Questions: No-Cloning Theorem

Question 1

The no-cloning theorem says you cannot copy an unknown quantum state. Can you copy a KNOWN quantum state?

• A. No — the theorem forbids copying any quantum state
• B. Yes — if you know the state, you can prepare as many copies as you want by running the appropriate state-preparation circuit
• C. Only if the state is a computational basis state
• D. Only if you have a quantum copier calibrated for that specific state

Answer: (B)

The no-cloning theorem applies to UNKNOWN states — you cannot build a device that copies an arbitrary input state without knowing what it is. But if you know the state (e.g., someone tells you it is (|0> + |1>)/sqrt(2)), you can prepare as many copies as you want by applying Hadamard to fresh |0> qubits. The theorem constrains devices that must work for all input states, not the preparation of specific known states.

Question 2

The no-cloning theorem follows from the unitarity of quantum mechanics — specifically, from the fact that quantum evolution is linear.

• T. True
• F. False

Answer: True

The proof is elegant. Suppose a unitary U clones: U|a>|0> = |a>|a> and U|b>|0> = |b>|b>. Taking the inner product of both sides: <a|b> = (<a|b>)^2. This equation has only two solutions: <a|b> = 0 or <a|b> = 1, meaning |a> and |b> must be either identical or orthogonal. A cloner that works for two non-orthogonal states leads to a contradiction. The linearity of U is what forces the inner product equation, so no-cloning is a direct consequence of quantum mechanics being a linear theory.

Question 3

How does the no-cloning theorem make quantum key distribution secure? What would happen if cloning were possible?

Model answer: If cloning were possible, an eavesdropper could intercept each qubit in transit, make a copy, send the original to the receiver, and later analyze the copy in the correct basis (learned from the public basis reconciliation). This would allow perfect undetectable eavesdropping, destroying QKD security. No-cloning prevents this: Eve must measure the qubit directly, which disturbs it and introduces detectable errors. The impossibility of cloning forces any information-gathering strategy to be invasive.

No-cloning is essential to QKD but also to the broader structure of quantum information. It means quantum information cannot be broadcast, cannot be backed up, and cannot be passively observed without disturbance. These properties are limitations from a computing perspective (making error correction harder) but resources from a cryptographic perspective (making eavesdropping detectable).