Superdense Coding Concept

Classical communications send information strictly one-to-one. To send two bits of data, such as 00 or 11, you physically transmit two electrical pulses or radio waves. Superdense coding challenges this constraint by allowing transmission of two classical bits of information by sending only a single qubit.

This protocol serves as a primary example of how entanglement acts as a computational resource. While a single qubit can only collapse to 0 or 1 upon measurement, the shared state between two parties creates a larger mathematical structure. By manipulating this shared structure, we can encode more information than the individual particle seems to hold.

The Role of Entanglement

The superdense coding protocol involves two parties, typically named Alice and Bob. Before any communication occurs, they must share an entangled pair of qubits. This is often prepared by a third party or by one of the users and distributed ahead of time.

The pair is usually in the Bell state $|\Phi^+\rangle$:

$|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}$

In this state, Alice holds qubit $A$ and Bob holds qubit $B$. Neither qubit has a defined value on its own, but their correlation is perfect. If Alice measures hers, she instantly knows what Bob will measure. However, superdense coding does not use measurement immediately. Instead, Alice manipulates her qubit locally to change the global state of the system.

Encoding Information

Alice wants to send a two-bit message to Bob: 00, 01, 10, or 11. Because she has access to qubit $A$, she can apply single-qubit quantum gates to it. Even though she is physically separated from Bob, applying a gate to her qubit transforms the specific Bell state they share into a different, orthogonal Bell state.

Here is how Alice encodes her two classical bits into the quantum system:

1. Message 00: Alice applies the Identity gate ($I$). She effectively does nothing. The state remains $|\Phi^+\rangle$.
2. Message 01: Alice applies the Pauli-Z gate ($Z$). This flips the relative phase. The state becomes $|\Phi^-\rangle$.
3. Message 10: Alice applies the Pauli-X gate ($X$). This flips the bit value. The state becomes $|\Psi^+\rangle$.
4. Message 11: Alice applies the Pauli-Z gate followed by the Pauli-X gate (equivalent to $iY$). This flips both phase and bit. The state becomes $|\Psi^-\rangle$.

Notice that Alice only interacts with her own qubit. She does not touch Bob's qubit. Yet, the mathematical description of the pair $|\psi\rangle_{AB}$ changes completely based on her local action.

The workflow of superdense coding. Alice manipulates her half of an entangled pair and sends it to Bob, who then possesses the entire system to decode the message.

Transmission and Decoding

Once Alice has applied the gate corresponding to her message, she sends her qubit to Bob. This is the only physical transmission in the protocol.

Bob now holds both qubit $A$ (received from Alice) and qubit $B$ (which he held from the start). While Alice encoded the data using single-qubit gates, the information is now stored in the relationship between the two qubits. To access it, Bob must disentangle them.

Bob performs a Bell State Measurement, which is essentially the reverse of the circuit used to create entanglement:

1. CNOT Gate: Bob uses qubit $A$ as the control and qubit $B$ as the target.
2. Hadamard Gate: Bob applies a Hadamard gate to qubit $A$.

This sequence maps the four entangled Bell states back onto the four standard computational basis states.

$$\begin{aligned} |\Phi^+\rangle & \xrightarrow{\text{Decode}} |00\rangle \\ |\Phi^-\rangle & \xrightarrow{\text{Decode}} |01\rangle \\ |\Psi^+\rangle & \xrightarrow{\text{Decode}} |10\rangle \\ |\Psi^-\rangle & \xrightarrow{\text{Decode}} |11\rangle \end{aligned}$$

When Bob measures the two qubits, the result corresponds exactly to the two classical bits Alice intended to send.

The Physics of the Gain

It is important to understand why this does not violate physics. We are not compressing two bits into one qubit in isolation. The protocol consumes one pair of entangled qubits to transmit two classical bits.

• Classical Resource: 1 physical qubit transmitted.
• Quantum Resource: 1 entangled pair consumed.

If Alice and Bob did not share entanglement, Alice would have to send two physical qubits to convey two bits of information (using standard dense coding limits). Superdense coding trades previously established quantum correlation for communication bandwidth. This trade-off is a recurring theme in quantum algorithms: we expend entanglement to perform tasks that classical systems find inefficient or impossible.