The no-cloning theorem of quantum mechanics can be understood from Fisher information.

The individuality of objects in quantum mechanics refers to quantum information described by wave functions or state vectors. Furthermore, this quantum information as individuality generally possesses a strong identity that cannot be replicated. If a quantum system is in a state vector state |ψ> that is unknown to an observer, that observer cannot create an exact copy of that state. In other words, one "object" |ψ> cannot be turned into two "objects" |ψ>. This is known as the "no-cloning theorem." While classical information described by bit strings of 0s and 1s can be replicated exactly as many times as desired, general quantum information cannot be replicated exactly.

This fact can actually be understood from the Cramér-Rao inequality, which is satisfied by Fisher information J, which was also an indicator of "surprise."

Let us assume that a quantum system of interest transitions from a known initial state |0> to a state |ψ(θ)> through a physical operation specified by the value of a real parameter θ that is unknown to the observer. The observer measures the quantum system in this state to estimate the value of θ. The observer may use any experimental apparatus, but the Cramér-Rao inequality for the estimation error Δθ of θ always holds. And while the reciprocal of the square root of the Fisher information J gives the fundamental lower bound of Δθ, J is calculated solely from |ψ(θ)> regardless of the measurement method.

Interestingly, using this Cramér-Rao inequality leads to the aforementioned no-cloning theorem. To do this, we use proof by contradiction. First, assume that the no-cloning theorem does not hold, and that it is possible to create two quantum systems in the state |ψ(θ)> from a single quantum system in an arbitrary state |ψ(θ)>. Then, we show the contradiction.

If two |ψ(θ)> states can be created from one |ψ(θ)> even while the value of θ remains unknown, then the two resulting |ψ(θ)> states can each be copied to create four |ψ(θ)> states. If this is repeated many times, an infinite number of quantum systems in the state |ψ(θ)> can be generated. Since each quantum system is independently in the pure state |ψ(θ)> on its own, if one measures them one by one and collects information about θ, the error Δθ becomes zero, and the value of the unknown θ can be determined accurately.

However, according to the Cramér-Rao inequality, that is impossible. Δθ can never be smaller than the reciprocal of the square root of the Fisher information J. Therefore, a contradiction arises, and the initial assumption that "it is possible to create two quantum systems in the state |ψ(θ)> from a quantum system in an arbitrary state |ψ(θ)>" was incorrect. Thus, the no-cloning theorem is derived. When one firmly understands that quantum mechanics is information theory, one can grasp the no-cloning theorem intuitively in this way.