# Writing Product States: When to Use a Sum?

• StevieTNZ
In summary, the quantum state of a system can be written as a sum of product states for two disjoint subsystems. However, the state can also be factorized into a product of states for the subsystems depending on the Hamiltonian. When a quantum state is written as a sum of product states, it may or may not be entangled. This depends on whether it can be factorized into a single product state or not. When a sum of product terms is required, it indicates entanglement, but the possibility of writing it as a sum does not necessarily mean entanglement.
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?

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.

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?)?

StevieTNZ said:
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.

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

StevieTNZ said:
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.

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

StevieTNZ said:
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.

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

## 1. When should I use a sum to write product states?

The sum notation should be used when the state of a system can be described as a combination of two or more separate states. This is often the case in quantum mechanics, where a system can exist in multiple states simultaneously.

## 2. How do I write a product state using a sum?

To write a product state using a sum, you first need to identify the separate states that make up the system. Then, use the sum notation to combine these states into one expression. For example, if a system can exist in two states, A and B, the product state can be written as A + B.

## 3. Can a product state be written using other mathematical operations besides a sum?

Yes, a product state can also be written using a tensor product, denoted by the symbol ⊗. This is often used when the separate states are not independent of each other, but rather interact with each other.

## 4. Are there any advantages to using a sum to write product states?

Yes, using a sum to write product states can make calculations and equations simpler and more intuitive. It also allows for a more elegant representation of a system's state, especially in quantum mechanics where states can be complex and abstract.

## 5. Are there any situations where a sum should not be used to write product states?

There are certain cases where a sum may not be the most appropriate notation for writing product states. For example, if the states are not discrete and can take on continuous values, other mathematical operations such as integration may be more suitable.

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