Superdense coding
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.
[1][2] This protocol was first proposed by Charles H. Bennett and Stephen Wiesner in 1970[3] (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.
[2] 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.
[2] 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.
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.
The necessity of having both qubits to decode the information being sent eliminates the risk of eavesdroppers intercepting messages.
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 can be split into five different steps: preparation, sharing, encoding, sending, and decoding.
The protocol starts with the preparation of an entangled state, which is later shared between Alice and Bob.
By applying a quantum gate to her qubit locally, Alice can transform the entangled state
Let's now describe which operations Alice needs to perform on her entangled qubit, depending on which classical two-bit message she wants to send to Bob.
There are four cases, which correspond to the four possible two-bit strings that Alice may want to send.
If Alice wants to send the classical two-bit string 00 to Bob, then she applies the identity quantum gate,
If Alice wants to send the classical two-bit string 01 to Bob, then she applies the quantum NOT (or bit-flip) gate,
If Alice wants to send the classical two-bit string 10 to Bob, then she applies the quantum phase-flip gate
If, instead, Alice wants to send the classical two-bit string 11 to Bob, then she applies the quantum gate
After having performed one of the operations described above, Alice can send her entangled qubit to Bob using a quantum network through some conventional physical medium.
These operations performed by Bob can be seen as a measurement which projects the entangled state onto one of the four two-qubit basis vectors
Now, the Hadamard gate is applied only to A, to obtain For simplicity, subscripts may be removed: Now, Bob has the basis state
General dense coding schemes can be formulated in the language used to describe quantum channels.
Alice and Bob share a maximally entangled state ω.
Alice then sends her subsystem to Bob, who performs a measurement on the combined system to recover the message.
The protocol of superdense coding has been actualized in several experiments using different systems to varying levels of channel capacity and fidelities.
In 2004, trapped beryllium-9 ions were used in a maximally entangled state to achieve a channel capacity of 1.16 with a fidelity of 0.85.
[6] High-dimensional ququarts (states formed in photon pairs by non-degenerate spontaneous parametric down-conversion) were used to reach a channel capacity of 2.09 (with a limit of 2.32) with a fidelity of 0.98.
