In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James L. Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by William Wootters and Wojciech H. Zurek as well as Dennis Dieks the same year). The aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem (as generally understood) concerns only pure states whereas the generalized statement regarding mixed states is known as the no-broadcast theorem. The no-cloning theorem has a time-reversed dual, the no-deleting theorem.

According to Asher Peres and David Kaiser, the publication of the 1982 proof of the no-cloning theorem by Wootters and Zurek and by Dieks was prompted by a proposal of Nick Herbert for a superluminal communication device using quantum entanglement, and Giancarlo Ghirardi had proven the theorem 18 months prior to the published proof by Wootters and Zurek in his referee report to said proposal (as evidenced by a letter from the editor). However, Juan Ortigoso pointed out in 2018 that a complete proof along with an interpretation in terms of the lack of simple nondisturbing measurements in quantum mechanics was already delivered by Park in 1970.

Suppose we have two quantum systems A and B with a common Hilbert space ${\displaystyle H=H_{A}=H_{B}}$. Suppose we want to have a procedure to copy the state ${\displaystyle |\phi \rangle _{A}}$ of quantum system A, over the state ${\displaystyle |e\rangle _{B}}$ of quantum system B, for any original state ${\displaystyle |\phi \rangle _{A}}$ (see bra–ket notation). That is, beginning with the state ${\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}}$, we want to end up with the state ${\displaystyle |\phi \rangle _{A}\otimes |\phi \rangle _{B}}$. To make a "copy" of the state A, we combine it with system B in some unknown initial, or blank, state ${\displaystyle |e\rangle _{B}}$ independent of ${\displaystyle |\phi \rangle _{A}}$, of which we have no prior knowledge.

The state of the initial composite system is then described by the following tensor product: $${\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}.}$$ (in the following we will omit the ${\displaystyle \otimes }$ symbol and keep it implicit).

There are only two permissible quantum operations with which we may manipulate the composite system:

The no-cloning theorem answers the following question in the negative: Is it possible to construct a unitary operator U, acting on ${\displaystyle H_{A}\otimes H_{B}=H\otimes H}$, under which the state the system B is in always evolves into the state the system A is in, regardless of the state system A is in?

Theorem—There is no unitary operator U on ${\displaystyle H\otimes H}$ such that for all normalised states ${\displaystyle |\phi \rangle _{A}}$ and ${\displaystyle |e\rangle _{B}}$ in ${\displaystyle H}$ $${\displaystyle U(|\phi \rangle _{A}|e\rangle _{B})=e^{i\alpha (\phi ,e)}|\phi \rangle _{A}|\phi \rangle _{B}}$$ for some real number ${\displaystyle \alpha }$ depending on ${\displaystyle \phi }$ and ${\displaystyle e}$.

The extra phase factor expresses the fact that a quantum-mechanical state defines a normalised vector in Hilbert space only up to a phase factor i.e. as an element of projectivised Hilbert space.