Bell tests: Singlet vs Triplet State

[jed clampett]

This topic came up in another discussion and I said I’d start a new thread for it. I had speculated that to experimentally demonstrate Bell violations, you needed particles in the spin singlet state; in particular, photons in the spin triplet state would not be especially distinguishable from classical light. I haven’t actually worked it out for photons but I think I know how to do it for electrons, so I’m going to try and sketch it out here.

Let’s recall what the singlet vs triplet states look like. It’s hard to believe that states which look so similar can have very different properties:

spin singlet: |up*down> - |down*up>
spin triplet: |up*down> + |down*up>

It’s apparent that if the first particle is measured spin up, the second one must be spin down. The two states appear to behave the same. But that’s only if you measure them in the preferred reference frame. Let’s see what happens if you rotate them 90 degrees.
In a rotated frame the states that were originally pure up become a mix of up and down:

up => up + down
down = down + up

(Obviously I haven’t bothered to put in the normalization factors.) If we substitute these changes into our expression or the spin triplet state, we get:

spin triplet: |(up + down)*(up – down)> - |(up – down)*(up + down)>

When you multiply this out and re-normalize, it turns out that every combination of up and down occurs with equal probability:

|up*up> - |up*down> + |down*up> - |down*down>

So measured in this direction, you get no correlation. But a very funny thing happens when you apply the same transformation to the singlet state. Terms cancel out (you can try it yourself) and you end up with exactly the same state you started out with.

For the triplet state, you get different correlations at different angles, and to calculate the expected result you just have to average over all angles. What it comes out to is basically the same as what you’d expect realistically if you sent off two electrons with opposite but unknown spins. The result for singlet state is very different. It’s as though the spin is identically zero everywhere in space until one of the electrons is actually measured.

 

[alxm]

A triplet state has two spins with the same orientation.

 

[jed clampett]

Jed Clampett wrote:

Let’s recall what the singlet vs triplet states look like. It’s hard to believe that states which look so similar can have very different properties:

spin singlet: |up*down> - |down*up>
spin triplet: |up*down> + |down*up>

Alxm replied:

A triplet state has two spins with the same orientation.

I looked up triplet states on Wikipedia and you're right, some of them have two parallel spins. But the one in the middle (m=o) looks an awful lot like what I've written down above.

Regardless of what you want to call these states, it appears from the spin algebra argument I presented that the first one shows 100% correlation at any angle for aligned detectors, and the second one doesn't.