Why use superpositions to write states even when they are not entangled?

Question

I am reading book "Dancing with Qubits. How quantum computing works". I had learned the rule of entangled state: when quantum state is separable(i.e. it can be written as such a tensor product of elementary states) then it is not entangled and entangled if can't. By this I don't understand why we use superposition when qubits are not entangled. For example, we have 2 qubits in circuit in initial state |0⟩. After aplying the Hadamard gate to each qubit the state will $\frac{1}{2}$(|00⟩+|10⟩+|01⟩+|11⟩). But why? Why not $\frac{1}{\sqrt2}$(|0⟩+|1⟩) and $\frac{1}{\sqrt2}$(|0⟩+|1⟩)? (I had removed tensor product consciously) Why don't we use superposition with every particle in the universe then?

I have this question because size of the state's vector grows as $2^n$. It is to hard to simulate on home computer but why we have to use superposition of all qubits when thay are not entagled? I think it is not necessary till entanglement state will born.

Also superposition give us an opportunity to move some coefficients and ect between qubits. For example It happens in the Deutsch–Jozsa algorithm.

Formulas are copied from the wikipedia page. In the first row $f(x)$ belongs to the $|y⟩$'s state. But in next we move it to the $|x⟩$'s state. We can't do it without superposition, but this state is separable. It means one unentangled qubit can influence another by transferring some of its characteristics to it. I can't understand why we don't include some photon from the sun in this superposition...

Answer 1

Not sure if this is exactly what you're asking, but:

1. Any state can be written as a superposition of elements of any corresponding orthonormal basis. For example, any two-qubit state can be written as a superposition of elements in $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$, or equivalently as a superposition of Bell states, or equivalently as a superposition of any set of four orthonormal two-qubit states. This is akin to how vectors can be decomposed as linear combinations of elements of an arbitrary basis. There is nothing physical about this, it's just a matter of how you choose to describe your state. Different descriptions can be useful in different circumstances. For example, the state $|00\rangle$ can be written as a superposition of Bell states: $$|00\rangle = \frac12\left(|\Psi^+\rangle+|\Psi^-\rangle+|\Phi^+\rangle+|\Phi^-\rangle\right),$$ where $|\Psi^\pm\rangle\equiv \frac1{\sqrt2}(|01\rangle\pm |10\rangle)$ and $|\Phi^\pm\rangle\equiv\frac1{\sqrt2}(|00\rangle\pm|11\rangle)$.

2. The fact that you can write a state $|\psi\rangle$ as a superposition of other states has nothing to do with it being entangled or not. Yes, any pure separable state is also a product state, meaning it can be written as $|\psi\rangle=|\psi_1\rangle\otimes|\psi_2\rangle$ for some pair of states $|\psi_1\rangle,|\psi_2\rangle$.

The bit about simulations is an entirely different question and should probably be asked separately.