# No-Cloning Theorem

### What is No-Cloning Theorem

In 1982, the team of Wootters, Zurek, and Dieks co-authored the “no cloning theorem” of quantum mechanics. The theorem states that because quantum operations are unitary linear transformations of quantum states, arbitrary unknown states cannot be perfectly copied.

Interestingly, the impetus behind the theorem was a paper that asserted that cloning was possible. It was understood at the time that it was not possible, but it hadn’t been formalized as a theorem. In response to the paper, the theorem was formalized. Although the first paper’s author hadn’t pushed the field in the desired direction, the consequences advanced the field nonetheless.

Because of the no cloning theorem quantum computing cannot make exact copies of quantum information. This is in stark contrast to classical computing, which allows exact copies of data to be made up to the limits of memory and storage. This restriction on quantum computing has implications for quantum algorithm design.

#### What is No-Cloning Theorem

The no-cloning theorem has implications beyond quantum “copy-and-paste.” Some of its broader influence includes:

- In quantum cryptography, specifically quantum key distribution (QKD), eavesdroppers are prohibited from copying quantum cryptographic keys.
- With quantum teleportation for communication, quantum information must be destroyed by the sender to be received by the receiver.
- With quantum error correction (QEC), time-tested classical error correction techniques cannot be implemented.

Despite the QEC prohibition, novel quantum error correction codes (QECC) have been developed that are capable of detecting and correcting errors without violating the no cloning theorem. For more information on the need for QEC, approaches to QEC, types of QECC, and other challenges to implementing QEC, be sure to check out “Quantum Error Correction: Mastering the Art of Error-Free Quantum Computing.”

#### Understanding Quantum States and Cloning

A quantum state represents information about a quantum system. A measurement of the system extracts classical information, but probabilistically. If the state can be prepared again, a subsequent measurement could extract different information. Consequently, quantum states are usually prepared and measured many thousands of times, which provides more information about the state. But that still only provides limited information; the state may be prepared many thousands more times and measured in different bases to more fully describe the state. This is called quantum state tomography.

Because of the probabilistic nature of quantum measurement, each experiment is likely to have similar but different results. Consequently, an unknown quantum state cannot be precisely determined. And because the unknown state cannot be precisely determined, it cannot be precisely recreated. In other words, it cannot be copied; it cannot be cloned.

One of the key words in this prohibition, however, is “precisely.” Although perfect cloning is prohibited, Buzek and Hillery introduced the notion of imperfect cloning, or approximate cloning. By preparing and measuring a quantum state many thousands of times, tomography can determine a state with considerable accuracy. It can’t be perfect, but it can be really close.

This prohibition also does not prevent the preparation of identical quantum states. If a state is unknown, tomography cannot determine it precisely. But if a state is known, it can be prepared infinite times. It is not the state that is being copied but the preparation algorithm, and there are no prohibitions on the repeated execution of algorithms. Consequently, it is possible to prepare infinite copies of a quantum state, even though it is not possible to precisely copy the quantum state after it is prepared.

The no cloning theorem has a few related theorems, such as virtual quantum broadcasting. As this particular theory’s name suggests, it applies to quantum communication. Its proposition is that quantum information can be transmitted in such a way so as to not violate the no cloning theorem.

For more information on the theory behind the no-cloning theorem, the theorem’s related theorems, as well as ample links to source material, check out the article “no-cloning theorem” by nLab.

#### The No-Cloning Theorem Proof

The no cloning theorem proof states that an unknown quantum state cannot be precisely copied. Oversimplified, this is proven with a thought experiment in which we try to do exactly that: make an identical copy of an unknown quantum state:

- Start with two quantum systems with a common Hilbert space; in other words, they consist of the same number of qubits.
- Assume that the first quantum system has information that needs to be precisely copied.
- Make an identical copy of that information on the second quantum system without disturbing the information on the first quantum system, resulting in two identical copies.

The theorem states that because U is unitary, the cloning procedure is written as:

⟨ψ_1∣ψ_2⟩=⟨ψ_1 ψ_1∣U^† U∣ψ_2 ψ_2⟩

Because the quantum state is normalized, this becomes:

⟨ψ_1∣ψ_2⟩=⟨ψ_1∣ψ_2⟩⟨ψ_1∣ψ_2⟩

Again, oversimplified, the inequality proves that it’s not possible to make the desired copy. For more information on the proof, as well as its consequences, check out “The no-cloning theorem” on Quantiki.