Are entangled and Bell states the same thing?

Question

I am a little confused whether the entangled state and Bell state are the same thing? If they have a bit of contrast, what is the difference between them?

Answer 1

A bipartite state $\rho_{AB}$ is called entangled if it cannot be written as $$ \rho_{AB} = \sum_i \lambda_i \sigma_A^i \otimes \sigma_B^i, $$ for some $\lambda_i\geq 0$, $\sum_i \lambda_i = 1$ and $\sigma_A^i$, $\sigma_B^i$ are states on systems $A$ and $B$ respectively, for all $i$. In other words, a state is entangled if it cannot be written as a convex combination of product states.

The four Bell-states are usually referring to the states $$ | \Phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle) \qquad |\Psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle) $$ The Bell states are all entangled in the sense above with for example $\rho = |\Phi^{\pm}\rangle \langle \Phi^{\pm}|$ or similarly for the other states. However, not all entangled states are Bell-states. For example, the states $|\psi\rangle = \cos(\theta) |00\rangle + \sin(\theta) |11\rangle$ for $\theta \in (0, \pi/4)$ are all entangled but are not Bell-states.

Answer 2

Put in simply words you could say that $$bell \implies entangled$$ but not the opposite. Namely the Bell state is a precise case of the entangled state.

We have that an entangled state is such that it cannot be expressed as a tensor product, i.e. $$\nexists |x⟩ |y⟩: |\phi⟩ = |x⟩\otimes |y⟩$$ in the two qbits case.

On the other hand, the Bell State is a precise state obtained by a precise sequence of transformations resulting in: $$|\psi⟩ = \dfrac{1}{\sqrt{2}}|00⟩+\dfrac{1}{\sqrt{2}}|11⟩$$ Which is indeed entangled (you can prove it) but not the only case of entanglement.

Answer 3

A simpler answer with fewer mathematical symbols:

The Bell states are examples of entangled states, but there's other examples of entangled states that are not Bell states, for example the W state:

$$ \tag{1} \frac{1}{\sqrt{3}}\left(|001\rangle + |010\rangle + | 100\rangle\right), $$

the GHZ states for example:

$$ \tag{2} \frac{1}{\sqrt{2}}\left(|000\rangle + |111\rangle\right), $$

and many more.