Quantum Nonlocality - Example

Example

Imagine two experimentalists, Alice and Bob, situated in separate laboratories. They conduct a simple experiment in which Alice chooses and pushes one of two buttons, A0 and A1, on her apparatus, and Bob observes on his apparatus one of two indicating lamps, b0 and b1, lighting. In this case there are four possible events that could occur in the experiment: (A0,b0), (A0,b1), (A1,b0) and (A1,b1). Suppose that after many runs of the experiment, only the events (A0,b0) and (A1,b1) occur; this is good evidence that A has an influence on B. Indeed, Alice could easily send messages to Bob by encoding those messages into sequences of 0's and 1's, and causing the b0 or b1 lamp to light up respectively.

More realistically, suppose that the four events occur with (conditional) probabilities P(b0|A0), P(b1|A0) = 1 - P(b0|A0), P(b0|A1) and P(b1|A1) = 1 - P(b0|A1). Here P(b0|A0) is the probability that Bob's b0 lamp lit up, given that Alice pushed the button A0. We can still rigorize the notion that A has an influence on B in this setting: if P(b0|A0) differs from P(b0|A1) then Alice's choice of button still affects the probabilistic outcome on Bob's side, and it is still possible for Alice to send Bob messages with low probability of error. For example, if P(b0|A0) = and P(b0|A1) =, then after 100 runs of the experiment in which Alice pushed the same button, Bob can tell with high probability which button it was by looking at how often b0 occurred.

Here is a more complicated scenario: Alice pushes one of two buttons, A0 and A1, and Bob also pushes one of two buttons, B0 and B1. Alice observes one of two outcomes, a0 and a1, and Bob also observes one of two outcomes, b0 and b1. There are 24 = 16 possible combinations of these 4 events:

where each of X,Y,x,y is 0 or 1. Suppose that of these 16, only 8 combinations actually occur, with the following (conditional) probabilities:

where denotes addition modulo 2.

Then if A1 and B1 are both pressed ( (A1,B1) chosen) the outcomes are perfectly anticorrelated, either (a0,b1) or (a1,b0), with an equal probability for both occurrences. In all other cases the two outcomes are perfectly correlated (either (a0,b0) or (a1,b1), again, equiprobably.

Do these outcomes imply that some influence exists (A on B, or B on A), or not? The question is important, since the answer depends on our fundamental assumptions about how mathematical theories describe physical reality.

On the one hand, Alice cannot send a message to Bob, using her buttons A0, A1 and his indicators b0, b1 (nor Bob to Alice). In this sense there is no influence of A on B, or of B on A, since it is easily checked that P(bx|A0) = P(bx|A1) for both x = 0 and x = 1 in the above example. That is to say, this particular set of probabilities is non-signalling.

On the other hand, it is provably impossible for two separated parties to simulate this outcome without any kind of interaction or communication between them. Thorough logical analysis reveals that the above outcome can only occur if there is some direct influence between A and B, if we assume local realism and, arguably, counterfactual definiteness. These fundamental assumptions, deeply rooted in our physical intuition, are incompatible with quantum theory. Different interpretations of quantum mechanics reject different parts of local realism and/or counterfactual definiteness (for detail, see Principle of locality). A classical definition of nonlocality, i.e. direct influence of one object on another, distant object, normally takes local realism and counterfactual definiteness for granted.