Entanglement

Consider the case of an electron detected by our SG(z) detector. The spin of this electron is measured to be either "up" (u) or "down" (d). Let a and b denote the probabilities of it being detected as u or d states, respectively. Then, before we measure the electron's spin, we can write the state of the electron, Y_i, as:

Y_i = a·u + b·d

Now, consider the case of two electrons. Each of these will have a state that is described by the above expression, when these electrons are considered individually. But if we were to describe the state of the system of the two electrons, then we must have three (four) possibilities to consider: (one is up, the other also up), (one is down, the other also down), and (one is up, but the other is down). The last possibility, could happen two separate ways, if we could distinguish these particles, i.e. label them as say numbers 1 and 2. So, there are four possibilities.

It is interesting to note that there are two different ways for describing this combined state of the two electrons. In case 1, let us write down each electron's individual state and then use it to construct the combined state of the two electrons. We write:

Y₁ = a₁u + b₁d for the first electron and Y₂ = a₂u + b₂d for the second electron. For the combined state we write the product of the two individual states, i.e.

Y = a₁u(1)·a₂u(2) + a₁u(1)·b₂d(2) + b₁d(1)·a₂u(2) + b₁d(1)·b₂d(2)

This shows that, for example, the probability that both electrons are to be found as "up", i.e. each have had an u-state, is related to the product: a₁a₂, the first term of the above relationship, and so on.

In case 2 we can just say that given two electrons, then their combined state will be the sum of the four possible product states: u(1)u(2), u(1)d(2), d(1)u(2), and d(1)d(2). Specifically, then

Y = γu(1)u(2) + δu(1) d(2) + αd(1)u(2) + βd(1)d(2)

Notice that the above two expressions for the state of a two-electron system would be the same, if we could write that

γ = a₁a₂, δ = a₁b₂, etc.

In a sense, the state of the two-electron system is just made of the individual states of two unrelated electrons. If we cannot make this assertion, however, then we say that the state of the two electron system is mixed, and that these electrons are entangled. In the mixed state, then, even if we knew the values of a₁, b₁, a₂, and b₂, we still would not know the values of δ and α.