Writing Product States: When to Use a Sum?

[StevieTNZ]

How do we know when we can write a product state for two systems, and situations when you need to use a sum of product states?

If you have a product state for two systems, does it evolve into a sum?

 

[lugita15]

In general, the quantum state of the whole system is a sum of product states for the two (disjoint) subsystems, but often this quantum state can be factorized into a product of states for the two subsystems. And yes, depending on the Hamiltonian it is in principle possible for a product of quantum states to evolve in time into an entangled state, but usually the Hamiltonian is nicer than that.

 

[StevieTNZ]

Okay so if we have two pairs of entangled photons:
We'd write the whole state of both pairs as the sum of the product state (which would be two photons TENSOR two photons)?

I don't even know if tensor is the right word (circle with x in it?)?

 

[lugita15]

Okay so if we have two pairs of entangled photons:
We'd write the whole state of both pairs as the sum of the product state (which would be two photons TENSOR two photons)?

I don't even know if tensor is the right word (circle with x in it?)?

Yes, exactly. And that symbol is a tensor product.

If you want to see this all done in detail, you can read Sakurai, the standard graduate text on QM. Or at an undergraduate level Townsend does a good job of covering this ground, and it's relatively short.

 

[StevieTNZ]

And when we write a sum of product states, they're entangled?

 

[lugita15]

And when we write a sum of product states, they're entangled?

If we write a quantum state as a sum of products of arbitrary states (they could be linearly dependent, for instance), then we may still be able to factor this state as a product of states, so there's not entanglement. If, however, it cannot be factored into a single product, then it's entangled.

 

[StevieTNZ]

Now I'm confused, because Erich Joos is saying "When you have to use a sum of product terms, you have an entangled state"

 

[lugita15]

Now I'm confused, because Erich Joos is saying "When you have to use a sum of product terms, you have an entangled state"

That's the point, when you have to use a sum, then it's entanglement. But if it's merely possible to write it using a sum, that need not be entanglement.

 

[StevieTNZ]

Ah yes. That makes more sense. Thanks for pointing that out!