State and prove the no-cloning principle

[quantumadmin]

The "no-cloning" principle is a fundamental concept in quantum mechanics, which states that it is impossible to create an exact copy of an arbitrary unknown quantum state. In other words, you cannot produce an identical copy of a quantum state without destroying the original state in the process. This principle has important implications for the field of quantum information theory and is closely related to the concept of quantum entanglement.

To provide a proof for the no-cloning principle, let's consider a general quantum state represented as a normalized vector in a Hilbert space, denoted as |ψ⟩. Suppose we have another quantum state |φ⟩ and an arbitrary quantum cloning operation that tries to clone |ψ⟩ onto a new state |ψ'⟩:

Clone: |ψ⟩ ⊗ |φ⟩ → |ψ'⟩ ⊗ |ψ'⟩.

Here, |ψ⟩ is the original quantum state, |φ⟩ is an ancillary quantum state (used for copying), and |ψ'⟩ is the cloned state.

The no-cloning principle can be proven using the linearity of quantum operations and the properties of inner products in a Hilbert space. The proof involves a contradiction, assuming the existence of a perfect quantum cloning operation, and then showing that it leads to inconsistencies with the fundamental principles of quantum mechanics.

Proof by Contradiction:

Assume there exists a quantum cloning operation that can copy an arbitrary quantum state |ψ⟩ onto another state |ψ'⟩:

Clone: |ψ⟩ ⊗ |φ⟩ → |ψ'⟩ ⊗ |ψ'⟩.

Consider an orthonormal basis {|0⟩, |1⟩} for a qubit system. Let |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex coefficients.

Apply the cloning operation to |ψ⟩:

Clone: (α|0⟩ + β|1⟩) ⊗ |φ⟩ → (α|0⟩ + β|1⟩) ⊗ (α|0⟩ + β|1⟩).

Expand the right-hand side:

(α|0⟩ ⊗ (α|0⟩ + β|1⟩)) + (β|1⟩ ⊗ (α|0⟩ + β|1⟩)).

Distribute the tensor product:

α²|0⟩ ⊗ |0⟩ + αβ|0⟩ ⊗ |1⟩ + βα|1⟩ ⊗ |0⟩ + β²|1⟩ ⊗ |1⟩.

Now, compare this with the original state |ψ⟩ ⊗ |φ⟩:

α|0⟩ ⊗ |φ⟩ + β|1⟩ ⊗ |φ⟩.

Since the cloning operation should be able to copy any arbitrary state |ψ⟩ onto another state |ψ'⟩, these two states should be equal:

α|0⟩ ⊗ |φ⟩ + β|1⟩ ⊗ |φ⟩ = (α|0⟩ + β|1⟩) ⊗ |ψ'⟩.

This implies:

α|0⟩ + β|1⟩ = α|0⟩ + β|1⟩ ⊗ |ψ'⟩.

However, the left-hand side represents the original state |ψ⟩, and the right-hand side represents the copied state |ψ'⟩, which is a contradiction. This shows that the assumption of a perfect quantum cloning operation is false.

Therefore, the no-cloning principle holds: It is impossible to create an exact copy of an arbitrary unknown quantum state.

This proof demonstrates that the linearity of quantum mechanics, combined with the principles of superposition and the no-cloning principle, prevents the perfect copying of arbitrary quantum states. The proof essentially relies on the fundamental axioms of quantum mechanics and the mathematical properties of quantum states in a Hilbert space.