Entangled States and the Mystery of Wave Function Collapse

[edpell]

Is spin the only physical parameter that can be entangled? If not, what else? If so, it seems there is something special about spin?

 

[Nugatory]

Is spin the only physical parameter that can be entangled? If not, what else? If so, it seems there is something special about spin?

No, not at all. In principle just about any observable can be entangled, and in practice we're limited only by what states are easily prepared. Most of the Bell experiments have been done with polarized photons, because they are easier to produce and to measure than spin pairs. The original EPR paper considered position and momentum.

 

[tom.stoer]

Is spin the only physical parameter that can be entangled? If not, what else? If so, it seems there is something special about spin?

I would not call it "parameter" but obervable.

An observable is a quantum mechanical operator with eigenvalues and eigenstates which can be used to label the results of measuments. In quantum mechanics we can have several obsvables, not all of them are mutually "compatible", but assume we have a maximal set of compatible observables {A, B, C, ...}. Then a quantum state is something like a set of labels |a, b, c, ...) with eigenvalues a, b, c, ... of the observables. Having such a state means that we can be sure that measuring A, B, C, ... will have the results a, b, c, ...

An entangled state w.r.t. an observable A is nothing else but a linear combination p|a, b, c, ...) + q|a', b, c, ...) where we have the probability p^2 to find a and the probability q^2 to find a' as result of the measurement on A.

So entanglement depends on the set of obsvables you chose to describe the system; and this depends e.g. on the experimental setup, what you want to measure. Observables can be energy, momentum, angular momentum, spin, ... (not all of them being mutually compatible!)

 

[Khashishi]

Entanglement has more to do with our abstract representation than reality itself.

When we describe some state in quantum mechanics, we have to choose some basis to label the state with. For a simple non-quantum analogy, consider colors. We might describe the color brown as "0.3*red + 0.25*green + 0.03*blue", using the red-green-blue basis. Or, this color can be expressed as something like "0.4*yellow + 0.1*magenta - 0.04*cyan" (I made up the numbers. Don't check them for correctness.). By changing the basis, we didn't change the state at all, only our abstract representation.

Now, consider we have a hypothetical green filter, which blocks the green portion of a color and let's the blue and red pass through. (We are talking about the color green, not the wavelength green, so such a filter probably doesn't exist in real life.) If we pass a random color through the green filter, we gained some information about the state. We know 1 out of three numbers that describe the state, and so there are 2 unknowns. If we switch over to the cyan-magenta-yellow basis, we can no longer write the state in terms of 1 number we know and 2 unknowns, because we don't know the value of any of the three numbers. But, we didn't actually lose any information about the state since we only made an abstract representation change. So we represent the state as a linear combination of states--an entangled state.

All of the mysteries regarding entangled states are actually mysteries of wave function collapse, and this weirdness persists whether or not we use entangled states. Using entangled states just makes it easier to see this weirdness in some circumstances.