The n-Category Café

We often describe the space of states of a classical system using a symplectic or Poisson manifold. But just like the category of Hilbert spaces, the categories of symplectic or Poisson manifolds are not cartesian!

When teaching a course on classical mechanics in 2008, this observation led me to suggest that cloning isn’t possible in classical mechanics, either. In my course notes the last sentences are:

I believe the non-Cartesian nature of this product means there’s no classical machine that can ‘duplicate’ states of a classical system:

[picture of classical machine where you feed a system into the hamper and two identical copies come out the bottom]

But, strangely, this issue has been studied less than in the quantum case!

Aaron Fenyes contacted me about this, and in 2010 he came out with a paper studying the issue:

• Aaron Fenyes, Limitations on cloning in classical mechanics.

Abstract. In this paper, we show that a result precisely analogous to the traditional quantum no-cloning theorem holds in classical mechanics. This classical no-cloning theorem does not prohibit classical cloning, we argue, because it is based on a too-restrictive definition of cloning. Using a less popular, more inclusive definition of cloning, we give examples of classical cloning processes. We also prove that a cloning machine must be at least as complicated as the object it is supposed to clone.

Feynes’s idea is that yes, if $X$ is a symplectic manifold of dimension > 0 it’s impossible to find a symplectomorphism $F$ that does this:

$$\begin{array}{cccl} F \colon & X \times X & \to & X \times X \\ & (x,y) & \mapsto & (x,x) \end{array}$$

But suppose we use a more general definition of cloning where we allow another system to get involved — the ‘cloning machine’, with its own symplectic manifold of states $M$, and look for a symplectomorphism

$$F \colon M \times X \times X \to M \times X \times X$$

that copies any state $x$ in the first copy of our original system if the machine starts out in the right state $m \in M$ and the second copy of our system starts out in the right state $x' \in X$. That is, for some $m \in M$ and $x' \in X$ and some function $f \colon X \to M$ we have

$$F(m,x,x') = (f(x), x, x) \qquad \qquad (\star)$$

for all $x \in X$.

With this definition, Feynes shows that cloning is possible classically — at least under some conditions on $M$ and $X$. For example, he shows the dimension of $M$ must be at least the dimension of $X$. That is, very roughly speaking, the machine needs to be at least as complex as the system it’s cloning!

But the analogous sort of cloning is not possible quantum mechanically. So there’s a real difference between classical and quantum mechanics, when it comes to cloning!

At the end of February, Yuan Yao contacted me with some new ideas on this issue. He had a nice result and I asked if he could generalize it. He did, and here it is:

• Yuan Yao, Phase spaces that cannot be cloned in classical mechanics.

Yao’s idea is to demand that our cloning map

$$F \colon M \times X \times X \to M \times X \times X$$

not only obeys $(\star)$ but is connected to the identity by a continuous 1-parameter family of symplectomorphisms. This is saying we can accomplish the cloning by a continuous process of time evolution — a very natural constraint to consider, physically speaking. And Yao shows that if this is true, the space $X$ needs to be contractible!

In short, only classical systems with a topologically trivial space of states can be cloned using a continuous process.

An interesting fact about Yao’s result is that it doesn’t really use symplectic geometry — only topology. In other words, we could replace symplectic manifolds by manifolds, and symplectomorphisms by diffeomorphisms, and the result would still hold.

All this suggests that classical cloning is a deeper subject than I thought. There’s probably a lot more left to discover. Yao has some suggestions for further research. And for a careful analysis of some of these issues, read this:

• Nicholas Teh, On classical cloning and no-cloning.

Maybe I can push things forward by formulating a challenge:

The Classical Cloning Challenge. Define a smooth cloning machine to consist of smooth manifolds $M$ and $X$ and a diffeomorphism

$$F \colon M \times X \times X \to M \times X \times X$$

such that for some $m \in M$ and $x' \in X$ and some function $f \colon X \to M$ we have

$$F(m,x,x') = (f(x), x, x)$$

for all $x \in X$. Define a symplectic cloning machine to be a smooth cloning machine where $M$ and $X$ are symplectic manifolds and $F$ is a symplectomorphism.

1) Find necessary and/or sufficient conditions on smooth manifolds $M$ and $X$ for there to exist a smooth cloning machine such that $F$ is connected to the identity in the group of diffeomorphisms of $M \times X \times X$.

2) Find necessary and/or sufficient conditions on symplectic manifolds $M$ and $X$ for there to exist a symplectic cloning machine.

3) Find necessary and/or sufficient conditions on symplectic manifolds $M$ and $X$ for there to exist a symplectic cloning machine such that $F$ is connected to the identity in the group of symplectomorphisms of $M \times X \times X$.

I’m also interested in Poisson manifolds because they include symplectic manifolds and plain old smooth manifolds as special cases: a Poisson manifold with nondegenerate Poisson tensor is a symplectic manifold, while any smooth manifold becomes a Poisson manifold with vanishing Poisson tensor. I expect that cloning becomes easier when the Poisson tensor has more degenerate directions, and easiest of all when it’s zero.

So, define a Poisson cloning machine to be a smooth cloning machine where $M$ and $X$ are Poisson manifolds and $F$ is an invertible Poisson map.

4) Find necessary and/or sufficient conditions on Poisson manifolds $M$ and $X$ for there to exist a Poisson cloning machine.

5) Find necessary and/or sufficient conditions on Poisson manifolds $M$ and $X$ for there to exist a Poisson cloning machine such that $F$ is connected to the identity in the group of Poisson diffeomorphisms of $M \times X \times X$.