Separable and Entangled States, Bell States

Quantum entanglement is a very interesting property of quantum theory and unique to the quantum world (as opposed to our everyday life).

Here is what it means mathematically for pure state qubits:

Not every $4$-dimensional quantum pure state $|\Psi\rangle \in \mathbb{C}^4$ can be written as the Kronecker product of two qubits $|\psi_1\rangle, |\psi_2\rangle \in \mathbb{C}^2$.

That is, there exist some states $|\Psi\rangle \in \mathbb{C}^4$ such that:

$$|\Psi\rangle \neq |\psi_1\rangle \otimes |\psi_2\rangle$$

no matter how we choose $|\psi_1\rangle, |\psi_2\rangle$.

These states are called entangled states.

Two qubit states $|\Psi\rangle \in \mathbb{C}^4$ for which there do exist states $|\psi_1\rangle, |\psi_2\rangle \in \mathbb{C}^2$ such that:

$$|\Psi\rangle = |\psi_1\rangle \otimes |\psi_2\rangle$$

are called product states (or separable states).

Entangled states and quantum entanglement in general are the subject of much controversy (and excitement!) among physicists.

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