nLab no-cloning theorem

Idea

In quantum physics the no-cloning theorem is the statement that there cannot be physical processes which produce “copies” of quantum states in the way known from classical physics.

Quantum theoretically, this effect is ultimately due to the non-cartesian nature of the tensor product of vector/Hilbert spaces of quantum states (category theoretically it refers to this tensor product lacking natural diagonal morphisms) and as such is a direct cousin of the fundamental phenomenon of quantum entanglement.

In other words 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 quantum information theory via dagger-compact categories.

Details

Elementary explanation

In its original formulation, the statement of the “no-cloing theorem” 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 $\langle\phi | \psi \rangle = \langle\phi | \psi \rangle^2$).

Category theoretic explanation

More category theoretically, the no-cloning theorem comes down to the statement that the tensor product of vector spaces (and similarly its refinement to Hilbert spaces) which makes VectorSpaces a symmetric monoidal category

$$(-) \otimes (-) \;\colon\; Vect \times Vect \longrightarrow Vect$$

does not admit a natural transformation of the form

(1)$$A \xrightarrow{\;\;} A \otimes A$$

satisfying basic properties expected of the diagonal morphisms in a cartesian monoidal category.

Without even requiring projection maps and hence “deleting” operations (which are part of a cartesian product) two such properties that the would-be diagonal (1) would have to satisfy are coassociativity and cocommutativity – and one finds that such cannot hold in VectorSpaces [Abramsky (2009), Thm. 11].

Remark

(relation to classical outcomes of quantum measurement)
Notice that the condition that (1) be a natural transformation is crucial for the above conclusion of “no cloning”:

Namely, for a fixed vector space $\mathscr{H}$, any choice of linear basis $\big\{ \vert b \rangle \big\}_{b \colon B}$ does induce a single linear map of the required form, given by duplicating (hence “cloning”) the fixed basis elements:

(2)$$\array{ \mathscr{H} &\longrightarrow& \mathscr{H} \otimes \mathscr{H} \\ \vert b \rangle &\mapsto& \vert b \rangle \otimes \vert b \rangle }$$

On the other hand, beware that, by the laws of linear maps, this means that no other elements except the basis elements are exactly cloned even by this operation; for example the superposition of two distinct basis elements goes to:

(3)$$\begin{array}{lcl} \vert b \rangle + \vert b' \rangle &\mapsto& \vert b \rangle \otimes \vert b \rangle + \vert b' \rangle \otimes \vert b' \rangle \\ &=& \big( \vert b \rangle + \vert b' \rangle \big) \otimes \big( \vert b \rangle + \vert b' \rangle \big) \,-\, \underset{correction}{ \underbrace{ \vert b \rangle \otimes \vert b' \rangle \,+\, \vert b' \rangle \otimes \vert b \rangle } } \end{array}$$

Nonetheless, the linear map (2) does satisfy coassociativity and cocommutativity in the evident sense; in fact it also satisfies counitality with respect to the unit element $1 \,\coloneqq\, \underset{b \colon B}{\sum} \vert b \rangle \;\in\; \mathscr{H}$. In summary this means that (2) defines a cocommutative coalgebra-structure on $\mathscr{H}$, which hence certainly exists on any (finite-dimensional) space of quantum states.

While such structures exist, they do not represent a general process of “cloning” of quantum states: Due to effects as in (3), these structures are not natural transformations and hence they “clone” not arbitrary quantum states, but just the chosen linear basis-states that define them.

Conversely, such coalgebra-structure (at least when regarded as the special symmetric Frobenius algebra-structure to which they always extend) in fact define a choice of linear basis of a quantum space of states. Since such a choice may encode the set of possible classical state outcomes under a quantum measurement, these “restricted cloning”-operations are also known as “classical structures” (at least in the literature on quantum information theory in terms of dagger-compact categories).

In summary: While a natural cloning operation on quantum states does not exist, enforcing one on a select linear basis of quantum states is tantamount to “turning these into classical states” (in a sense that can be made more precise, see discussion at quantum reader monad and quantum circuits via dependent linear types).

Remark

(Converse statement)
One may turn the issue around 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, asserting that the category must be “essentially classical”:

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 (2009), Thm. 11 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.

No broadcasting

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

Related concepts

• deferred measurement principle

• quantum superposition

• entanglement

• Bell's inequalities

• Gleason's theorem

• interpretation of quantum mechanics

• quantum cryptography

References

The original articles:

• 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 (1982) 802-803 [doi:10.1038/299802a0]

See also:

• Wikipedia, No-cloning theorem

Discussion from the point of view of monoidal category-theory (quantum information theory via dagger-compact categories):

• Samson Abramsky, No-Cloning in categorical quantum mechanics, in: I. Mackie, S. Gay (eds.), Semantic Techniques for Quantum Computation, Cambridge University Press (2009) 1-28 [arXiv:0910.2401, doi:10.1017/CBO9781139193313.002]

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)