The GHZ Gate

The garden gate

This gate depicts the famous “GHZ” experiment (named after its inventors, Greenburger, Horne and Zeilinger), which demonstrates a very strange property of quantum mechanics, called non-locality. This means that quantum measurements carried out in different locations can show spooky correlations (“spooky”, or more precisely “spukhafte”, is Einstein’s adjective).

To understand this, the first thing we need to know is that quantum mechanical measurements (at least those that interest us here) give binary answers. One can think of the spin of a particle as a little arrow pointing in some direction in space. In traditional physics, we could measure this direction and get an answer like “20 degrees to the vertical in a westerly direction”. Quantum mechanically, we can ask if the spin is “up” or “down” and get a definite answer; if it is “up” that means the arrow is more likely to be pointing in a general upwards than downwards direction. Or we can ask if the spin is “east” or “west” and get a definite answer (and some statistical information about the arrow’s westward component). However — like a fairy-tale game — we can only ask ONE question about a given particle. This is because the direction of the arrow is changed when the measurement is made, so the original information is lost.

It is possible to prepare two particles so that measurements of the two of them always give opposite results. If one particle has its spin up, the other has its spin down. If one gives east the other gives west, and so on. There is nothing spooky about this yet, because the particles could have been pre-arranged on creation so that one was up and the other down (and likewise east and west). However, it turns out that there are certain ingenious experiments that allow one to rule out such a pre-arrangement. John Bell showed this, in a revolutionary paper he published in 1964. After this, there was no escape from spookiness. What it meant was that the particles cannot have their responses prescribed at creation but must communicate with each other to decide how to respond in a coordinated way to measurements. This would still be true if they were light years apart, which seems to contradict relativity (though by a saving clause — the randomness of quantum mechanics — it actually does not do so).

The garden gate portrays not Bell’s original experiment, but another experiment that leads to the same conclusion in a particularly beautiful and compelling way. Instead of two particles, it uses three. To make its predictions vivid, we imagine experiments carried out on three planets: Earth, Mars and Venus, say. The three particles are created in a certain special way on Earth. One is kept on Earth, another sent to Mars and the last one to Venus. On each planet we can carry out one of two kinds of measurement, called “X” and “Y”. These are measurements of spin in particular directions. Instead of calling the X measurement outcomes up or down, we shall just call them +1 and -1. Likewise we also call the Y outcomes +1 and -1, instead of east and west.

Let us assume we only carry out one measurement, either an X or a Y, on each particle on each planet. Whether we choose X or Y is decided at random, but only certain combinations of measurements are considered. To be specific, we count measurements where X is measured on Earth, Y on Mars and Y on Venus, denoted by the triple “XYY”, and also triples YXY, YYX and XXX. We ignore cases XXY, XYX, YXX and YYY. On the gate, the four triples we count are shown in four rectangles.

The GHZ experiment asks for the three particles to be put into a special quantum state initially (before being sent off on their interplanetary journeys), which is shown in silver on the gate. The notation for the state is due to Dirac, and is written with “kets”, for instance |000>. (A ket is the right-hand half of a bra(c)ket.) Now, the calculational rules of quantum mechanics tell us that when the particles are in this state, and when we measure the XYY triple, if we multiply together the individual +1 or -1 measurement results from the three planets we will always get a -1. Similarly, we will get a -1 by multiplying the results from the triples YXY and YYX. From XXX, however, we will get a +1.

Suppose therefore that the outcomes of measurements were prearranged before separation. Could we find a particular set of assignments of X and Y values to each planet that would explain the quantum mechanical predictions for each of the four triples? The following table shows a possible set of preassignments.

From this table, the measurement of XYY would be the product of X on Earth (+1), Y on Mars (+1) and Y on Venus (-1), which is (+1) x (+1) x (-1) = -1. Carrying out the same procedure for the other triples we find that YXY gives -1, YYX gives -1 and XXX gives -1. This is correct for XYY, YXY and YYX, but incorrect for XXX.

There are 64 ways of assigning +1 or -1 to the six positions in the table. Surely we can choose one that makes the answer come out the way quantum mechanics predicts for all four of the triples? Surprisingly, we cannot. The easiest way to see this is to note that in the four triples XYY, YXY, YYX, XXX, X occurs twice in the first position (in the first and fourth triple) and so does Y (in the second and third). The same is true for all three positions. Thus when we multiply all the values, +1 or -1, for each triple, XYY, YXY, YYX and XXX, we must get +1. (This is true for the example given by our table where we get -1 for all four triples, which multiply together to give +1.) However, for our quantum mechanical predictions the product is -1. Thus no assignment of values to X or Y at each location can explain the quantum mechanical result. Thus it is not possible to preassign the measurement results, and so somehow the particles must collaborate in giving the correct results, even though they lie at vast, interplanetary, distances apart. This is indeed a very strange conclusion.

The gate was made by Neil Fisher, who has also made other pieces of notable ironwork in Cambridge.

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