Absolute amateur here with a question that was probably answered eighty years ago somewhere. Suppose we have three particles of a type such that if two are entangled, then if either particle is observed to be in state 0, then the other must be in state 1. Now suppose that A and B become entangled, then separate. Later, B and C become entangled, so that C is now bound up with both A and B. Then if A is observed to be in state 0, then B must be in state 1, whence C must be in state 0, whence A must be in state 1: uh-oh.

I see four possibilities here:

  1. Another amateur is ignorant of, or has failed to appreciate, subtleties. What are those?

  2. Some mechanism prevents the entanglement. What mechanism is that?

  3. The entanglement can happen, and the resulting object becomes in principle unobservable—dark matter.

  4. Quantum theory is broken.

Can anyone enlighten me? Thanks in advance.

  • $\begingroup$ Quantum theory is broken, of course. What else. $\endgroup$ – Norbert Schuch Dec 30 '20 at 23:40
  • $\begingroup$ Alternatively: 1, and thus 2. $\endgroup$ – Norbert Schuch Dec 30 '20 at 23:41
  • $\begingroup$ Why bring quantum mechanics into it? Imagine that I have three shoes, all of which are either black or white. A and B are opposite colors. B and C are opposite colors. A and C are opposite colors. But if A is black, then B is white, so C is black. Uh oh! What could possibly explain this? $\endgroup$ – WillO Dec 31 '20 at 4:03

The problem is that when particle B later becomes entangled with particle C, this new entanglement implies that its correlations with particle A have necessarily changed, and similarly when C becomes entangled with A its correlations with particle B must change.

Another way of stating the problem is to point out that I simply can't write down the final state which you try to create. A state with particle's A and B entangled, and not entangled with C, could be written as $(|01\rangle+|10\rangle)|\psi_C\rangle$. I can't there's no way to write down the state you describe because it's self-contradictory.

As @WillO points out you don't need quantum mechanics in order to write down self-contradictory correlations, but in quantum you can formalize the concept into something known as "The monogomy of entanglement" (see e.g. here or here). From the latter link,

If two qubits A and B are maximally quantumly correlated they cannot be correlated at all with a third qubit C. In general, there is a trade-off between the amount of entanglement between qubits A and B and the same qubit A and qubit C.

The initial entanglement you describe is I believe maximum entanglement, and so you can't further entangle particles with others without sacrificing some of the original entanglement. Hope this clarifies the issues at hand and gives you some routes for further reading.

  • $\begingroup$ Thank you, and thanks for the lack of sarcasm. To those who could not refrain from sarcasm: if the question wasted your time, surely the snarky answer wasted much more---unless, of course, your intention was to prevent further innocent questions. $\endgroup$ – devlin Dec 31 '20 at 6:28

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