Wednesday, 2 August 2017

Schroedinger's cat: not one thing and the other.

Schroedinger's cat is not alive, is not dead, and is not simultaneously alive and dead. It's more interesting than that.

Schroedinger's Cat

We're all familiar with the Schroedinger's cat (thought) experiment. A cat is placed in a box along with a sample of a deadly poison and a radioactive sample all hooked together in such a way that at a certain time the probability of the sample decaying, and releasing the poison resulting in a dead cat is exactly \(1/2\). The question then is, what is the state of the cat at just that time? The standard understanding of quantum mechanics tells us that if we have a lot of identical boxes of this type and we look inside them all, then half of them will contain a live cat and the other half will contain a dead cat. But what is the state of the cat before the box is opened?

It is quite common to see statements to the effect that the cat is simultaneously dead and alive up until the moment the box is opened and it is observed. But observing the cat forces it into one state or the other, and so we never see a cat simultaneously alive and dead: but if we do the experiment many times, we expect just to see a live cat half of the time and a dead cat the other half.

There are a couple of questions which spring pretty irresistibly to mind. The first is, what on earth does it mean for the cat to be simultaneously dead and alive? And the second is, if it is in fact simultaneously dead and alive, why don't we ever see that?

Decoherence-a digression

Let's address the second question first. One current understanding is that when the interior of the box interacts with the outside world, the state of the interior is forced into a well-defined classical state by a phenonemon known as decoherence. This gives a purely physical explanation for the 'collapse of the state', a central aspect of the Copenhagen interpretation of quantum mechanics, avoiding the earlier speculations about a conscious observer being required, and the consequent problems raised by Wigner's friend.

But this is not an easy question, nor one with a universally accepted answer. The issue has been vigorously debated for many years and will almost certainly continue to be for many to come. I am going to take the coward's way out, and avoid the question entirely. Instead, I will consider the notion of a state and what kind of state describes the cat in the box.

From now on, then, when I talk about 'observation', you can think about 'what I see when I look at the system' or 'the state of the system when decoherence due to interaction with the exterior world results in a classical state', as you prefer.

Quantum states

In the quantum mechanics picture of the world, the state of a system is described by a vector: this contains all the information that there is about the system, and the results of any measurement that can be made. In (fairly) general, the state of a system can be written \[ |\psi\rangle = \sum_i \alpha_i | \psi_i \rangle \] where each of the \(|\psi_i\rangle\) is a vector corresponding to a classical state of the system, such as 'alive' or 'dead' for the cat, or 'decayed' or 'not decayed' for the radioactive sample. The \(|\psi_i\rangle\) have unit length and are orthogonal to each other, and the \(\alpha_i\) are complex numbers whose squared moduli add up to \(1\). Finally, the probability of the system being in the \(i\)th state when observed is \(|\alpha_i|^2\).

The use of the notation \(| \;\; \rangle\) to represent a state-vector is due to Dirac, and is part of a fairly awful pun he was very fond of. If you think of \(|\psi\rangle\) as a column vector, then the corresponding row vector is \(\langle \psi |\). Dirac called the row vector a bra, and the column vector a ket, so that when you take the inner product of two vectors \(|\psi\rangle\) and \(|\phi\rangle\) you get \(\langle \psi | \phi \rangle\), a bra-ket or bracket. I said it was awful.

There's a long story here which I'm not going into. As time passes, the \(\alpha_i\) evolve according to Schroedinger's (differential) equation, which relates the rate of change of the \(\alpha_i\) to the surroundings. The details of how this works are, fortunately, irrelevant here.

In addition, there can be more than one way of expressing the state \(|\psi\rangle\) as a combination of classical states, and the relationship between these expressions is tied to the famous uncertainty principle of Heisenberg. We are doubly fortunate, because we don't need to worry about that here either.

The cat's state

So, what's actually going on inside the box? We have the radioactive sample, which at a certain time is in a quantum state which means that an observation will show decayed or not decayed with equal probability. Such a microscopic system being in a weird quantum state doesn't raise any immediate alarm bells, it's when the state of a macroscopic object (such as a cat, and whether the cat is alive or dead) is weird in this way that we start to worry.

Looking at this again, it seems reasonable that we can identify the two classical states of the radioactive samples as 'decayed' and 'not decayed', and the relevant aspects of the cat's state as 'dead' (if the sample has decayed) and 'alive' (if the sample has not yet decayed). I'll use \(|a\rangle\) to represent the state 'alive' and \(|d\rangle\) to represent 'dead'.

So if the cat is in the state \(|a\rangle\) then the cat will certainly be alive when it is observed; likewise, if it is in the state \(|d\rangle\) then it will certainly be dead when observed. So you can see that it is not true that observation of a quantum system unavoidably changes its state. That depends on the state and the type of observation: the uncertainty principle is much more fundamental than the idea that poking a system to see where it is pushes it about a bit. But that's another story.

But in general the cat is not in either of these states. Instead, it is in some state of the form \[ |\psi\rangle = \alpha |a\rangle + \beta |d\rangle) \] where \(|\alpha|^2+|\beta|^2=1\). In this state, the probability that the observation reveals a live cat is \(|\alpha|^2\), and the probability of a dead cat is \(|\beta|^2\). In the classical thought experiment, we have both of these probabilities equal to \(1/2\), so one possible state would be \[ |\psi\rangle = \frac{1}{\sqrt{2}}(|a \rangle + |d \rangle) \]

So what is the state of the cat here? It is not alive. That would mean it was in the state \(|a\rangle\). It is not dead. That would mean it was in the state \(|d\rangle\). It certainly isn't simultaneously alive and dead, any more than I can be simultaneously in the living room and in the hallway. It also isn't in some kind of intermediate state, such as when I am standing in the doorway with one foot in the living room and one in the hallway. It is in a different kind of state, one without a classical analogue, and this kind of state is called a superposition.

To say that the cat is simultaneously alive and dead is like saying that somebody pointing north-east is simultaneously pointing north and east. They aren't: but the direction they are pointing in has a northern and an eastern component. In the same way, the cat isn't simultaneously alive and dead: but the state it is in has an 'alive' and a 'dead' component.

This actually has enormous consequences. The different components of the state can evolve indepenently, almost as if there are two universes involved, in one of which the cat is alive, and in the other of which it is dead. Developing this idea leads to the many worlds, or Everett-Wheeler model of quantum mechanics, which Sean Caroll describes here.

In any case, if we take quantum mechanics seriously, and the evidence for it it is pretty compelling, then we have to learn to live with and accept (even if we can never be entirely comfortable with) the idea that the state of a system might not be a classically recognized state, but rather a superposition of such states. We even have to accept that the acceptable states may not quite correspond to what we think of as classically acceptable states. They are more general in some respects (we have to allow for superpositions of classical states), but more restricted in others (there is no quantum state for a particle with a simultaneously well-defined position and momentum).

Decoherence again

All this has been about what the quantum state of the (unobserved) cat in the box is. But maybe you aren't any happier with the cat being in a superposition of states than with the cat somehow being in two states at once. There is a possible way of avoiding this, and it is to argue that when the radioactive sample interacts with the rest of the apparatus in the box (including the cat), that already causes decoherence, so at box opening time what we actually have is a box with either a dead cat or a live cat in it: or again, if we did it many times, a collection of boxes of which about half contain live cats and half contain dead one.

It would be easy to give the impression that decoherence is supposed to be the cure for all our problems with why macroscopic systems look classical, rather than quantum. It would also be wrong: I'm not going to go into any detail at all here, but if you've made it this far you can probably get quite a lot from reading (at least substantial chunks of) Schlosshauer's review article.

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