[jeffreyH (OP)]

If this is so can no two electrons in the entire universe share the same quantum state?

[evan_au]

Sean Carroll has discussed similar questions on his Mindscape podcast. As I understand his comments:
A quantum-entangled system coming in contact with an outside particle becomes entangled with the state of that particle.
- But because you generally don't know the quantum state of the outside particle, your quantum system goes into an unknown state.

Extending that, if your carefully controlled quantum-entangled system comes in contact with an uncontrolled outside system, your controlled system becomes entangled with the state of the uncontrolled system, and also enters an unknown state
- That's why quantum states are very delicate and are isolated from all external influences (as far as possible)

can no two electrons in the entire universe share the same quantum state?

I would place a constraint on this:  since a particle cannot influence another particle outside its light cone, I would suggest that the question be rephrased as "can no two electrons in the same light cone can share the same quantum state?".

For a single, isolated atom, the Pauli Exclusion Principle ensures this.
- For atoms that are brought close together (in a gas), the spectral lines are broadened, as the outer electrons adopt very slightly different energy levels
- In a metal, the electrons in the conduction band can take on quite a wide range of energies - the conduction band.

This broadening of energy levels does not apply so much to inner electrons, which are partially shielded from adjacent atoms by the outer electrons.
See: https://en.wikipedia.org/wiki/Spectral_line#Line_broadening_and_shift

Can the whole universe be considered a quantum system?

According to the Big Bang Theory, at one time the universe existed within the same light cone, so there was the possibility that all particles could have been entangled with every other particle in the universe. That produces a quantum state that is too complex for us to understand...

I can't comment on whether the universe actually contained electrons at that time, or perhaps some other, more exotic particles (such as the LHC researchers have been seeking).

So even if we manage to create a controlled quantum state in a 50-odd qubit quantum computer (as Google claims)...
- We must carefully isolate it from the rest of the universe, as any interaction will create an unknown quantum state in our quantum computer   

The dream of quantum computer designers is to develop quantum error correction, so that if the quantum computer does come in contact with (say) 1 stray electron, there will be enough redundant state in the computer to deduce what impact that electron had on the quantum computer, and undo the change, without losing the integrity of the ongoing quantum computation.
- Of course, if the quantum computer comes into contact with a sufficiently complex uncontrolled system, it will overwhelm the  capacity of the error correction, and the quantum computation will need to be restarted.
See: https://en.wikipedia.org/wiki/Quantum_error_correction

[alancalverd]

I think it all depends on what you mean by "the same". Suppose we polarise the unpaired spins of a sample in a homogeneous magnetic field. Then with respect to the field, they are all in the same state, but because they are in different positions in the sample, they are in individual states. If we reverse the field, the spins do not all flip at the same time but follow a relaxation curve.

[jeffreyH (OP)]

@evan_au I wasn't thinking in terms of entanglement, just generally.

[jeffreyH (OP)]

Does this sum things up?
https://en.m.wikipedia.org/wiki/Relaxation_(NMR)

[Paul25]

From a frame of reference isolated causally like a black hole or inside the atom.