# B Can a particle be in two entanglements at the same time?

1. Jun 24, 2017

### Lunct

Can one particle be entangled with multiple particles at the same time?
Let's say we have particle A, and particle B, then particle C.
Can particle A be entangled with particle C and B at the same time?

Sorry if this is a stupid question.

2. Jun 24, 2017

### Simon Phoenix

Not a stupid question at all.

The entanglement of 2 particles (bi-partite entanglement) is, perhaps, reasonably well-understood and characterized. Multi-partite entanglement is much less easy to understand and to characterize. It's still an active area of research on how best to characterize the notion of 'entanglement' when more than 2 particles, or objects, are concerned and to figure out the properties.

The answer to your question is yes, it's possible to entangle 3 or more particles.

It's a bit beyond a B level but basically the definition of entanglement, for pure states, says that objects are entangled if their overall state cannot be written as a product of states from the individual subspaces of the objects. So if we could write a general state for object (particle) $A$ as $| \psi_A \rangle$ and a general state $| \psi_B \rangle$ for object $B$ then there exist overall states for the combined $AB$ system that cannot be written as $| \psi_A \rangle | \psi_B \rangle$.

The states that can't be written in this product form are the entangled states. This definition generalizes to any number of objects or particles.

It's not really possible to give the correct definition of entanglement at B-level, but I hope you get something of the flavour, at least.

3. Jun 24, 2017

### Lunct

I do find entanglement so odd. Throughout physics we have been trying to stop making interactions instant and have nothing in between them. Like with Michael Faraday's magnetic field, and then Einstein's GR. Then we discover entanglement and it goes against everything. No wonder Einstein said it was wrong.

4. Jun 24, 2017

### Simon Phoenix

Lol - indeed. It's all part of the difficulty of trying to find an 'intuitive' picture drawn from everyday experience that will fully explain QM. We can understand it all at the 'formalism' level - or even say something like "QM is just a generalization of probability" but ultimately it is indeed very difficult to understand in anything but more formalistic approaches. This is often expressed using Mermin's pithy and only slightly tongue-in-cheek remark "just shut up and calculate".

A lot of the difficulties - like imagining an instantaneous effect - are actually due to an interpretation we consciously, or unconsciously, impose. In a different interpretation of what's going on we don't have to view some of the features of entanglement as being an instantaneous non-local effect. I share Mermin's view that none of the interpretations of QM are wholly satisfactory - but you can't argue with the formalism which seems to work whatever interpretation we choose to impose - hence the advice to "shut up and calculate".

In a sense entanglement is just the superposition principle of QM applied to 2 or more things. So if we imagine a particle that could be in 2 states $| \psi \rangle$ or $| \phi \rangle$ and we can't distinguish (without measurement) which of these 'paths' - then we'd have to write the state of the particle as $a | \psi \rangle + b | \phi \rangle$ where $a$ and $b$ are complex numbers.

So apply the same reasoning now but to 2 particles which could be in the states $| \psi_1 \rangle | \psi_2 \rangle$ or $| \phi_1 \rangle | \phi_2 \rangle$ then if we couldn't distinguish the 'paths' here we'd have to write the overall state of the 2 particles as something like $$| \psi_{12} \rangle = a~| \psi_1 \rangle | \psi_2 \rangle + b~ | \phi_1 \rangle | \phi_2 \rangle$$which is an entangled state of the 2 particles - just the same kind of 'path' indistinguishability going on here. In reference to my previous post there's no way to write this state $| \psi_{12} \rangle$ as a product state.

Bit beyond B-level again - sorry - but I don't really know how to explain some of this stuff without a few more advanced concepts from QM. Maybe others will be able to do better than me