No Cloning Theorem: Why Quantum States Cannot Be Copied

The no cloning theorem states that there is no physical operation that perfectly copies an arbitrary unknown quantum state.

Statement

There is no unitary $U$ such that for every $|\psi\rangle$:

$$U(|\psi\rangle|0\rangle) = |\psi\rangle|\psi\rangle$$

Proof Sketch

Assume cloning works for two states $|\psi\rangle$ and $|\phi\rangle$.

Then:

$$\langle\psi|\phi\rangle = \langle\psi\psi|\phi\phi\rangle = (\langle\psi|\phi\rangle)^2$$

So overlap $s = \langle\psi|\phi\rangle$ must satisfy $s = s^2$, which implies $s \in \{0, 1\}$.

That means cloning can only hold for identical or orthogonal states, not arbitrary unknown states.

Why This Matters

• Prevents perfect copying of quantum information.
• Supports security foundations of quantum key distribution.
• Changes how error correction and communication protocols are designed.

Approximate cloning and state dependent cloning exist, but universal perfect cloning does not.

Linearity View (Another Intuition)

Quantum evolution is linear. If cloning worked for basis states:

$$|0\rangle|0\rangle \to |0\rangle|0\rangle, \quad |1\rangle|0\rangle \to |1\rangle|1\rangle$$

then by linearity, input $(|0\rangle+|1\rangle)/\sqrt{2}$ would map to:

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

But true cloning would require:

$$\left(\frac{|0\rangle+|1\rangle}{\sqrt{2}}\right) \otimes \left(\frac{|0\rangle+|1\rangle}{\sqrt{2}}\right) = \frac{|00\rangle+|01\rangle+|10\rangle+|11\rangle}{2}$$

These are different states, so universal cloning is impossible.

What Is Still Allowed

• Copying known classical bits encoded as orthogonal basis states is allowed.
• Approximate cloning machines can exist for restricted goals.
• Quantum teleportation transfers a state, but the original is destroyed, so it is not cloning.

This distinction is central in quantum communication protocols.