Quantum mechanics talks of electrons being connected at a distance, sometimes kilometres apart. What makes a pair of electrons connected? Presumably it's not just any pair of electrons, or a nominated pair? What does the pairing?

Quantum mechanics talks of electrons being connected at a distance, sometimes kilometres apart. What makes a pair of electrons connected? Presumably it's not just any pair of electrons, or a nominated pair? What does the pairing?
Ignore ralfcis; he has no clue what he's talking about. Most of what he's written is either meaningless gibberish, a random assortment of semirelevant terms mixed with irrelevant ones, or just completely wrong. You'll never sort through it, so just pay no attention.
Don't bother googling "cooper pairs." They're not particularly relevant to your general question (they do not exist in a vacuum).
If you're going to google anything, google "quantum entanglement" first. The wikipedia article is not a bad place to start. You'll see that if particles are prepared or otherwise arranged in a way that places constraints on their states, measurement of some of the states may suffice to make definite statements about the asyet unmeasured states. A canonical example is a pair of particles created in a way that their total spin is zero. If one particle is then found to have a clockwise spin, we immediately know that the other must have a counterclockwise spin.
Or a nonquantum example: I have a box with a silver coin and a gold coin (we have prepared an initial entangled state). I blindly pick one and send it to Bob, and send the other to Mary. Bob tells you that he has received a gold coin, and thus you know immediately that Mary has the silver one. Having prepared the initial state limits the possible outcomes to the point where a single measurement suffices to make a determination about both "particles". There's more to the story, but that should help you get started.
The answer to your question is that when particles form they form in pairs and are entangled by the fact that they are opposite of each other ie a particle and an anti particle, an positive charge and a negative charge, a number of other opposite aspects too.
What makes the entanglement unlike the left shoe right shoe or gold coin silver coin examples is that to find out what particle one has it has to be subject to a magnetic field. That tells you if it is a + or  it is 50/50 it could be either. However that's not all there is to it. When you apply the field and make the measurement you randomize the particle. The next time you measure it it could still be either + or . Like if you put a pair of shoes in two boxes and check one, and it's left then you check it again could be a right.So the idea is that every application of the field randomizes the particle. The thing that happens is that if you apply the field 10 times we assume the particle has randomly changed from + to . We don't know how many times it changed. Maybe it was + 8 times. Maybe ir was + 3 times. Maybe 5/5. We don't know. We didn't check every time but we do know that if we had it would be a random result. But when after we have applied the field 10 times and on the tenth time we check the result the other particle when we check it it is opposite.
There are a couple of interpretations of this. One is entanglement. They are connected in some unknown way that allows them to transmit their state to each other even faster than light. That's why it's unknown. We don't know of anything that allows anything to go faster than light. Our models of physics forbid anything from going faster than light.
The other interpretation is called the many worlds interpretation. That's where any random event, a flip of a coin for instance, Is destined from the beginning of that universe to come up heads. In another universe the result was destined to be tails. It looks random to anyone in any universe but it really isn't. It requires infinite universes, and absolute determinism in all of them. That any flip of a coin was determined by the initial conditions in the formation of the universe billions or trillions of years ago. There are an infinite number of universes that are exactly like this one except for this time in this universe the coin flipped heads and in another one it flipped tails. In fact there are an infinite number that are exactly like each other except for the ons where the couns flip heads and the others flip tails. That's the many worlds model.
Entangled particles effectively share the same wave function, until they decohere. They behave as if they are the same particle until as pointed out above are measured or disturbed in anyway.
No. That's tecnobabble bs that poorly describes something else. The dual slit thing.
There are some normal scale things with no "wave function collapse" involved that act in exactly like entangled particles except for the faster than light spookiness. Consider the rear wheels of a toy car. The kind that has the wheels connected by one axle. Turn the right one clockwise the left turns counter clock. Put both in separate boxes with a magnet on one of the wheels and pass a magnet by the box we know depending on which way the magnet on the wheel in the box is facing will cause the wheel to spin either clockwise or counter clock. We also know that if we look at either wheel that the other will be spinning the opposite. We can replace the axle with a long cable and still the wheels will spin opposite. Problem is that the change of spin has to go from the wheel with the magnet to the other wheel through the axle or cable and can only go at the speed of light. So they're both going the same way for a period of time depending on how far apart they are. But particles must always be paired with an opposite. Otherwise the universe is unbalanced, more particles than anti particles, and that hasn't happened since the big bang. The only way to solve that problem is for the paired particles to connect their state by some unknown means faster than light. That's why we say the particles are entangled rather than connected.
That's by the entangled interpretation
In the many worlds interpretation there is no connection between the particles. We just happen to be in a universe where when we check the particle pairs they happen to be opposite. In some universes they happen to be the same. In some they happen to be randomly unrelated. It's like we are in a universe where no matter how many times we flip a coin it always comes up tails. But in other universes the coins come up random. Like when I flip a coin I always get all the luck. There is no physical reason for it. Nothing special about my relationship with coins. Could just as easily be someone else who gets all the luck. Just happens to our universe that gets all the opposite paired particles when we check. That's the many worlds.
So there's your choices. Faster then light entanglement or many worlds unlikely million, and billions, and trillions of head flips in a row. Some people get all the luck.
You do that a lot. Throw out phrases you heard somewhere that you obviously don't understand because you apply them in to situations they don't apply to. You don't explain what you think they mean because you think you can't be told what you think they mean is wrong. Holographic principle is a black hole thing. Wave function decoherence collapse is a dual slit thing as I already said. This question is about spooky action.
The many worlds is a proven optional interpretation that has the advantage over the proven optional interpretation of spooky action in that it does not violate the impossibility of faster than light. Spooky action has the advantage of not requiring the unproven existence of multiple universes. Like we don't have to flip a coin ten googleplexes of times to prove that it is not impossible for all of them to come up heads. That's not how math works. In fact in mathematical infinity it is certain that there would be ten googlplexes of heads in a row. Even an infinite number of sets of ten googolplexes of heads in a row. That's how mathematical infinity works.
Because photons don't come in pairs of photon anti photon so are not entangled with anything. There's no wave function involved in entanglement. Totally different subject When words and concepts from one subject are ignorantly applied to another it show he understands neither. It's technobabble because it's baby talk.
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