nLab no-cloning theorem

Idea

In quantum physics the no-cloning theorem is the statement that one cannot produce a second copy of an arbitrary given quantum state by a quantum physical process.

More in detail, in its original version the statement is that given a quantum system with Hilbert space $H$ and with a chosen initial pure quantum state $e \in H$, then there is no unitary operator on the tensor product $H \otimes H$ which would take states of the form $(\psi, e)$ to $(\psi,\psi)$.

Of course this cannot exist, because such a map would not even be linear. Often the statement is relaxed to observing that if there is a unitary that takes $(\psi,e)$ to $(\psi,\psi)$ and $(\phi,e)$ to $(\phi,\phi)$ for any two states $\phi,\psi \in H$, then either $\phi= \psi$ or $\phi \perp \psi$ (since it follows from unitarity that in this case $\langle\phi | \psi \rangle = \langle\phi | \psi \rangle^2$).

There is a slightly more substantial generalization of this observation to mixed quantum states, then called the no-broadcasting theorem. Dually, there is also a no-deleting theorem.

Formally, when quantum physics is axiomatized by quantum logic in the guise of linear logic/linear type theory, the content of “no-cloning” and “no-deleting” is the very “linearity” of this logic, the absence of a diagonal map and of a projection map for the non-cartesian tensor product in the categorical semantics given by non-cartesian symmetric monoidal categories such as that of Hilbert spaces. See also at finite quantum mechanics in terms of dagger-compact categories.

From this perspective one gets a more interesting statement if one turns this around and asks which conditions are imposed on a symmetric monoidal category if its tensor product is assumed to admit a diagonal map (for “cloning”) or projection maps (for “deleting”) or both. These turn out to be very strong conditions. For instance in order for a diagonal to exist for the tensor product of a compact closed category implies that every endomorphism on every object in the category is a multiple of the identity (Abramsky 09, in generalization of “Joyal’s lemma”). This is clearly false in any category in which one could find interesting quantum mechanics, certainly in that of (finite dimensional) Hilbert spaces.

Related concepts

• quantum superposition

• entanglement

• Bell's inequalities

• Gleason's theorem

• interpretation of quantum mechanics

• quantum cryptograpgy?

References

The original articles are

• Dennis Dieks, Communication in EPR devices, Physics Letters A 92, 271-272 (1982)

• William Wooters, Wojciech Zurek, A single quantum cannot be cloned, Nature 299, 802-803 (1982)

General reviews include

• Wikipedia, No-cloning theorem

Discussion from the point of view of monoidal category-theory (finite quantum mechanics in terms of dagger-compact categories) includes

• Samson Abramsky, No-Cloning in categorical quantum mechanics, (2008) in I. Mackie and S. Gay (eds), Semantic Techniques for Quantum Computation , Cambridge University Press (arXiv:0910.2401)

The original suggestion to use the no-cloning theorem for quantum cryptography:

• Stephen Wiesner, Conjugate Coding circulated in the 1960s, finally published in ACM SIGACT News 15 1 (1083) 78–88 (original pdf, doi:10.1145/1008908.1008920)