Send 2 classical bits by transmitting 1 qubit
How can sending a single qubit convey two classical bits of information?
Superdense coding, proposed by Bennett and Wiesner in 1992, is the complement of quantum teleportation. While teleportation uses 2 classical bits + entanglement to send 1 qubit of information, superdense coding uses 1 qubit + entanglement to send 2 classical bits. Together, they reveal the precise exchange rate between quantum and classical information in the presence of entanglement. Superdense coding was one of the earliest demonstrations that entanglement is a genuine information-theoretic resource.
Alice and Bob share a Bell pair. Alice wants to send Bob a 2-bit message (00, 01, 10, or 11). She applies one of four operations to her qubit (I, X, Z, or XZ), each producing a different Bell state. She sends her qubit to Bob. Bob now has both halves of the Bell pair and performs a Bell measurement to determine which of the four Bell states he has — recovering Alice’s 2-bit message from a single transmitted qubit.
The four Bell states form an orthonormal basis, so Bob can perfectly distinguish them. Alice’s local operation on one qubit of a Bell pair changes the joint state to one of four orthogonal states, encoding 2 bits in the pre-shared entanglement.
Prepare |Φ⁺⟩ = (|00⟩+|11⟩)/√2. Alice keeps qubit 0, Bob keeps qubit 1.
Alice applies one of four gates to her qubit depending on her 2-bit message: (00)→I, (01)→X, (10)→Z, (11)→XZ. This transforms the Bell state into one of four orthogonal Bell states.
Alice physically sends her qubit to Bob. Now Bob holds both qubits of the (modified) Bell pair.
Bob applies CNOT(q0,q1) then H(q0) to reverse the Bell preparation. This maps each Bell state to a unique computational basis state.
Measuring both qubits gives Bob Alice’s 2-bit message directly: |00⟩, |01⟩, |10⟩, or |11⟩.
Bob’s measurement result is Alice’s original 2-bit message. The Z gate in the circuit encodes the message “10”; using X would encode “01”, I would encode “00”, and XZ would encode “11”.
Starting from |Φ⁺⟩, Alice’s four operations produce: I→|Φ⁺⟩, X→|Ψ⁺⟩, Z→|Φ⁻⟩, XZ→|Ψ⁻⟩. These are orthogonal and span the two-qubit space. Bob’s Bell measurement (CNOT then H, followed by computational-basis measurement) uniquely identifies which Bell state was received, giving 2 classical bits per qubit transmitted.
Classically, sending 2 bits of information requires transmitting 2 bits. One bit can carry at most 1 bit of information (Holevo bound without entanglement).
With pre-shared entanglement, 1 qubit can carry 2 classical bits. The entanglement acts as a pre-established correlation that doubles the channel capacity.
Superdense coding does not violate Holevo’s bound, which says 1 qubit carries at most 1 bit without entanglement. The extra bit comes from the pre-shared Bell pair, which is an additional resource consumed in the process.
Superdense coding and teleportation together show that 1 qubit + 1 ebit (shared Bell pair) = 2 classical bits, and 2 classical bits + 1 ebit = 1 qubit. These are the fundamental exchange rates of quantum information theory.
In superdense coding, why can a single qubit carry 2 classical bits?