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StevieTNZ

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

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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.

- #1

StevieTNZ

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

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- #2

lugita15

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- #3

StevieTNZ

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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?)?

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lugita15

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Yes, exactly. And that symbol is a tensor product.StevieTNZ said:

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?)?

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.

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StevieTNZ

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

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lugita15

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

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StevieTNZ

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- #8

lugita15

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That's the point, when youStevieTNZ said:

- #9

StevieTNZ

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

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.

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.

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.

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.

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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