That doesn't give results that match how entangled particles behave.

Imagine you and a friend found a mysterious device that consisted of some sort of two port docking station with two hand-held devices docked. There is a button on the docking station that if pressed with both devices docked beeps and then both devices turn on. On each device a screen lights up, displaying the number 1000.

Each device has three buttons, labeled A, B, and C, and a green LED, and a red LED.

You and your friend each take one of the devices and go home to experiment with them, agreeing to meet the next day and compare notes.

What you both find out is that if you press more than one button at a time nothing happens. If you press a single button one of the LEDs flashes once, and the number on the screen goes down by one. While the LED is on the buttons appear to do nothing. When the number on the screen reaches 0, the device appears to shut off, so the displayed number is evidently a count of how many presses you have left.

You both keep a log of what buttons you pressed, at what times, and which LED flashed.

Just looking at your results, for each button you get red half the time and green half the time. When you look at N-tuples of consecutive presses, each of the 2^N combinations of red/green seem to be equally represented. Same when you look at groups formed any other way you can think of.

Every statistical test you try fails to distinguish whatever the devoce is doing from a hypothetical device that is using a uniformly distributed random bit generator to pick the color when you press the button.

The next day when you compare notes you find that when the number on the counter was the same:

1. If you both pressed the same button, you got the same color light.

2. If one of you pressed B and one of you pressed A or C, you got the same color 85% of the time.

3. If one of you pressed A and the other C, you got the same color 50% of the time.

If you compare presses when the counter is not the same, such as comparing your odd presses to your friend's even presses you got the same color 50% of the time.

You both put the devices back in the dock, press the dock button, and the devices are reset. You both take them and go home for another round of testing.

The first thing you do is 20 presses in the same sequence you used for round 1, and compare the results to round 1. You find today's Nth press matches yesterday's Nth press 50 % of the time. So whatever the dock did, it didn't just set the device to repeat yesterday.

The obvious guess at how these devices work is that the dock loads each of them with a 1000 entry table that contains three columns, one for each button, that says what color to show. But if you try construct such a table you run into problems. The tables have to be identical in both devices to get 100% color matching when the same button is pressed. If you construct tables that get AA, BB, CC right, and that also get AC right (50% match rate), then you can't get AB and BC both right. If you make AB and BC the same they end up at 67.5% which is well short of the 85% we want. You can fiddle the tables to raise one of them without breaking AB, BB, CC, and AC, but that lowers the other one.

Next guess would be the devices communicate. For a given counter value the device that first gets a button pushed might pick its result (from a static table or via a random number generator) and communicate that somehow to the other device. The other device can then take that into account, matching if the same button was pressed, and using a random generator otherwise with the probability set based on the button pair.

To rule out communications you and your friend try a few rounds where before pressing any buttons you take the devices far away from each other--far enough that you each finish your 1000 presses before anything non-FTL could reach the other device from yours. This makes no difference.

OK, so how the heck do the devices work? Answer: entanglement.

The dock generates 1000 pairs of entangled photons. Each device gets a photon from each pair, and remembers which pair number they came from.

When you press a button the device takes the photon from the pair number corresponding to the counter, and checks to see if it is polarized in a certain direction. If it is the device lights the green LED and if it isn't the red LED lights.

Which button you press determines the direction of the polarization check. If you press A the check is at 0 degrees from a reference direction. If you press B it is 22.5 degrees from the reference direction. If you press C it is 45 degrees from the reference direction. The way polarization works with entangled photons is that if I measure one of them at some angle and you measure the other one at an angle that differs from mine by θ the probability that we get the same result is cos^2(θ). If we pick the same direction, that's 1, if our directions differ by 22.5 degrees is is 0.8536, and if our directions differ by 45 degrees it is 0.5.

Note that if it was somehow pre-defined when the entangled pairs were created how they were going to respond when measured after a button press they could not give the above results for the same reason that the preloaded table approach failed.

And that's not even trying to pre-define for every angle. It's only 3 and it already has fallen apart.