Lucy and Ricardo are going to encounter many pairs of cakes, and for each pair they will make a random choice as to whether to open the oven halfway along and look at the cake or to wait and taste the cake at the end. They are only allowed to make one observation each per pair of cakes; they can either open the oven halfway along and check whether their cake has risen, or they can taste their cake at the end, but not both. Their task is to record the results.
For the first pair of cakes, Lucy tastes her cake at the end of her conveyor and it tastes good. Ricardo does the same and finds that his cake tastes bad. For the second pair of cakes, Ricardo opens his oven halfway along and sees that the cake has risen. Lucy tastes her cake at the end and finds that it tastes good. And so on. After many cakes, they find:
Whenever Lucy’s cake rose early, Ricardo’s cake always tasted good.
Whenever Ricardo’s cake rose early, Lucy’s cake always tasted good.
In cases where both Lucy and Ricardo checked the ovens halfway along, 1/12 of the time both cakes had risen early.
If we use our common sense, honed by our experience of the world, these three observational facts lead us to infer that both cakes should taste good at least 1/12 of the time. We can infer this because:
when Ricardo and Lucy checked both ovens, they found that in 1/12 of the cases both cakes had risen, and
we know that when Ricardo’s cake has risen, Lucy’s tastes good and vice versa.
There is nothing surprising here, other than that they are rubbish bakers. However, here comes a shocking observation. In the quantum kitchen:
Both cakes never taste good.
How can that be? Using what seems like unimpeachable logic, we have concluded that in at least 1/12 of cases both cakes should taste good, and yet they never do.
It’s fun to play around and try to work out what is wrong with this reasoning – what possible mechanism could be in play to produce these strange results? Physics students like to do this in the pub. Long ago, the authors spent an evening figuring out why attaching a rod to the Moon and tapping it in morse code could not violate the principles of relativity by sending signals faster than light.
Figure 12.1. The quantum kitchen, from Paul Kwiat and Lucien Hardy’s ‘The Mystery of the Quantum Cakes’. (Figure 1 from ‘The Mystery of the Quantum Cakes’, by P. G. Kwiat and L. Hardy, American Journal of Physics, 68:33–36 (2000), https://doi.org/10.1119/1.19369. Reproduced here by permission of the authors and the American Association of Physics Teachers.)
There could be some mechanism whereby the only way to make a good-tasting cake is if the other person opens their oven at the midway point. Maybe the act of opening one oven midway causes a sound that shakes the other oven in just such a way as to make its cake taste good. Or perhaps a tiny, heat-resistant culinary gnome is sitting in each oven watching what is happening to the other oven who ensures that their cake tastes good if they see the other oven door opened. Both these possibilities (which exploit a causal link) can be ruled out if we make the conveyor belts sufficiently long and fast-moving that no signal can travel between the ovens before the observations are made. The signal would need to depart after Lucy/Ricardo has decided to open their oven at the midway point and arrive at the other oven before Ricardo/Lucy tastes their cake. We can arrange things so that there isn’t enough time for this to happen. Or perhaps the chef who makes the cake mixture somehow knows in advance the measurement choices that Ricardo and Lucy are going to make. The chef could then produce a bad mixture at just the right time. This possibility can be eliminated if Ricardo and Lucy make their random decisions after the ovens have left the kitchen. And so on. There is one logical possibility that could explain the results without resorting to quantum theory: every event in the Universe is predetermined, freewill does not exist and the results were baked in at the beginning of time. Putting that aside, we are left with quantum mechanics.
The results we’ve quoted above can be explained if the cakes are produced in an entangled quantum state. In this case, the quantum state of the cakes is such that both cakes can never taste good. This is like the situation we encountered earlier with our entangled system of qubits; there was no chance that both qubits could be 0 or 1.
Here is a quantum state for the cakes that reproduces the results obtained by Lucy and Ricardo:
where B and G mean ‘tastes bad’ and ‘tastes good’ respectively and the subscripts refer to Lucy and Ricardo’s cakes. This is a more complicated state than we’ve seen before, but you can see that, because there is no |GL⟩|GR⟩ term, both cakes never taste good. You can also see that both cakes taste bad one third of the time. To explain the numerical results associated with opening the ovens and observing whether the cakes have risen, we need a bit more quantum theory that won’t be necessary for what follows, but it is interesting, so we’ve moved the discussion to Box 12.1.
BOX 12.1. More from the quantum kitchen
You might be wondering where the ‘risen’ or ‘not risen’ measurements fit into the quantum state of the cakes. To reproduce the results obtained by Lucy and Ricardo, the ‘tastes bad’ and ‘tastes good’ states of the cakes are given by:
where R and N mean ‘risen’ and ‘not risen’. How should we interpret these states? Let’s take |B⟩ as an example. If a cake is observed to taste bad, then this means it must be in state |B⟩. If we now make a subsequent observation and ask whether it has risen or not, there is a 50 per cent chance it will not have risen, because 1/√2 squares to ½. If you fancy a little bit of mathematics, you can substitute these expressions for |B⟩ and |G⟩ into the state |Q⟩ to ascertain the coefficient of the |RL⟩|RR⟩ term. You should find it’s –1/√12 which gives a probability of 1/12 that both cakes will have risen. With less effort, you can also see that there is no |RL⟩|BR⟩ and |BL⟩|RR⟩ piece in |Q⟩, which explains facts 1 and 2. Our quantum chef is responsible for preparing the cakes in these very specific states.
For our purposes, the most important feature of the quantum kitchen is that, because the cakes are in an entangled state, they do not possess the qualities of ‘tastes good’, ‘tastes bad’, ‘risen’ or ‘nor risen’ independently. Rather the whole two-cake system is produced by the quantum kitchen in a state that mixes all these possible measurement outcomes with correlations that give the results we quoted above. And yet the quantum state is also such that in every case, each individual cake has the potential to taste good or bad, or to rise or not rise, when it leaves the kitchen. To reiterate a very important point, the probabilities observed by Lucy and Ricardo are not the result of a lack of knowledge about the state of the quantum cake system. They can know the state and yet before a measurement is made they cannot know whether an individual cake will taste good or bad, or is risen or not because each cake is all of these, until it’s observed.‡
Where does the information about the correlations in the entangled state reside? It is not stored locally by each individual cake. Returning to our simple two-qubit system:
a measurement of one of the qubits will reveal 0 or 1 with equal probability. That’s just like tossing a coin. Half the time it will come up heads and half the time it will come up tails. There is no information in that; it’s completely random. And yet, if the coins were quantum coins entangled in this way, if one of the coins came up heads, we would know for certain that the other came up tails. This would be true even if the coins were on opposite sides of the Universe. There is information stored in the state that prevents both the coins from ever coming up heads or tails at the same time, but it’s not stored in a way that is familiar to us. In a book, information is stored locally on each page and as we read more pages, the story gradually unfolds. In a quantum book each individual page would be gibberish and the story would reside in the correlations between the pages. We’d therefore have to get through a good portion of the book before we gained any insight into the story at all. No correlations, and therefore no information, would be discernible from a single page. The lack of information stored in a small part of a large, entangled quantum system is a very important property, which is central to the black hole information paradox.