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Comment by bmnmasdas | Hacker News Reader

I guess I was being generous when interpreting your statement that "all components of a quantum system are always entangled". For pure states, such a statement is trivially true in the sense any kind of interaction in the Hamiltonian will generically create entanglement. For mixed states, your statement is of course false. It is unfortunately often the case in the real world that two subsystems are in a non-entangled quantum state.

Regarding your general theme: I understand that confusion can arise if seemingly informal language is taken verbatim as a formal statement. I do not think that the solution is to abolish the former, which can be extremely efficient to reason in and communicate with (you gave some examples in your post), but rather to educate on the interpretation. It is unfortunate that this is not always done, as your experiences suggest.

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You have good points. However I'm going to be a little nitpicky here. Mixed states aren't exactly "real" in the sense that what is a pure state for Alice can be a mixed state for Bob (I can provide an example if you don't know what I mean). So in that case the "lack" of entanglement is simply due to ignorance of which pure state the system is actually in.

The point is that entanglement always refers to an a priori choice of subsystems (say, Alice and Bob). This is the part that makes the phenomenon non-trivial. If there are other systems around (say, Eve the environment) then the joint state of Alice and Bob will be usually be mixed as a consequence of for the "trivial" reason that we discussed in the previous posts (to adapt a famous saying, almost all components of a quantum system are always mixed ;-). There is nothing "unreal" about mixed states, and not all mixed states lack entanglement. However, for mixed states, being entangled is no longer the generic behavior. The unavoidable interactions with the environment are the reason why it is hard to maintain entanglement between subsystems.

To say that we should do better and bring the environment back into the picture is missing the point if we are interested in the correlations between Alice and Bob. These do exclusively depend on their joint state (mixed or not).

Going to watch interstellar. Let's see what entanglement really is!

(Sorry for continuing the rather long discussion, but I'd like to achieve some resolution here.)

By "real", I mean that the concept of a density matrix is derived completely on top of the postulates of quantum mechanics in combination with the Born rule. There's nothing fundamental about mixed states; you can also have classical mixed states (e.g. deriving thermodynamics from classical statistical mechanics). It's essentially just taking the postulates for pure states and applying a layer of statistics on top of it. I believe it was von Neumann that originally did this? In other words, the density matrix formulation does not add any additional predictive capability to physics that the original QM from the 1920s did not already provide. It's just a more convenient tool for connecting QM to experimentally realizable systems. Do you disagree with this?

When you lose entanglement due to decoherence (specifically, the off-diagonal terms of the density matrix approaching zero), these correlations are lost because you're essentially performing a measurement. But they still exist in the whole Alice + Bob + you and your measurement device system! But then, this starts treading into the discussion of the whole unsolved measurement problem which I kind of wanted to lurk around, since no one ever gets anywhere with those discussions.

Indeed your last point ("To say that we should do better and bring the environment back into the picture is missing the point") is essentially the whole picture I'm focusing on. Perhaps my background with quantum chemistry has slanted the way I explain things on here, because you're never collapsing these systems when you perform simulations of them to calculate their properties.

The density matrix is as real as the wave function when it comes to describing the corresponding subsystem. In the situation I was sketching, there is no measurement, no collapse, and the "measurement problem" does not play a role. Here is a concrete example: Suppose that you have three spins that are in a superposition of |000> and |111>. Alice has one the spins, Bob the other spin, and the third one belongs to the environment. The reduced state of any two of the three spins is NOT entangled. Therefore, Alice and Bob which will not be able violate any Bell inequality, win a CHSH game, distill Bell pairs, etc. if they only control two of the three spins. It is irrelevant that Alice is entangled with the joint system of Bob and Eve.

Again, the basic point is that the notion of entanglement refers to a choice of subsystems. Your statement that "All components of a quantum system are always entangled" is either trivializing the discussion or demonstrably false.

See my message to Lisper.

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