Why are interacting states necessarily entangled?

[metroplex021]

It just occurred to me that I don't know why composites of interacting particles are always in entangled -- as opposed to mere product -- states. Obviously if they are not interacting we will just represent them as being in a product state; but why is it that being in a product state entails not interacting?

Any comments appreciated! Thanks!

 

[Zarqon]

You have to clarify what you mean I think. I don't see any problem with particles being interacting and in a product state at the same time. A straightforward example is 2 ions trapped in the same ion trap. They would have to be considered as interacting because they affect each other with mutual coulomb forces, but they can surely be in a product state.

Where you think of some special situations or interactions?

 

[metroplex021]

Huh! I was wrong about that then. Thanks for putting me straight. Can I then ask: does anyone know what physical conditions have to be fulfilled for a pair of particles to go into an entangled state upon interaction? Thanks to anyone who can help!

 

[kith]

does anyone know what physical conditions have to be fulfilled for a pair of particles to go into an entangled state upon interaction?

There are no special physical conditions. Most states in the composite Hilbert space of two single systems are entangled states. If the particles interact and we initially prepare a product state, it is simply much more likely that at an arbitrary later time, the product state has evolved into an entangled state instead of another product state.

In the example of the ion trap, we have to consider an additional third system: the trap. If the particles are in a product state at all times, this means that the interaction with the trap is such that it destroys the entanglement of the particles on a short time scale (decoherence). Entanglement then exists between the particles and the trap and not between the particles themselves.

 

[metroplex021]

Perfect, thank you. Much appreciated!