A2/Discussion of the story
Do these Quantum-lottery-tickets exist?
At the end of the story we stated, that Bob has encountered a kind of state that can only exist within quantum physics, and not within classical physics. We will now draw conclusions from the story and discuss the notion of (quantum-) entanglement. We start by discussing how to realise these quantum states in practise. “Quantum-scratch-cards” as depicted in the story cannot (yet) be manufactured in this robust form, but can be realised for instance as state of two photons. One photon, travelling to Bob, represents the left side of the scratch-card, the other photon, travelling to Alice, represents the right side. Opening either the front or the back side of the ticket corresponds to either performing a polarisation measurement in linear or in circular basis. Such pairs of photons can be prepared in a state, which displays the behaviour mentioned in the story. We will come back to the layout of such an experiment later in section B.2.
We note here, that the numbers given in the story (e.g. the 15% observed non-coincidence) are realistic. In fact, the minimum number compatible with quantum mechanics is slightly lower, namely . Even within quantum mechanics it is not possible to go to smaller values, and the bound on the quantity is also called a Tsirelson-bound .
Are there hidden variables?
When making his deducations in the story, Bob first assumes that the behaviour of the scratch cards can be explained by hidden variables. This would be the case, if the cards had been manufactured by simply printing ink on a card and then hiding the symbols under the scratch-material. This would also imply that the value of each field would be predetermined and opening the fields would simply reveal what had been hidden there before. We have also called this a “passive scratch-card”. The term “hidden variable” refers to the fact, that Bob cannot reveal both sides of the ticket due to the rules of the game, so some of the parameters will always remain hidden to him. In quantum physics, restrictions on which parameters can be jointly measured occur regularly. One example of such a pair of non-jointly measurable parameters are “position” and “momentum” of particles. Such pairs are called “complemental” to each other and the quantitative relation of how well they can be jointly measured is called an “uncertainty relation” (compare course A.1).
In the early year of quantum physics, it had been a question if the physical quantities would allow for a similar description with hidden variables. If this was possible, then the value of any measurement result would in principle be determined before the measurement, and the measurement would simply reveal the value. But as we have seen in the story, any such model would be in conflict with the observations. To make this point more clearly, we will in the next section construct a hypothetical machine, called the Bell telephone, which shows that the existence of hidden variables in quantum physics would lead to a violation of another fundamental principle, namely locality.
The Bell telephone – an impossible machine
In this section we assume, that Bob has at his disposal a machine, which allows him to simultaneously open both sides of his lottery ticket. We will show, that this would allow superluminal information transfer from Alice to Bob. From this we can follow that such a machine is impossible.
The protocol is as follows: Suppose Alice holds a message that she wants to transmit to Bob, and that this message is binary coded as a string of 0s and 1s. They both share many of the lottery tickets from the story, which they open on a predetermined schedule. Whenever Alice wants to transmit a 0 to Bob she opens the test field on her card, and when she wants to transmit a 1 she opens the winning field. Suppose now, that Bolds this special machine that enables him to open both fields on his card. To reproduce the correlations from the story, the symbols on his card have to coincide with the symbol on Alice’s side most of the time, whenever she opens the test field. This also implies that the two symbols on Bob’s side have to coincide. Likewise, when Alice opens the winning side of her ticket, the symbols on Bob’s side have to be different. So, whenever Bob opens both the fields on his ticket and observes identical symbols, he will interpret this as a 0, while he interprets different symbols as a 1. Using this protocol, it would be possible to send a message to Bob. As the coincidence probabilities in the story were not 100%, but 85%, Alice will have to add some redundancy to the communication to make it reliable.
From the last paragraph we know, that Alice can send a message to Bob, using the quantum lottery tickets from the story and provided that he possesses a machine that allows him to open both sides of his ticket. For this communication to happen, it would not even be necessary for Alice to send any physical signal to Bob – she can choose to open either the front or the back side of her ticket at any point, and the observation on Bob’s side has to match her decision instantaneously. With this, they would be able to communicate without loss of time over arbitrary distances.
Even though this might sound nice, it would not be compatible with the physical principle of relativity, which states that no signal can travel faster than the speed of light. This is a strong argument, why such a measurement device capable of measuring both sides of the ticket cannot exist.