Tuesday, June 3, 2014

Entangled and Unentangled States

Let's take a simple example of electron spin states. The reason it is 'simple' is that you only have two states to consider: either an electron's spin is 'up' or it is 'down'. We can use the notation ↑ to stand for the state that its spin is 'up' and the down arrow ↓ to indicate that its spin is down.

If we write, say, two arrows together like this ↑↓ it means we have two electrons, the first one has spin up and the second one spin down. So, ↓↓ means both particles have spin down.

Now one way in which the two particles can be arranged experimentally is in an entangled form. One state that would describe such a situation is a wavefunction state like this:

Ψ = ↑↓ - ↓↑.

This is a superposition state combining (in some mysterious fashion!) two basic states: the first one ↑↓ describes is a situation where the first particle has spin up and the second spin down, and the second state ↓↑ describes a situation where the first particle has spin down and the second particle has spin up. But when the two particles are in the combined superposition state Ψ (as above), it's in some sort of mix of those two scenarios. Like the case of the cat that is half dead and half alive! :-)

Why exactly is this state Ψ 'entangled' -- and what exactly do we mean by that? Well, it means that if you measure the spin of the first electron and you discover that its spin is down ↓, let's say, that picks out the part "↓↑" of the state Ψ! And this means that the second electron must have spin up! They're entangled! They're tied up together so knowing some spin info about one tells you the spin info of the other - instantly! This is so because the system has been set up to be in the state described by Ψ.

Now what about an unentangled state? What would that look like for our 2-electron example. Here's one:

Φ = ↑↑ + ↓↓ + ↑↓ - ↓↑.

Here this state is made up of two electrons that can have both their spins up (namely, ↑↑), both their spins down (↓↓), or they could be in the state Ψ (consisting of the opposite spins). In this wavefunction state Φ (called a "product state" which are generally not entangled), if you measure the spin, say, of the first electron and you find that it is up ↑, then what about the spin of the other one? Well, here you have two possibilities, namely ↑↑ and ↑↓ involved in Φ, which means that the second electron can be either in the up spin or the down spin. No entanglement, no correlation as in the Ψ case above. Knowing the spin state of one particle doesn't tell you what the other one has to be.

You can illustrate the same kind of examples with photon polarization, so you can have their polarizations entangeled or unentangled - depending on how the system is set up by us or by nature.

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