In quantum information theory, superdense coding (also referred to as dense coding) is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the assumption of sender and receiver pre-sharing an entangled resource. In its simplest form, the protocol involves two parties, often referred to as Alice and Bob in this context, which share a pair of maximally entangled qubits, and allows Alice to transmit two bits (i.e., one of 00, 01, 10 or 11) to Bob by sending only one qubit. This protocol was first proposed by Charles H. Bennett and Stephen Wiesner in 1970 (though not published by them until 1992) and experimentally actualized in 1996 by Klaus Mattle, Harald Weinfurter, Paul G. Kwiat and Anton Zeilinger using entangled photon pairs. Superdense coding can be thought of as the opposite of quantum teleportation, in which one transfers one qubit from Alice to Bob by communicating two classical bits, as long as Alice and Bob have a pre-shared Bell pair. The transmission of two bits via a single qubit is made possible by the fact that Alice can choose among four quantum gate operations to perform on her share of the entangled state. Alice determines which operation to perform accordingly to the pair of bits she wants to transmit. She then sends Bob the qubit state evolved through the chosen gate. Said qubit thus encodes information about the two bits Alice used to select the operation, and this information can be retrieved by Bob thanks to pre-shared entanglement between them. After receiving Alice's qubit, operating on the pair and measuring both, Bob obtains two classical bits of information. It is worth stressing that if Alice and Bob do not pre-share entanglement, then the superdense protocol is impossible, as this would violate Holevo's theorem. Superdense coding is the underlying principle of secure quantum secret coding. The necessity of having both qubits to decode the information being sent eliminates the risk of eavesdroppers intercepting messages.
Overview
Suppose Alice wants to send two classical bits of information (00, 01, 10, or 11) to Bob using qubits (instead of classical bits). To do this, an entangled state (e.g. a Bell state) is prepared using a Bell circuit or gate by Charlie, a third person. Charlie then sends one of these qubits (in the Bell state) to Alice and the other to Bob. Once Alice obtains her qubit in the entangled state, she applies a certain quantum gate to her qubit depending on which two-bit message (00, 01, 10 or 11) she wants to send to Bob. Her entangled qubit is then sent to Bob who, after applying the appropriate quantum gate and making a measurement, can retrieve the classical two-bit message. Observe that Alice does not need to communicate to Bob which gate to apply in order to obtain the correct classical bits from his projective measurement.
The protocol The protocol can be split into five different steps: preparation, sharing, encoding, sending, and decoding.
Preparation The protocol starts with the preparation of an entangled state, which is later shared between Alice and Bob. For example, the following Bell state
| Φ + ⟩ = 1 2 ( | 0 ⟩ A ⊗ | 0 ⟩ B + | 1 ⟩ A ⊗ | 1 ⟩ B ) {\displaystyle |\Phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B})}
is prepared, where ⊗ {\displaystyle \otimes } denotes the tensor product. In common usage the tensor product symbol ⊗ {\displaystyle \otimes } may be omitted:
| Φ + ⟩ = 1 2 ( | 0 A 0 B ⟩ + | 1 A 1 B ⟩ ) {\displaystyle |\Phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0_{A}0_{B}\rangle +|1_{A}1_{B}\rangle )} .
Sharing After the preparation of the Bell state | Φ + ⟩ {\displaystyle |\Phi ^{+}\rangle } , the qubit denoted by subscript A is sent to Alice and the qubit denoted by subscript B is sent to Bob. Alice and Bob may be in different locations, an unlimited distance from each other. There may be an arbitrary period between the preparation and sharing of the entangled state | Φ + ⟩ {\displaystyle |\Phi ^{+}\rangle } and the rest of the steps in the procedure.
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