No Cloning Theorem

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The No-Cloning Theorem is a fundamental result in quantum mechanics that states it is impossible to create an exact copy of an arbitrary unknown quantum state. Unlike classical information, which can be copied freely, quantum information obeys stricter rules due to the principles of superposition, linearity, and measurement.

The no-cloning theorem states that there is no universal quantum operation that can copy an arbitrary quantum state. Mathematically, we assume that there exists a unitary transformation such that, for an arbitrary quantum state and an initial blank state , the cloning process would satisfy: for all possible states . The theorem proves that such a unitary transformation does not exist.

Proof of No cloning theorem

Suppose there exists a unitary operator 𝑈 that can clone quantum states, meaning that for any two arbitrary states ∣𝜓⟩ and ∣𝜙⟩, we have:

𝑈∣𝜓⟩|0> = ∣𝜓⟩∣𝜓⟩

𝑈∣𝜙⟩|0> = ∣𝜙⟩∣𝜙⟩

This assumption states that 𝑈 transforms an unknown state ∣𝜓⟩ and an auxiliary blank state ∣0⟩ into two identical copies of ∣𝜓⟩.

Apply the Cloning Operator to a Superposition State: Consider a general superposition of two quantum states,

∣𝜓′⟩= √ ½ (∣𝜓⟩+∣𝜙⟩)

If cloning were possible, applying 𝑈 should result in:

𝑈∣𝜓′⟩∣0⟩ = ∣𝜓′⟩∣𝜓′⟩

Expanding ∣𝜓′⟩∣𝜓′⟩ we get:

√ ½ (∣𝜓⟩+∣𝜙⟩) √ ½ (∣𝜓⟩+∣𝜙⟩)

= ½ (∣𝜓⟩∣𝜓⟩+∣𝜓⟩∣𝜙⟩+∣𝜙⟩∣𝜓⟩+∣𝜙⟩∣𝜙⟩)

Because 𝑈 is linear, we should also have:

𝑈∣𝜓′⟩∣0⟩ = √ ½ (𝑈∣𝜓⟩∣0⟩+𝑈∣𝜙⟩∣0⟩)

𝑈∣𝜓′⟩∣0⟩ = √ ½ (∣𝜓⟩∣𝜓⟩+∣𝜙⟩∣𝜙⟩)

Clearly, the two results are not equal unless ∣𝜓⟩ and ∣𝜙⟩ are identical or orthogonal. Since quantum states are generally not orthogonal, the assumption that 𝑈 can clone all states leads to a contradiction.