If I want do describe two particles #1 and #2 within Quantum Mechancis - let them be different in some way (Spin for example) - how do "I" decide if the total state of the system should be an entangled state, a product state or some density matrix?
I am concerned about this because I read that for some reason the universe is technically in an entangled state with everything and if we ignore it we should write every state as a density matrix (because we end up with a statistical mixture, if we ignore the universe). But this would mean that writing states as Dirac Brakets would be always wrong.
Edit: I can build up a two particle system ether as product-state or as an entangled state. Say I want to mix an electron with the rest of the universe.
If I choose a product state I can ignore the universe and just look at the electron without losing any information. My electron description is complete without the universe.
If I choose an entangled state for the electron and the universe and I again want to focus only on the electron I end up with a statistical mixture. That I can ONLY write as a density matrix because I choose to ignore the universe I "forget" some information about the state. I add to the quantum probability the probability due to my incomplete knowledge of the actual electron state. This is not possible with Diracs Braket.
My question is how do I choose with what kind of state I start?