alice would like to send classical information to Bob using qubits, instead of classical bits. Alice would encode the classical information in a qubit and send it to Bob. After receiving the qubit, Bob recovers the classical information via measurement. The question is: how much classical information can be transmitted per qubit? Since non-orthogonal quantum states cannot be distinguished reliably, one would guess that Alice can do no better than one classical bit per qubit. Holevo's theorem discusses this bound on efficiency. Thus, there is no advantage gained in using qubits instead of classical bits. However, with the additional assumption that Alice and Bob share an entangled state, two classical bits per qubit can be achieved. The term superdense refers to this doubling of efficiency. Also, it can be proved that the maximum amount of classical information that can be sent (even while using entangled state) using one qubit is two bits.

Crucial to this procedure is the shared entangled state between Alice and Bob, and the property of entangled states that a (maximally) entangled state can be transformed into another state via local manipulation.

Suppose parts of a Bell state, say

are distributed to Alice and Bob. The first subsystem, denoted by subscript A, belongs to Alice and the second, B, system to Bob. By only manipulating her particle locally, Alice can transform the composite system into any one of the Bell states (entanglement cannot be broken using local operations):

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