What happens to bells inequality when you disregard the free will assumption?

You find that you can predict experimental results in a deterministic hidden variable allowing way.

Quantum mechanics may be even spookier than we thought
Quantum mechanics is inherently statistical in that it can tell you the probability of something like a nucleus emitting an alpha particle in a given time, but it can’t tell you exactly when or how. In the early days of quantum mechanics this caused great consternation for many scientists, including Einstein whose dislike of this apparent randomness prompted him to protest “God does not play dice!”

Einstein and others proposed what’s now known as hidden variable theory, to get some causality back into the quantum world. In essence this says that there are mechanisms within the nucleus that lead to the emission of the alpha particle in a deterministic way, but we can’t see them so they appear random to us. However in 1964, the physicist John Bell published a famous paper in which he argued that no hidden variable theory can reproduce all of the observed quantum phenomena.

A well-known and intriguing aspect of Bell’s work are what’s known as Bell inequalities. Bell considered a situation in which something like the decay of a nucleus emits two particles simultaneously that move in opposite directions.
One of the key features of quantum mechanics is that each such emitted particle exists in a superposition of every possible state until a measurement is made, at which point they condense into a single real state. In this way, it’s the actual process of making measurements that in effect “creates” reality – strange but true.

Conservation rules also dictate that if the spin of one such emitted particle is up, then that of its twin must be down. But of course until a measurement is actually made, each particle has both spin up and down at the same time. Physicists call this quantum entanglement - two particles whose states depend on each other but where both particles still exist in all possible states because no measurement has yet been made on them.

In a Bell inequality scenario, a measurement made on one such entangled particle causes both it and its twin to instantly condense into a single real state, even if the two are separated by vast distance. This notion was also intensely disliked by Einstein, who termed it “spooky interaction at a distance.”

However, Bell showed Einstein could not have his cake and eat it too. His “Bell inequality” paper proved that any hidden variable theory for quantum systems, in which observers had free will to choose what they measured, was forced to have such spooky interactions!

Of course people immediately began to ask if a Bell inequality scenario, or something like it, could be used to create faster than light communication? The basic idea is that a pair of quantum entangled particles are sent in opposite directions from a location midway between two distant observers. If a measurement is made on one, perhaps by an alien civilization in a distant galaxy, that would instantly affect the state of its twin here on Earth so surely that’s faster than light communication right?

Unfortunately the universe doesn’t work that way. The problem is, that although the effect may be instant, if we measure the spin of our particle as up, we don’t know if the aliens caused that by measuring theirs as down, or if it was our own measurement that condensed the sate. There are exactly as many ups as downs so there’s no way to tell. To find out you’d have to send a message to the aliens and ask them what they did and of course that would have to be at sub-light speed. It would seem therefore that no faster than light signalling or information transfer is possible. However that’s not to say that the physics of Bell inequalities isn’t interesting.

One scientist looking into the interpretation of quantum mechanics and what might or might not be possible within Bell inequalities is Dr Michael Hall, a visiting fellow at the ANU Department of Theoretical Physics.
“It’s been shown that you can’t have no-signalling, determinism and experimental free will all together in a world described by quantum mechanics, “ he says, ”you have to give up some or all of at least one. But how much of each one is a really interesting question.”

Dr Hall has recently published a paper in the prestigious journal Physics Review Letters suggesting that if you give up just a little experimental free will you can accurately model a Bell inequality using the kind of deterministic hidden variable physics Einstein might have loved.

“I’m looking at what’s known as a relaxed Bell inequality, that is one in which the need for absolute free will is relaxed slightly. Do that, and you can make it work deterministically without evoking faster than light signalling.”
If this is indeed how the universe operates, the very strange implication would be that the apparent randomness in the spin orientation of entangled particles may be weakly coupled to the apparent randomness with which an experimenter chooses when and where to measure. In other words your free will might not be as free as you think!

“My model sidesteps Bell’s theorem, by allowing the same underlying variable that predetermines the measurement outcomes to have a small statistical influence on the choices of measurement made for each particle.” Dr Hall explains, “This influence, called ‘measurement dependence’, is not directly observable but leads to the correct quantum correlations”

Of course that slight statistical shift in the “free” choices made by the measurer are not limited to human experimenters. If a random number generator were used, there may also be a shift in its choices though statistically the numbers it created would appear perfectly random.

How could this still appear random? Well imagine tossing a coin four times. Heads, heads, tails, tails is statistically 50% so is heads, tails, heads, tails. The first might be influenced by measurement dependence the second not – how would you know?

Such an effect may have implications for quantum cryptography which relies on entanglement to send secure signals. If someone taps into the system, the entanglement is lost and the eavesdropper is sprung.

“Quantum cryptography is basically sound from a physics point of view,” Dr Hall says, “but our work opens up the question of the sending and receiving devices being tampered with. If someone has ‘monkeyed’ with your equipment, you may be exposed to data leaks hidden within the seemingly random statistics without that being at all obvious.”