Experiments in no cloning in stochastic mechanics (Rev #5)

Copy machines are useful to say the least. Perhaps as a child you wrote on one side of a piece of paper with a pencil, say

You then might have flipped this over, placed it on top of another sheet of paper and scribble on the other side, like this.

Some of you might be laughing, but scribbling is no joke. As you can see from the etymology, the first known use of the noun dates to the 1570s. The case that is likely to apply here would then be *scribbler* from the 1550s, meaning “petty author”.

Now when you look underneath, you’ll notice that you’ve produced the mirror image of what you had originally written.

This is not unlike at least in principle to the first copy machine, invented by James Watt around 1779.

These days we are so used to copy machines that we hardly even notice what life would be like without them. For instance, when we turn on our computers and head straight to read the latest Azimuth post, we all view our own copy of the contents. This is a very familiar experience but in quantum mechanics things are not so simple however.

- There is a fundamental result about quantum mechanics, proven by Wootters, Zurek, and Dieks in 1982, asserting that a quantum state can not be cloned.

For those of you following the Network Theory Series, we’ve been considering a theory called, stochastic mechanics. This theory looks remarkably similar to quantum mechanics. See for instance, Part 5.

Unlike other posts in this series, today’s post will be rather short. What we will do is show that the proof showing that quantum states can’t be cloned also holds in the stochastic case. It’s not so surprising, as we will see, since both theories are linear. However, it makes for good reading and gives us yet another chance to illustrate some of this machinery being put to use.

As we recall, in both quantum and stochastic mechanics, given an initial state $\psi(0)$ at time $t=0$ the state at a later time $t$ is given by

(1)$\psi(t) = e^{t H} \psi(0)$

The first difference to notice between these two theories is that the operator $H$ has different properties in each case:

- in stochastic mechanics, $H$ is infinitesimal stochastic
- in quantum mechanics, $H$ is hermitian (note that we absorbed the complex number $i$ into the definition of $H$ to illustrate the similarities).

The other salient difference between quantum and stochastic mechanics, is that quantum states have amplitudes and stochastic states have probabilities for coefficients. Amplitudes are complex numbers which square to probabilities and probabilities are well, probabilities. In other words, in both theories, we can expand a state as a formal series

(2)$\psi = \sum_n c_n z^n$

Here, as John explained in Part 6, $z^n$ means $n$ things of type $z$. The coefficient $c_n$ is a probability in stochastic mechanics: the probability of there being $n$ things of type $z$. In quantum mechanics, $c_n$ is an amplitude. An amplitude multiplied by its complex conjugate yields a probability.

Here we are going to be concerned with cloning: said another way, we want to build a copy machine to copy states. At first sight, we have to build two copies machines. One of these must copy stochastic states and this machine must itself be governed by stochastic mechanics. Likewise, the second copy machine will be designed to copy quantum states, and the stochastic copy machine itself will be governed by quantum mechanics.

There are two typical ways in which one can prove no-cloning in quantum mechanics. One of these ways relies on the linearity of the theory: something that quantum and stochastic mechanics have in common. This method of prof works for both theories concurrently! That’s right, the similarities allow us to prove both cases concurrently. In other words, the quantum proof applies directly to the form of stochastic mechanics we’ve been considering in the Network Theory Series.

Let us consider an operator

(3)$U_t = e^{t H}$

This operation is our copy machine which we assert to exist. We expect it to work as follows.

(4)$U_t : z^k \otimes z^0 \mapsto z^k\otimes z^k$

(5)$U_t: z^m \otimes z^0 \mapsto z^m \otimes z^m$

In the first case, we consider $k$ copies of ‘things’ of type $z$ and in the second case $m$ copies of things of type $z$. We copy them to a slack system that initially has zero things of type $z$. Having picked this slack system to initially contain zero things is not crucial, we could have instead picked it to contain $q$ things of type $z$.

In both cases, as per definition of the map, $U_t$ copies these two states. Let us then consider the state

(6)$\phi = \alpha z^k + \beta z^m$

By linearity, we know exactly how $U_t(\phi\otimes z^0)$ should look.

(7)$U_t(\phi\otimes z^0) = \alpha z^k\otimes z^k + \beta z^m \otimes z^m$

however, to copy the state $\phi$ we would instead expect

(8)$\phi\otimes \phi = (\alpha z^k + \beta z^m )\otimes (\alpha z^k + \beta z^m )$

which is not equal to $U_t(\phi\otimes z^0)$. We arrive at a contradiction. In other words, we assumed that the copy machine would exist, but found that it can’t!

Now cloning is something that John has talked about before. See for example the following paper.

- There’s no cloning in symplectic mechanics, by Aaron Fenyes (2011).

What we have shown here is not so unexpected. The part I like is that the structure of the theories make the proofs identical in both cases. In other words, the quantum proof did not rely on anything that was not allowed in stochastic mechanics! This after all follows the spirit of the game we’ve been playing: constructing a theory that is classical (stochastic) in a way that resembles quantum theory.