Questions: Superdense Coding

Question 1

In superdense coding, Alice applies the Z gate to her qubit to encode message '10'. What Bell state does the shared pair become after this operation?

• A. (|00> + |11>)/sqrt(2)
• B. (|00> - |11>)/sqrt(2)
• C. (|01> + |10>)/sqrt(2)
• D. (|01> - |10>)/sqrt(2)

Answer: (B)

Starting from the Bell state (|00> + |11>)/sqrt(2), applying Z to Alice's qubit (the first qubit) gives Z|0>|0>/sqrt(2) + Z|1>|1>/sqrt(2) = |0>|0>/sqrt(2) - |1>|1>/sqrt(2) = (|00> - |11>)/sqrt(2). Each of the four Pauli operations maps the initial Bell state to a different, orthogonal Bell state, which is why Bob can distinguish all four messages perfectly.

Question 2

Superdense coding allows transmitting information faster than light because only a qubit needs to travel.

• T. True
• F. False

Answer: False

Alice still sends a physical qubit to Bob, which travels at most at the speed of light. The advantage is bandwidth, not speed: one qubit carries two classical bits of information (with the help of pre-shared entanglement). The entanglement distribution also required prior physical communication. No part of the protocol violates relativistic causality.

Question 3

Why does superdense coding require a pre-shared entangled pair? What would happen if Alice just applied one of four unitaries to a single qubit in state |0> and sent it to Bob?

Model answer: Without entanglement, Alice can only encode information in the state of a single qubit — a two-dimensional space. Four orthogonal states cannot exist in two dimensions, so Bob cannot perfectly distinguish four messages from a single qubit measurement. The entangled pair provides a second qubit at Bob's location, giving him access to a four-dimensional space (two qubits) where the four Bell states are orthogonal and perfectly distinguishable.

This connects to the Holevo bound: a single qubit, without entanglement assistance, can convey at most one classical bit reliably. Superdense coding achieves two bits because the entangled pair effectively provides a pre-positioned second qubit that doubles the accessible Hilbert space dimension at Bob's end. The entanglement is a resource that is consumed in the process.