Quantum Superdense Coding is an attractive quantum communication protocol that permits two parties, regularly referred to as Alice and Bob, to transmit two classical bits of information by sending one quantum bit (qubit), provided they share a special entangled pair of qubits early. This is a counter-intuitive result compared to classical information theory, where one bit can only consistently transmit one bit of information. Superdense coding highlights the power of quantum entanglement as a resource for communication.
Superdense coding was introduced by Charles Bennett and Stephen Wiesner in 1992. It is regularly defined as the reverse process of quantum teleportation, where one qubit of quantum information is transmitted using two classical bits and a shared entangled pair. Both protocols fundamentally rely on the non-local correlations provided by quantum entanglement.
Quantum superdense coding is an amazing protocol that showcases the power of quantum entanglement for improving communication efficiency. By pre-sharing an entangled pair of qubits, two parties can use a single qubit transmission to convey two classical bits of information, highlighting a fundamental difference between classical and quantum information processing. The protocol includes entanglement preparation, encoding classical information using Pauli gates, sending the qubit, and decoding the information through a Bell state measurement. While practical implementations face challenges, superdense coding remains a cornerstone of quantum communication theory and establishes the potential of quantum mechanics to reform information transfer.
How Does Superdense Coding Work?
1. Entanglement Preparation and Distribution:
- The protocol starts with Alice and Bob sharing a pair of entangled qubits, naturally in one of the Bell states. The most common Bell state used for superdense coding is the |β₀₀⟩ state, which is a maximally entangled state represented as:
|β₀₀⟩ = (|00⟩ + |11⟩) / √2
This state indicates that the two qubits are associated such that if one is measured to be |0⟩, the other will also be |0⟩, and if one is |1⟩, the other will also be |1⟩, with equal probability.
- Alice takes control of one qubit of the entangled pair, and Bob holds the other. This sharing of the entangled resource must occur before Alice wants to send the classical information.
2. Encoding Classical Information:
- Alice requests to send two classical bits of information to Bob. There are four possible two-bit messages: 00, 01, 10, and 11. To encode these messages, Alice applies one of the four Pauli operations to her qubit of the entangled pair:
- To send the classical bits “00”, Alice applies the Identity (I) gate (or does nothing) to her qubit. This leaves the shared state unchanged:
- (I ⊗ I) |β₀₀⟩ = (|00⟩ + |11⟩) / √2
- To send the classical bits “01”, Alice applies the Pauli-X (σₓ) gate to her qubit. This flips the state of her qubit:
- (σₓ ⊗ I) |β₀₀⟩ = (|10⟩ + |01⟩) / √2
- To send the classical bits “10,” Alice applies the Pauli-Z gate to her qubit. This introduces a phase flip if her qubit is in the |1⟩ state:
- (σ<0xE2><0x82><0x9A> ⊗ I) |β₀₀⟩ = (|00⟩ – |11⟩) / √2
- To send the classical bits “11,” Alice applies both the Pauli-Z and Pauli-X gates (or the Pauli-Y gate) to her qubit:
- (σ<0xE2><0x82><0x9A> σₓ ⊗ I) |β₀₀⟩ = (iσ<0xE2><0x82><0x97> ⊗ I) |β₀₀⟩ = (|10⟩ – |01⟩) / √2
- Therefore, by applying one of these four quantum gates to her qubit, Alice encodes two bits of classical information into the joint entangled state.
3. Sending the Qubit:
- After applying the suitable Pauli operation, Alice sends her qubit to Bob through a quantum communication channel. This transmission contains physically transferring the qubit while preserving its quantum state.
4. Decoding the Information:
- Once Bob receives the qubit from Alice, he now possesses both qubits of the original entangled pair, which are in one of the four possible Bell states depending on the classical information Alice intended to send.
- To decode the two classical bits, Bob performs a Bell state measurement on the two qubits he holds. A Bell state measurement is a joint measurement that can distinguish between the four orthogonal Bell states:
- |β₀₀⟩ = (|00⟩ + |11⟩) / √2 (Corresponds to “00”)
- |β₀₁⟩ = (|01⟩ + |10⟩) / √2 (Corresponds to “01”)
- |β₁₀⟩ = (|00⟩ – |11⟩) / √2 (Corresponds to “10”)
- |β₁₁⟩ = (|01⟩ – |10⟩) / √2 (Corresponds to “11”)
- The outcome of this Bell state measurement directly reveals the two classical bits that Alice intended to send.
Quantum Superdense Coding and Quantum Teleportation
The differences between quantum superdense coding and quantum teleportation are in what is being transmitted and the resources used for the transmission, despite both depending on pre-shared entanglement:
- Superdense coding uses a pre-shared entangled qubit to send more classical information (2 bits) than a single qubit could normally carry.
- Quantum teleportation uses a pre-shared entangled qubit and classical communication (2 bits) to send quantum information (1 qubit) without physically transmitting it in its original form.
Quantum Superdense Coding
- Goal: To transmit two classical bits of information.
- Transmission Medium: One qubit is sent from the sender (Alice) to the receiver (Bob).
- Resource Requirement: Requires a pre-shared entangled pair of qubits between Alice and Bob. This is a maximally entangled Bell state like (|00⟩ + |11⟩) / √2.
- Mechanism: Alice encodes her two classical bits by applying one of the four Pauli operations to her half of the entangled pair. She then sends this qubit to Bob, who executes a Bell state measurement on the two qubits in his control to decode the two classical bits.
- Efficiency: Reaches a transmission of two classical bits per qubit sent, which is more than what is possible with only classical means. This efficiency is dependent on the shared state being excellently entangled. If the entanglement is zero, the transmission capacity reduces to one bit per qubit.
- Analogy: Repeatedly considered the reverse of quantum teleportation.
Quantum Teleportation
- Goal: To transmit an unknown quantum state (a qubit) from the sender (Alice) to the receiver (Bob).
- Transmission Medium: Two classical bits of information are sent from Alice to Bob. No qubit carrying the unknown quantum state is physically sent between them in the final stage.
- Resource Requirement: Requires a pre-shared entangled pair of qubits between Alice and Bob. Again, a Bell state like (|00⟩ + |11⟩) / √2 is typically used.
- Mechanism: Alice interrelates the unknown qubit she wants to send with her half of the entangled pair and performs a Bell measurement on these two qubits. The result of this measurement (two classical bits) is then sent to Bob. Based on these two classical bits, Bob applies one of the Pauli operations to his half of the entangled pair, which transforms it into the original unknown quantum state.
- Efficiency: Transmits one qubit of quantum information using two classical bits and a shared entangled pair.
- Significance: Establishes how entanglement acts as a fundamental substrate in quantum computation and permits for the transmission of quantum information without physically sending the qubit itself. The two classical bits are necessary for Bob to perform the correct recovery operation.
Both protocols are important for various quantum information processing tasks, and the best part is the exclusive capabilities presented by quantum entanglement. For example, quantum teleportation can be used as a route for quantum computation, enabling non-local multi-qubit operations. Superdense coding establishes the potential for increased communication capacity using quantum resources.