Well, yes and no. In certain games you do gain an advantage this way over a classical opponent, it's called quantum pseudo-telepathy: https://en.wikipedia.org/wiki/Quantum_pseudo-telepathy. It sounds like quackery but it's a real thing.
If I understand it correctly, that's a really obtuse name for what is actually quite simple and has nothing to do with telepathy or information transfer. Basically the fact that both parties have access to the SAME random number generator can be useful in some schemes.
As an even simpler example, let's say Alice and Bob need to solve a really hard problem that can be cleanly split into two parts, A and B. In round 1 Alice and Bob can't communicate but they have access to an entangled photon. In round 2 they meet and compare notes. If they choose which part to work on randomly, then in only 50% of the outcomes are both parts A and B solved when round 2 starts. If, on the other hand, they use the polarity of the entangled photon to decide which part to solve, then quantum mechanics guarantees that they both solve different parts. It's random and unpredictable which part gets assigned to whom, but in 100% of outcomes when round 2 starts, Alice has solved one of A or B, and Bob has solved the other one.
I’m not sure if the simple example is faulty, but to me it seems like this can easily be done with a classical physics system. For example a white light beam passing through a randomly oriented prism and using mirrors to either send the purple or red light beam to Alice/Bob, don’t really need entanglement
The correlations achieved by separated measurements on entangled particles measurably exceed what is classically achievable. That's what's meant by "Bell inequality violations".
We can't transmit information instantaneously, but there are nevertheless certain distributed tasks we can do better at when we have a source of entangled states.
To paraphrase, Alice, Bob and Carol play a game where they can't communicate (after deciding on a strategy) and the referee shows each of them a bit. After being shown the bit they must reply with 0 or 1. The four possible combinations the referee chooses uniformly from is:
If the first combination was shown, the answers must have an even sum, otherwise the answers must have an odd sum.
First, without using probability, if A0 is the answer Alice gives when shown 0 and A1 when shown 1 (and similarly for Bob and Carol), you get the following set of equations needed to win always:
However, each term occurs twice on the left hand side, so when you add all equations up (mod 2) you would find the left hand side is even. However the right hand side sums to an odd number, thus all four equations can't hold simultaneously.
Now, probability doesn't help here, because any mixed strategy can be shown to be equivalent to a combination of pure strategies, none of which can guarantee a win.
Yet with a shared entangled state, it can be won 100% of the time. This isn't a probability thing - you can actually guarantee a win using the quantum strategy, even over arbitrarily large distances.
It depends on what you mean by transmit information. It is possible for one party to read a true random number generator (the state of the photon) and transmit that reading faster than the speed of light to another party (the receiver of the entangled pair photon).
The physicist won’t call this transmitting information, but the information scientist has no qualms about acting on random data. And once you give data meaning, it is information. Maybe both parties pre-agree that N zero bits in a row from the digitized reading of the entangled photons is a start signal, and the bits that follow are used to make choices in whatever action is carried out. Now the first party has “sent” an instantaneous message informing of their actions.
To be clear I don’t think there is a clever gotcha here. But it is helpful in constraining what is meant by information locality here.
There are several exciting applications that rely on a source of entangled particles. Super dense coding and quantum teleportation to name a couple. This is a step towards achieving those.
As an even simpler example, let's say Alice and Bob need to solve a really hard problem that can be cleanly split into two parts, A and B. In round 1 Alice and Bob can't communicate but they have access to an entangled photon. In round 2 they meet and compare notes. If they choose which part to work on randomly, then in only 50% of the outcomes are both parts A and B solved when round 2 starts. If, on the other hand, they use the polarity of the entangled photon to decide which part to solve, then quantum mechanics guarantees that they both solve different parts. It's random and unpredictable which part gets assigned to whom, but in 100% of outcomes when round 2 starts, Alice has solved one of A or B, and Bob has solved the other one.