Question

So I know that two particles can be entangled in a quantum way, but is it possible that more than two particles be entangled in a quantum way? Most descriptions provide with two-particles cases, so I wonder. (It's hard to think of three particles entangled in spin, or so.)

Answer 1

Yes, you can have as many entangled particles as you want. It might be rather cumbersome to achieve it but it can in principle be done. Multipartite entangled states actually lie at heart of a special type of quantum computation, called measurement-based quantum computation. Here, you start from a large entangled state of many parties (usually called cluster state) and by performing certain measurements on certain parties of the state achieve required state of the rest of the system. You might want to google it up, there is quite a lot of literature on this topic.

The multipartite entangled states, however have to major drawbacks - as I already said, they are not always easy to prepare, and secondly, it quickly becomes difficult to classify their entanglement. Let me illustrate this on a system of two and three qubits.

With two qubits, it is easy to decide whether a given system is entangled or not - the positivity of the partial trace is a necessary and sufficient condition for separability. But with three qubits (let's denote them by A, B and C) things start to get a little messy. You can consider three bipartitions of the whole system, A|BC, B|CA, C|AB, and look at their separability properties. Now, it may happen that the state will be separable with respect to the A|BC partitioning but not to the C|AB partition. (I am not completely sure about this, but this is the way it works for continuous-variable Gaussian states). You might even find that all three partial traces are positive but you won't be able to find a separable state of all three systems (such states are called bound entangled). So in principle, you can have states completely inseparable, separable with respect to one or two bipartitions, states separable with respect to all three bipartitions but not completely separable, and fully separable states.

And now, imagine going to four qubits. Now you can separate the system in 2+2 or 1+3 subsystems and the possibilities grow. So it becomes almost impossible to classify the entanglement of the given state. And entanglement quantification of such complex systems is even more problematic.